This document summarizes a presentation on rigorously verifying the accuracy of numerical solutions to semi-linear parabolic partial differential equations using analytic semigroups. It introduces the considered problem of finding the solution to a semi-linear parabolic PDE. It then discusses using a piecewise linear finite element discretization in space and time to obtain an initial numerical solution. The goal is to rigorously enclose the true solution within a radius ρ of this numerical solution in the function space L∞(J;H10(Ω)). Key steps involve using properties of the analytic semigroup generated by the operator A and estimating discretization errors to compute the enclosure radius ρ.
This document summarizes methods for estimating average treatment effects in nonlinear models with endogenous switching using variably parametric regression. It describes two estimators: 1) a minimally parametric estimator that specifies an exponential conditional mean, and 2) a fully parametric estimator that specifies a generalized gamma conditional density. A Monte Carlo study shows the fully parametric estimator has lower bias even in small samples. The methods are then applied to a real dataset on birthweight and maternal smoking.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
11.solution of linear and nonlinear partial differential equations using mixt...Alexander Decker
This document discusses solving linear and nonlinear partial differential equations using a combination of the Elzaki transform and projected differential transform methods.
[1] It introduces the Elzaki transform and defines its properties for handling partial derivatives.
[2] It then introduces the projected differential transform method and its fundamental theorems for decomposing nonlinear terms.
[3] As an example application, it shows how to use the two methods together to solve a general nonlinear, non-homogeneous partial differential equation with initial conditions. The Elzaki transform is used to obtain an expression in terms of the nonlinear terms, which are then decomposed using the projected differential transform method.
Solution of linear and nonlinear partial differential equations using mixture...Alexander Decker
This document discusses solving linear and nonlinear partial differential equations using a combination of the Elzaki transform and projected differential transform methods.
[1] It introduces the Elzaki transform and defines its properties for solving partial differential equations. Properties include formulas for the Elzaki transform of partial derivatives.
[2] It then introduces the projected differential transform method and defines its basic properties and theorems for decomposing nonlinear terms.
[3] As an example application, it shows how to use the two methods together to solve a general nonlinear, non-homogeneous partial differential equation with initial conditions. The Elzaki transform is used to obtain an expression for the solution, then the projected differential transform decomposes the
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
11.[104 111]analytical solution for telegraph equation by modified of sumudu ...Alexander Decker
1) The document presents a new integral transform method called the Elzaki transform to solve the general linear telegraph equation.
2) The Elzaki transform is used to obtain analytical solutions for the telegraph equation. Definitions and properties of the transform are provided.
3) Several examples are presented to demonstrate the method. Exact solutions to examples of the telegraph equation are obtained using the Elzaki transform.
It is a new theory based on an algorithmic approach. Its only element
is called nokton. These rules are precise. The innities are completely
absent whatever the system studied. It is a theory with discrete space
and time. The theory is only at these beginnings.
11.homotopy perturbation and elzaki transform for solving nonlinear partial d...Alexander Decker
The document discusses using a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. It begins by introducing the homotopy perturbation method and Elzaki transform individually. It then presents the homotopy perturbation Elzaki transform method, which applies Elzaki transform to reformulate the problem before using homotopy perturbation method to obtain approximations of the solution as a series. Finally, it applies the new combined method to solve an example nonlinear partial differential equation.
This document summarizes methods for estimating average treatment effects in nonlinear models with endogenous switching using variably parametric regression. It describes two estimators: 1) a minimally parametric estimator that specifies an exponential conditional mean, and 2) a fully parametric estimator that specifies a generalized gamma conditional density. A Monte Carlo study shows the fully parametric estimator has lower bias even in small samples. The methods are then applied to a real dataset on birthweight and maternal smoking.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
11.solution of linear and nonlinear partial differential equations using mixt...Alexander Decker
This document discusses solving linear and nonlinear partial differential equations using a combination of the Elzaki transform and projected differential transform methods.
[1] It introduces the Elzaki transform and defines its properties for handling partial derivatives.
[2] It then introduces the projected differential transform method and its fundamental theorems for decomposing nonlinear terms.
[3] As an example application, it shows how to use the two methods together to solve a general nonlinear, non-homogeneous partial differential equation with initial conditions. The Elzaki transform is used to obtain an expression in terms of the nonlinear terms, which are then decomposed using the projected differential transform method.
Solution of linear and nonlinear partial differential equations using mixture...Alexander Decker
This document discusses solving linear and nonlinear partial differential equations using a combination of the Elzaki transform and projected differential transform methods.
[1] It introduces the Elzaki transform and defines its properties for solving partial differential equations. Properties include formulas for the Elzaki transform of partial derivatives.
[2] It then introduces the projected differential transform method and defines its basic properties and theorems for decomposing nonlinear terms.
[3] As an example application, it shows how to use the two methods together to solve a general nonlinear, non-homogeneous partial differential equation with initial conditions. The Elzaki transform is used to obtain an expression for the solution, then the projected differential transform decomposes the
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
11.[104 111]analytical solution for telegraph equation by modified of sumudu ...Alexander Decker
1) The document presents a new integral transform method called the Elzaki transform to solve the general linear telegraph equation.
2) The Elzaki transform is used to obtain analytical solutions for the telegraph equation. Definitions and properties of the transform are provided.
3) Several examples are presented to demonstrate the method. Exact solutions to examples of the telegraph equation are obtained using the Elzaki transform.
It is a new theory based on an algorithmic approach. Its only element
is called nokton. These rules are precise. The innities are completely
absent whatever the system studied. It is a theory with discrete space
and time. The theory is only at these beginnings.
11.homotopy perturbation and elzaki transform for solving nonlinear partial d...Alexander Decker
The document discusses using a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. It begins by introducing the homotopy perturbation method and Elzaki transform individually. It then presents the homotopy perturbation Elzaki transform method, which applies Elzaki transform to reformulate the problem before using homotopy perturbation method to obtain approximations of the solution as a series. Finally, it applies the new combined method to solve an example nonlinear partial differential equation.
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Alexander Decker
The document presents a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. The homotopy perturbation method is used to handle the nonlinear terms, while the Elzaki transform is applied to reformulate the equations in terms of transformed variables, obtaining a series solution via inverse transformation. The method is demonstrated to be effective for both homogeneous and non-homogeneous nonlinear partial differential equations. Key steps include using integration by parts to obtain Elzaki transforms of partial derivatives and defining a convex homotopy to reformulate the equations for the homotopy perturbation method.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
The document discusses the quantified fragment of set theory called ∀π0. ∀π0 allows for restricted quantification over sets and ordered pairs. A decision procedure for the satisfiability of ∀π0 formulas works by non-deterministically guessing a skeletal representation and checking if its realization is a model of the formula. The document considers encoding the conditions on skeletal representations as first-order formulas to view ∀π0 as a first-order logic and leverage tools developed for first-order logic fragments.
This document presents a new coupled fixed point theorem for mappings having the mixed monotone property in partially ordered metric spaces. Specifically:
1) The theorem establishes the existence of a coupled fixed point for a mapping F that satisfies a contraction-type condition and has the mixed monotone property in a partially ordered, complete metric space.
2) It is shown that the coupled fixed point can be unique under additional conditions involving midpoint lower or upper bound properties.
3) An estimate is provided for the convergence rate as the iterates of the mapping F converge to the coupled fixed point.
This document discusses Bayesian nonparametric posterior concentration rates under different loss functions.
1. It provides an overview of posterior concentration, how it gives insights into priors and inference, and how minimax rates can characterize concentration classes.
2. The proof technique involves constructing tests and relating distances like KL divergence to the loss function. Examples where nice results exist include density estimation, regression, and white noise models.
3. For the white noise model with a random truncation prior, it shows L2 concentration and pointwise concentration rates match minimax. But for sup-norm loss, existing results only achieve a suboptimal rate. The document explores how to potentially obtain better adaptation for sup-norm loss.
This document introduces the concept of conditional expectation and stochastic calculus. It defines conditional expectation as the projection of a random variable X onto the sub-σ-algebra generated by another random variable or process Y. It must minimize the mean squared error between X and the projected variable. Properties like linearity and monotonicity are proven. Conditional expectation allows incorporating observable information to make optimal guesses about unobserved variables. Martingales, which generalize random walks, also play an important role in stochastic calculus.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as extensions of classical set theory and differential equations to account for uncertainty. Several examples of fuzzy initial value problems are analyzed, comparing their behaviors under different types of differentiability. The solutions exhibit very different properties, even though the original crisp equations were equivalent, showing that different fuzzy representations can model the same real-world problem very differently.
This document discusses kernel-based estimation methods for inequality indices and risk measures. It begins with an overview of stochastic dominance and related indices like first-order, convex, and second-order stochastic dominance. It then discusses nonparametric estimation of densities and copula densities using kernel methods. Specifically, it proposes using beta kernels and transformed kernels to improve estimation at the boundaries. The document explores combining these approaches and using mixtures of distributions like beta distributions within the kernels. It concludes by discussing applications to heavy-tailed distributions.
This document provides an introduction to stochastic calculus. It begins with a review of key probability concepts such as the Lebesgue integral, change of measure, and the Radon-Nikodym derivative. It then discusses information and σ-algebras, including filtrations and adapted processes. Conditional expectation is explained. The document concludes by introducing random walks and their connection to Brownian motion through the scaled random walk process. Key concepts such as martingales and quadratic variation are defined.
This document provides an overview and agenda for a master's level course on probability and statistics. It covers key topics like statistical models, probability distributions, conditional distributions, convergence theorems, sampling, confidence intervals, decision theory, and testing procedures. Examples of common probability distributions and functions are also presented, including the cumulative distribution function, probability density function, independence, and conditional independence. Additional references for further reading are included.
Numerical solution of boundary value problems by piecewise analysis methodAlexander Decker
This document presents a numerical method called Piecewise-Homotopy Analysis Method (P-HAM) for solving fourth-order boundary value problems. P-HAM is based on the Homotopy Analysis Method (HAM) but uses multiple auxiliary parameters, with each parameter applied over a sub-range of the domain for improved accuracy. The document outlines the basic steps of P-HAM, including constructing the zero-order deformation equation and deriving the governing equations. It then applies P-HAM to solve two example problems and compares the results to other numerical methods.
The document discusses using the programming language R for actuarial science applications. It presents R as a vector-based language suitable for working with life tables and performing actuarial calculations. Examples are given of how to model life contingencies like life expectancies, annuities, and insurance values using vectors and matrices in R. The document also discusses using R to fit prospective mortality models like the Lee-Carter model to data matrices.
Analysis and numerics of partial differential equationsSpringer
1) The document discusses Enrico Magenes' early research in partial differential equations in the 1950s, applying Picone's method to transform boundary value problems into integral equations.
2) It describes Magenes' collaboration with G. Stampacchia at the University of Genoa in the late 1950s, where they studied works by Schwartz and others on weak solutions and Sobolev spaces and published an influential paper applying these concepts.
3) It outlines Magenes' long collaboration with J.-L. Lions in the 1960s, where they developed a general framework for defining weak solutions and traces for non-homogeneous boundary value problems using duality and distribution theory.
International journal of engineering and mathematical modelling vol2 no3_2015_2IJEMM
The document presents a mixed finite element approximation for modeling reaction front propagation in porous media. The model couples equations for motion, temperature, and concentration. The semi-discrete problem is formulated using mixed finite element spaces. Existence and uniqueness of the semi-discrete solution is proven. Error estimates show that the temperature, concentration, velocity, and pressure errors converge with order h^σ, where h is the mesh size and σ is the solution regularity. Stability conditions on the time step and parameters are required.
Arthur Charpentier's presentation covered perspectives on predictive modeling. He discussed prediction versus estimation, parametric versus nonparametric models, linear models and least squares, modeling categorical variables, and prediction using covariates. Key points included defining prediction as estimating the expected value, providing confidence intervals to quantify uncertainty, using maximum likelihood to estimate parameters, and modeling conditional distributions based on covariates.
This document discusses quantum modes and the correspondence between classical and quantum mechanics. It provides three key principles of quantum mechanics: (1) quantum states are represented by ket vectors, (2) quantum observables are hermitian operators, and (3) the Schrodinger equation governs the causal evolution of quantum systems. It also outlines how classical quantities like position and momentum correspond to quantum operators and how they form Lie algebras through commutation relations. Representations of quantum mechanics are discussed through examples like the energy basis of the harmonic oscillator.
This document presents Joe Suzuki's work on Bayes independence tests. It discusses both discrete and continuous cases. For the discrete case, it estimates mutual information using maximum likelihood and proposes a Bayesian estimation using Lempel-Ziv compression. This Bayesian estimation is shown to be consistent. For the continuous case, it constructs a generalized Bayesian estimation that is also consistent. It also discusses the Hilbert Schmidt independence criterion (HSIC) and its limitations. Experiments show the proposed method performs well on both synthetic and real data, while HSIC shows poor performance in some cases. The proposed method has significantly better execution time than HSIC.
The document summarizes key points from a lecture on stochastic integration and Itô's formula. It introduces stochastic integration of semimartingales as the sum of integrating the martingale and continuous variation parts. It defines a "martingale-valued measure" using the Lévy-Itô decomposition, and properties of this measure. Finally, it unifies the usual stochastic integral with the Poisson integral by defining an integral with respect to the martingale-valued measure, and conditions on integrand functions.
This document discusses compactness estimates for nonlinear partial differential equations (PDEs), specifically Hamilton-Jacobi equations. It provides background on Kolmogorov entropy measures of compactness and covers recent results estimating the Kolmogorov entropy of solutions to scalar conservation laws and Hamilton-Jacobi equations, showing it is on the order of 1/ε. The document outlines applications of these estimates and open questions regarding extending the estimates to non-convex fluxes and non-uniformly convex Hamiltonians.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as a way to model dynamical systems involving uncertainties. It then examines three different fuzzy initial value problems and their solutions. The solutions exhibit very different behaviors despite being fuzzy representations of equivalent crisp differential equations. This shows that different fuzzy representations of the same crisp problem can lead to different outcomes.
This document summarizes a presentation on random entire functions. It discusses three classical families of random entire functions - Gaussian, Rademacher, and Steinhaus entire functions. It presents several theorems regarding inequalities concerning the maximum modulus and zeros of random entire functions. Specifically, it shows that the weighted zero counting function of random entire functions in family Y is close to the logarithm of the maximum modulus function, with an error term that is independent of the probability space. It also establishes an analogy of Nevanlinna's second main theorem for random entire functions in family Y.
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Alexander Decker
The document presents a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. The homotopy perturbation method is used to handle the nonlinear terms, while the Elzaki transform is applied to reformulate the equations in terms of transformed variables, obtaining a series solution via inverse transformation. The method is demonstrated to be effective for both homogeneous and non-homogeneous nonlinear partial differential equations. Key steps include using integration by parts to obtain Elzaki transforms of partial derivatives and defining a convex homotopy to reformulate the equations for the homotopy perturbation method.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
The document discusses the quantified fragment of set theory called ∀π0. ∀π0 allows for restricted quantification over sets and ordered pairs. A decision procedure for the satisfiability of ∀π0 formulas works by non-deterministically guessing a skeletal representation and checking if its realization is a model of the formula. The document considers encoding the conditions on skeletal representations as first-order formulas to view ∀π0 as a first-order logic and leverage tools developed for first-order logic fragments.
This document presents a new coupled fixed point theorem for mappings having the mixed monotone property in partially ordered metric spaces. Specifically:
1) The theorem establishes the existence of a coupled fixed point for a mapping F that satisfies a contraction-type condition and has the mixed monotone property in a partially ordered, complete metric space.
2) It is shown that the coupled fixed point can be unique under additional conditions involving midpoint lower or upper bound properties.
3) An estimate is provided for the convergence rate as the iterates of the mapping F converge to the coupled fixed point.
This document discusses Bayesian nonparametric posterior concentration rates under different loss functions.
1. It provides an overview of posterior concentration, how it gives insights into priors and inference, and how minimax rates can characterize concentration classes.
2. The proof technique involves constructing tests and relating distances like KL divergence to the loss function. Examples where nice results exist include density estimation, regression, and white noise models.
3. For the white noise model with a random truncation prior, it shows L2 concentration and pointwise concentration rates match minimax. But for sup-norm loss, existing results only achieve a suboptimal rate. The document explores how to potentially obtain better adaptation for sup-norm loss.
This document introduces the concept of conditional expectation and stochastic calculus. It defines conditional expectation as the projection of a random variable X onto the sub-σ-algebra generated by another random variable or process Y. It must minimize the mean squared error between X and the projected variable. Properties like linearity and monotonicity are proven. Conditional expectation allows incorporating observable information to make optimal guesses about unobserved variables. Martingales, which generalize random walks, also play an important role in stochastic calculus.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as extensions of classical set theory and differential equations to account for uncertainty. Several examples of fuzzy initial value problems are analyzed, comparing their behaviors under different types of differentiability. The solutions exhibit very different properties, even though the original crisp equations were equivalent, showing that different fuzzy representations can model the same real-world problem very differently.
This document discusses kernel-based estimation methods for inequality indices and risk measures. It begins with an overview of stochastic dominance and related indices like first-order, convex, and second-order stochastic dominance. It then discusses nonparametric estimation of densities and copula densities using kernel methods. Specifically, it proposes using beta kernels and transformed kernels to improve estimation at the boundaries. The document explores combining these approaches and using mixtures of distributions like beta distributions within the kernels. It concludes by discussing applications to heavy-tailed distributions.
This document provides an introduction to stochastic calculus. It begins with a review of key probability concepts such as the Lebesgue integral, change of measure, and the Radon-Nikodym derivative. It then discusses information and σ-algebras, including filtrations and adapted processes. Conditional expectation is explained. The document concludes by introducing random walks and their connection to Brownian motion through the scaled random walk process. Key concepts such as martingales and quadratic variation are defined.
This document provides an overview and agenda for a master's level course on probability and statistics. It covers key topics like statistical models, probability distributions, conditional distributions, convergence theorems, sampling, confidence intervals, decision theory, and testing procedures. Examples of common probability distributions and functions are also presented, including the cumulative distribution function, probability density function, independence, and conditional independence. Additional references for further reading are included.
Numerical solution of boundary value problems by piecewise analysis methodAlexander Decker
This document presents a numerical method called Piecewise-Homotopy Analysis Method (P-HAM) for solving fourth-order boundary value problems. P-HAM is based on the Homotopy Analysis Method (HAM) but uses multiple auxiliary parameters, with each parameter applied over a sub-range of the domain for improved accuracy. The document outlines the basic steps of P-HAM, including constructing the zero-order deformation equation and deriving the governing equations. It then applies P-HAM to solve two example problems and compares the results to other numerical methods.
The document discusses using the programming language R for actuarial science applications. It presents R as a vector-based language suitable for working with life tables and performing actuarial calculations. Examples are given of how to model life contingencies like life expectancies, annuities, and insurance values using vectors and matrices in R. The document also discusses using R to fit prospective mortality models like the Lee-Carter model to data matrices.
Analysis and numerics of partial differential equationsSpringer
1) The document discusses Enrico Magenes' early research in partial differential equations in the 1950s, applying Picone's method to transform boundary value problems into integral equations.
2) It describes Magenes' collaboration with G. Stampacchia at the University of Genoa in the late 1950s, where they studied works by Schwartz and others on weak solutions and Sobolev spaces and published an influential paper applying these concepts.
3) It outlines Magenes' long collaboration with J.-L. Lions in the 1960s, where they developed a general framework for defining weak solutions and traces for non-homogeneous boundary value problems using duality and distribution theory.
International journal of engineering and mathematical modelling vol2 no3_2015_2IJEMM
The document presents a mixed finite element approximation for modeling reaction front propagation in porous media. The model couples equations for motion, temperature, and concentration. The semi-discrete problem is formulated using mixed finite element spaces. Existence and uniqueness of the semi-discrete solution is proven. Error estimates show that the temperature, concentration, velocity, and pressure errors converge with order h^σ, where h is the mesh size and σ is the solution regularity. Stability conditions on the time step and parameters are required.
Arthur Charpentier's presentation covered perspectives on predictive modeling. He discussed prediction versus estimation, parametric versus nonparametric models, linear models and least squares, modeling categorical variables, and prediction using covariates. Key points included defining prediction as estimating the expected value, providing confidence intervals to quantify uncertainty, using maximum likelihood to estimate parameters, and modeling conditional distributions based on covariates.
This document discusses quantum modes and the correspondence between classical and quantum mechanics. It provides three key principles of quantum mechanics: (1) quantum states are represented by ket vectors, (2) quantum observables are hermitian operators, and (3) the Schrodinger equation governs the causal evolution of quantum systems. It also outlines how classical quantities like position and momentum correspond to quantum operators and how they form Lie algebras through commutation relations. Representations of quantum mechanics are discussed through examples like the energy basis of the harmonic oscillator.
This document presents Joe Suzuki's work on Bayes independence tests. It discusses both discrete and continuous cases. For the discrete case, it estimates mutual information using maximum likelihood and proposes a Bayesian estimation using Lempel-Ziv compression. This Bayesian estimation is shown to be consistent. For the continuous case, it constructs a generalized Bayesian estimation that is also consistent. It also discusses the Hilbert Schmidt independence criterion (HSIC) and its limitations. Experiments show the proposed method performs well on both synthetic and real data, while HSIC shows poor performance in some cases. The proposed method has significantly better execution time than HSIC.
The document summarizes key points from a lecture on stochastic integration and Itô's formula. It introduces stochastic integration of semimartingales as the sum of integrating the martingale and continuous variation parts. It defines a "martingale-valued measure" using the Lévy-Itô decomposition, and properties of this measure. Finally, it unifies the usual stochastic integral with the Poisson integral by defining an integral with respect to the martingale-valued measure, and conditions on integrand functions.
This document discusses compactness estimates for nonlinear partial differential equations (PDEs), specifically Hamilton-Jacobi equations. It provides background on Kolmogorov entropy measures of compactness and covers recent results estimating the Kolmogorov entropy of solutions to scalar conservation laws and Hamilton-Jacobi equations, showing it is on the order of 1/ε. The document outlines applications of these estimates and open questions regarding extending the estimates to non-convex fluxes and non-uniformly convex Hamiltonians.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as a way to model dynamical systems involving uncertainties. It then examines three different fuzzy initial value problems and their solutions. The solutions exhibit very different behaviors despite being fuzzy representations of equivalent crisp differential equations. This shows that different fuzzy representations of the same crisp problem can lead to different outcomes.
This document summarizes a presentation on random entire functions. It discusses three classical families of random entire functions - Gaussian, Rademacher, and Steinhaus entire functions. It presents several theorems regarding inequalities concerning the maximum modulus and zeros of random entire functions. Specifically, it shows that the weighted zero counting function of random entire functions in family Y is close to the logarithm of the maximum modulus function, with an error term that is independent of the probability space. It also establishes an analogy of Nevanlinna's second main theorem for random entire functions in family Y.
Compatible discretizations in our hearts and mindsMarie E. Rognes
This document discusses a total pressure augmented formulation for simulating fluid flow in porous media, such as modeling cerebral fluid flow in the brain. The formulation introduces total pressure as a variable to overcome issues with Poisson locking in the incompressible limit. The formulation results in a coupled system of equations that describes solid displacement, total pressure, and fluid pressures. Finite element methods are developed using this formulation that achieve optimal convergence rates, including in the incompressible limit, using Taylor-Hood elements. Numerical experiments demonstrate the improved convergence rates over standard formulations.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document summarizes a presentation on developing a natural finite element for axisymmetric problems. It introduces an axisymmetric model problem, defines appropriate axisymmetric Sobolev spaces, and presents a discrete formulation using a P1 finite element on triangles. Numerical results on a test problem show the method achieves the same convergence rates as classical approaches but with significantly smaller errors. The analysis draws on previous work to prove first-order approximation properties under certain mesh assumptions.
The document defines the trapezoidal rule for approximating definite integrals. It provides the trapezoidal formula, explains the geometric interpretation of dividing the region into trapezoids, and outlines an algorithm and flowchart for implementing the trapezoidal rule in Python. Sample problems applying the trapezoidal rule are included to evaluate definite integrals numerically.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This dissertation defense presentation summarizes Sawinder Pal Kaur's PhD dissertation on the existence and properties of positive solutions to certain nonlinear eigenvalue problems involving multi-dimensional arrays. The presentation outlines Kaur's work on proving the existence of a unique positive solution for nonlinear transformations in two and three variables, including the discretization of the Gross-Pitaevskii equation. It also summarizes the properties of these solutions, such as continuity and asymptotic behavior. The proofs utilize tools like Kronecker products, Perron-Frobenius theory, and the Kantorovich fixed point theorem.
The document discusses finite speed approximations to the Navier-Stokes equations. It introduces relaxation approximations and a damped wave equation approximation as two methods to derive finite speed approximations. It then discusses a vector BGK approximation, which takes a kinetic approach using a system of hyperbolic equations that approximates the Boltzmann equation and can be shown to converge to the incompressible Navier-Stokes equations in the diffusive limit. The document provides details on the vector BGK model, including compatibility conditions, conservation laws, and proofs of consistency, stability, and existence of global solutions.
A numerical method to solve fractional Fredholm-Volterra integro-differential...OctavianPostavaru
The Goolden ratio is famous for the predictability it provides both in the microscopic world as well as in the dynamics of macroscopic structures of the universe. The extension of the Fibonacci series to the Fibonacci polynomials gives us the opportunity to use this powerful tool in the study of Fredholm-Volterra integro-differential equations. In this paper, we define a new hybrid fractional function consisting of block-pulse functions and Fibonacci polynomials (FHBPF). For this, in the Fibonacci polynomials we perform the transformation $x\to x^{\alpha}$, with $\alpha$ a real parameter. In the method developed in this paper, we propose that the unknown function $D^{\alpha}f(x)$ be written as a linear combination of FHBPF. We consider the fractional derivative $D^{\alpha}$ in the Caputo sense. Using theoretical considerations, we can write both the function $f(x)$ and other involved functions of type $D^{\beta}f(x)$ on the same basis. For this operation, we have to define an integral operator of Riemann-Liouville type associated to FHBPF, and with the help of hypergeometric functions, we can express this operator exactly. All these ingredients together with the collocation in the Newton-Cotes nodes transform the integro-differential equation into an algebraic system that we solve by applying Newton's iterative method. We conclude the paper with some examples to demonstrate that the proposed method is simple to implement and accurate. There are situations when by simply considering $\alpha\ne1$, we obtain an improvement in accuracy by 12 orders of magnitude.
This document discusses whether and when the entropy of a spatially homogeneous Boltzmann equation system with infinite initial entropy will become finite. It presents examples showing that for some initial distributions, the entropy remains infinite over time, while for others it becomes finite after a finite time. The main tool used to estimate the entropy is the Duhamel formula for the solution. For hard potential and hard sphere models, rules are given for determining whether the entropy will appear finite based on properties of the initial distribution and collision kernel.
This document summarizes a research article that reviews integration in Banach spaces, with a focus on the Bochner integral, generalized derivatives, and generalized gradients. It presents key definitions related to strongly measurable functions, the Bochner integral, Lp spaces of functions from an interval to a Banach space, generalized derivatives, and monotone operators. It also states theorems regarding Banach and Hilbert space properties, generalized derivative properties under weak convergence, and finite extensions of the Bochner integral for sums and products. The main results develop these finite extensions of the Bochner integral.
This document discusses Newton's method for solving systems of nonlinear equations. It begins by introducing Newton's method in one dimension and extending it to multiple dimensions using a Jacobian matrix. It then proves that under certain conditions, Newton's method will converge quadratically to the solution. An example is provided to illustrate computing the Jacobian and using Newton's method. The document also discusses shooting methods for solving boundary value problems by converting them into initial value problems through an initial guess of a boundary condition.
This document summarizes Hidehiko Ichimura's 1993 paper on semiparametric least squares estimation of single-index models. It provides an overview of single-index models and assumptions required for identification and estimation. Key results discussed include:
1) The SLS estimator is consistent under certain regularity conditions on the model and kernel estimator.
2) Under additional moment and identification assumptions, the SLS estimator is asymptotically normal with the standard sandwich variance formula.
3) Optimal weighting of the SLS estimator achieves the semiparametric efficiency bound.
Stability analysis for nonlinear impulsive optimal control problemsAI Publications
We consider the generic stability of optimal control problems governed by nonlinear impulsive evolution equations. Under perturbations of the right-hand side functions of the controlled system, the results of stability for the impulsive optimal control problems are proved given set-valued theory.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...ijfls
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point
theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for
the approximate controllability of the system.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...Wireilla
This summary provides the key details about the document in 3 sentences:
The document discusses approximate controllability results for impulsive linear fuzzy stochastic differential equations under nonlocal conditions. It presents sufficient conditions for the approximate controllability of such systems using Banach fixed point theorems, stochastic analysis, and fuzzy processes. The main result establishes approximate controllability of impulsive linear fuzzy stochastic differential equations by verifying assumptions on the system using fixed point theorems.
1. The document presents a new approach to proving comparison theorems for stochastic differential equations (SDEs) using differentiation of solutions with respect to initial data.
2. It proves that if the drift term of one SDE is always greater than or equal to the other, and their initial values satisfy the same relation, then the solutions will also satisfy this relation for all time.
3. Two methods are provided: the first uses explicit solutions, the second avoids this by showing the difference process cannot reach zero in finite time based on its behavior.
- Hiroaki Shiokawa's research interests include graph mining, network analysis, and efficient algorithms. He was previously employed at NTT from 2011 to 2015.
- His current research focuses on developing clustering algorithms for large-scale networks and evaluating their performance on real-world network datasets.
- He has published highly cited papers in top data mining and network science conferences such as KDD, CIKM, and WSDM.
This document discusses information theory and related concepts such as entropy, Kullback-Leibler divergence, mutual information, independent component analysis, clustering algorithms, change point detection, kernel density estimation, and nonparametric regression. It provides mathematical definitions and formulas for these concepts. Figures are included to illustrate clustering and change point detection methods. The document contains information that could be useful for understanding techniques in machine learning, signal processing, and statistics.
This document presents an overview of optimization algorithms on Riemannian manifolds. It begins by introducing concepts such as vector transport and retraction mappings that are used to generalize algorithms from Euclidean spaces to manifolds. It then summarizes several classical optimization methods including gradient descent, conjugate gradient, and variants of quasi-Newton methods adapted to the Riemannian setting using these geometric concepts. The convergence of the Fletcher-Reeves method is analyzed under standard assumptions on the objective function. Overall, the document provides a conceptual and mathematical foundation for optimization on manifolds.
This document discusses methods for identifying the source node of information spread in networks based on the observed spread over time. It begins by introducing epidemic models like SIS and SI for modeling information spread over networks. It then discusses maximum likelihood methods for identifying the source node on regular tree networks based on the observed subgraph. The accuracy of these methods increases with network size and degree. Extensions to other network structures and SIR models are also proposed. Overall, the document reviews mathematical models and algorithms for source identification in networks from limited observations of information spread.
This document discusses pattern formation in crowd dynamics. It begins with an introduction to crowd dynamics and then discusses two specific patterns: lane formation and freezing-by-heating transition. Lane formation occurs when pedestrians walking in opposite directions spontaneously form lanes to allow for more efficient movement. Freezing-by-heating transition refers to the phenomenon where increasing noise or energy in a crowd leads to the formation of orderly lanes, rather than disorder. The document explores mathematical modeling of these patterns using particle simulation models.
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
5. 半線形放物型方程式
Let Ω be a bounded polygonal domain in R2
.
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(t0, x) = u0(x) in Ω,
▶ J := (t0, t1], 0 ≤ t0 < t1 < ∞ or J := (0, ∞),
▶ f : twice differentiable nonlinear mapping,
▶ u = 0 on ∂Ω is the trace sense,
▶ u0 ∈ H1
0 (Ω).
Lp
(Ω): the set of Lp
-functions,
H1
(Ω): the first order Sobolev space of L2
(Ω),
H1
0 (Ω) := {v ∈ H1
(Ω) : v = 0 on ∂Ω}.
5/55
6. 記号
A : D(A) ⊂ H1
0 (Ω) → L2
(Ω) is defined by
A := −
∑
1≤i,j≤2
∂
∂xj
(
aij(x)
∂
∂xi
)
,
where aij(x) = aji(x) is in W1,∞
(Ω) and satisfies
∑
1≤i,j≤2
aij(x)ξiξj ≥ µ|ξ|2
, ∀x ∈ Ω, ∀ξ ∈ R2
with µ > 0.
6/55
7. 記号
We endow L2
(Ω) with the inner product:
(u, v)L2 :=
∫
Ω
u(x)v(x)dx.
Use the usual norms:
∥u∥L2 :=
√
(u, u)L2 , ∥u∥H1
0
:= ∥∇u∥L2 ,
and
∥u∥H−1 := sup
0̸=v∈H1
0 (Ω)
∥v∥H1
0
=1
|⟨u, v⟩| ,
where ⟨·, ·⟩ is a dual product between H1
0 (Ω) and H−1
(Ω)1
.
1
The topological dual space of H1
0 (Ω).
7/55
8. 精度保証付き数値計算
(PJ ) の弱解: For t ∈ J, u(t) := u(t, ·) ∈ H1
0 (Ω) with the
initial function u0 such that
(∂tu(t), v)L2 + a(u(t), v) = (f(u(t)), v)L2 , ∀v ∈ H1
0 (Ω),
where a : H1
0 (Ω) × H1
0 (Ω) → R is a bilinear form:
a(u, v) :=
∑
1≤i,j≤2
(
aij(x)
∂u
∂xi
,
∂v
∂xj
)
L2
satisfying
a(u, u) ≥ µ∥u∥2
H1
0
, ∀u ∈ H1
0 (Ω),
|a(u, v)| ≤ M∥u∥H1
0
∥v∥H1
0
, ∀u, v ∈ H1
0 (Ω).
8/55
9. 精度保証付き数値計算
(PJ ) の弱解: For t ∈ J, u(t) := u(t, ·) ∈ H1
0 (Ω) with the
initial function u0 such that
(∂tu(t), v)L2 + a(u(t), v) = (f(u(t)), v)L2 , ∀v ∈ H1
0 (Ω),
where a : H1
0 (Ω) × H1
0 (Ω) → R is a bilinear form:
a(u, v) :=
∑
1≤i,j≤2
(
aij(x)
∂u
∂xi
,
∂v
∂xj
)
L2
satisfying
a(u, u) ≥ µ∥u∥2
H1
0
, ∀u ∈ H1
0 (Ω),
|a(u, v)| ≤ M∥u∥H1
0
∥v∥H1
0
, ∀u, v ∈ H1
0 (Ω).
8/55
10. 精度保証付き数値計算
(PJ ) の弱解: For t ∈ J, u(t) := u(t, ·) ∈ H1
0 (Ω) with the
initial function u0 such that
(∂tu(t), v)L2 + a(u(t), v) = (f(u(t)), v)L2 , ∀v ∈ H1
0 (Ω),
where a : H1
0 (Ω) × H1
0 (Ω) → R is a bilinear form:
a(u, v) :=
∑
1≤i,j≤2
(
aij(x)
∂u
∂xi
,
∂v
∂xj
)
L2
satisfying
a(u, u) ≥ µ∥u∥2
H1
0
, ∀u ∈ H1
0 (Ω),
|a(u, v)| ≤ M∥u∥H1
0
∥v∥H1
0
, ∀u, v ∈ H1
0 (Ω).
8/55
12. 精度保証付き数値計算に関する先行研究
M.T. Nakao, T. Kinoshita and T. Kimura,
“On a posteriori estimates of inverse operators for linear
parabolic initial-boundary value problems”, Computing
94(2-4), 151–162, 2012.
M.T. Nakao, T. Kimura and T. Kinoshita,
“Constructive A Priori Error Estimates for a Full Discrete
Approximation of the Heat Equation”, Siam J. Numer. Anal.,
51(3), 1525–1541, 2013.
T. Kinoshita, T. Kimura and M.T. Nakao,
“On the a posteriori estimates for inverse operators of linear
parabolic equations with applications to the numerical
enclosure of solutions for nonlinear problems”, Numer. Math,
Online First, 2013.
10/55
13. 力学系の観点からみた研究
P. Zgliczy´nski and K. Mischaikow,
“Rigorous Numerics for Partial Differential Equations: The
Kuramoto―Sivashinsky Equation”, Foundations of
Computational Mathematics, 1(3), 1615–3375, 2001.
P. Zgliczy´nski,
“Rigorous numerics for dissipative PDEs III. An effective
algorithm for rigorous integration of dissipative PDEs”, Topol.
Methods Nonlinear Anal., 36, 197–262, 2010.
11/55
14. 離散半群を用いた数値スキームの研究
H. Fujita,
“On the semi-discrete finite element approximation for the
evolution equation ut + A(t)u = 0 of parabolic type”, Topics
in numerical analysis III, Academic Press, 143–157, 1977.
H. Fujita and A. Mizutani,
“On the finite element method for parabolic equations, I;
approximation of holomorphic semi-groups”, J. Math. Soc.
Japan, 28, 749–771, 1976.
H. Fujita, N. Saito and T. Suzuki,
“Operator theory and numerical methods”, Elsevier(Holland),
308pages, 2001.
12/55
16. Considered problem
J := (t0, t1] : arbitrary time interval. τ := t1 − t0.
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(t0, x) = u0(x) in Ω,
where u0 is an initial function in H1
0 (Ω).
Vh ⊂ H1
0 (Ω) : a finite dimensional subspace.
Starts from: ˆu0, ˆu1 ∈ Vh
ω(t) = ˆu0ϕ0(t) + ˆu1ϕ1(t), t ∈ J,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
14/55
17. Considered problem
J := (t0, t1] : arbitrary time interval. τ := t1 − t0.
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(t0, x) = u0(x) in Ω,
where u0 is an initial function in H1
0 (Ω).
Vh ⊂ H1
0 (Ω) : a finite dimensional subspace.
Starts from: ˆu0, ˆu1 ∈ Vh
ω(t) = ˆu0ϕ0(t) + ˆu1ϕ1(t), t ∈ J,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
14/55
18. Considered problem
Let the initial function satisfy
∥u0 − ˆu0∥H1
0
≤ ε0.
We rigorously enclose the solution in a Banach space2
,
L∞
(
J; H1
0 (Ω)
)
:=
{
u(t) ∈ H1
0 (Ω) : ess sup
t∈J
∥u(t)∥H1
0
< ∞
}
.
Namely, we compute a radius ρ > 0 of the ball:
BJ (ω, ρ) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: ∥y − ω∥L∞
(J;H1
0 (Ω)) ≤ ρ
}
.
2
∥u∥L∞(J;H1
0 (Ω)) := ess supt∈J ∥u(t)∥H1
0
15/55
19. Considered problem
Let the initial function satisfy
∥u0 − ˆu0∥H1
0
≤ ε0.
We rigorously enclose the solution in a Banach space2
,
L∞
(
J; H1
0 (Ω)
)
:=
{
u(t) ∈ H1
0 (Ω) : ess sup
t∈J
∥u(t)∥H1
0
< ∞
}
.
Namely, we compute a radius ρ > 0 of the ball:
BJ (ω, ρ) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: ∥y − ω∥L∞
(J;H1
0 (Ω)) ≤ ρ
}
.
2
∥u∥L∞(J;H1
0 (Ω)) := ess supt∈J ∥u(t)∥H1
0
15/55
20. Considered problem
Let the initial function satisfy
∥u0 − ˆu0∥H1
0
≤ ε0.
We rigorously enclose the solution in a Banach space2
,
L∞
(
J; H1
0 (Ω)
)
:=
{
u(t) ∈ H1
0 (Ω) : ess sup
t∈J
∥u(t)∥H1
0
< ∞
}
.
Namely, we compute a radius ρ > 0 of the ball:
BJ (ω, ρ) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: ∥y − ω∥L∞
(J;H1
0 (Ω)) ≤ ρ
}
.
2
∥u∥L∞(J;H1
0 (Ω)) := ess supt∈J ∥u(t)∥H1
0
15/55
21. Analytic semigroup
The weak form of A, which is denoted by −A3
, generates the
analytic semigroup {e−tA
}t≥0 over H−1
(Ω). The following
abstract problem has an unique solution:
∂tu + Au = 0, u(0, x) = u0 =⇒ ∃u = e−tA
u0.
Fact
Let x ∈ D(A) and λ0 be a positive number. A satisfies
⟨−Ax, x⟩ ≤ 0, R(λ0I + A) = H−1
(Ω).
Then, there exists an analytic semigroup {e−tA
}t≥0 generated by −A.
Proofs are found in several textbooks.
3
A : H1
0 (Ω) → H−1
(Ω) s.t. ⟨Au, v⟩ := a(u, v), ∀v ∈ H1
0 (Ω).
16/55
22. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
17/55
23. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
17/55
24. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
17/55
25. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
17/55
26. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
17/55
27. Sketch of proof
Let z(t) ∈ H1
0 (Ω) for t ∈ J. We put u(t) = ω(t) + z(t).
For any v ∈ H1
0 (Ω),
(∂tz(t), v)L2 + a(z(t), v)
= (f(u(t)), v)L2 − ((∂tω(t), v)L2 + ⟨Aω(t), v⟩)
=: ⟨g(z(t)), v⟩ ,
where g(z(t)) = f(u(t)) − (∂tω(t) + Aω(t)). Note that by
the definition of the natural embedding L2
(Ω) → H−1
(Ω),
(ψ, v)L2 = ⟨ψ, v⟩ holds for ψ ∈ L2
(Ω).
Define S : L∞
(J; H1
0 (Ω)) → L∞
(J; H1
0 (Ω)) using the
analytic semigroup e−tA
as
S(z) := e−(t−t0)A
(u0 − ˆu0) +
∫ t
t0
e−(t−s)A
g(z(s))ds.
18/55
28. Sketch of proof
Let z(t) ∈ H1
0 (Ω) for t ∈ J. We put u(t) = ω(t) + z(t).
For any v ∈ H1
0 (Ω),
(∂tz(t), v)L2 + a(z(t), v)
= (f(u(t)), v)L2 − ((∂tω(t), v)L2 + ⟨Aω(t), v⟩)
=: ⟨g(z(t)), v⟩ ,
where g(z(t)) = f(u(t)) − (∂tω(t) + Aω(t)). Note that by
the definition of the natural embedding L2
(Ω) → H−1
(Ω),
(ψ, v)L2 = ⟨ψ, v⟩ holds for ψ ∈ L2
(Ω).
Define S : L∞
(J; H1
0 (Ω)) → L∞
(J; H1
0 (Ω)) using the
analytic semigroup e−tA
as
S(z) := e−(t−t0)A
(u0 − ˆu0) +
∫ t
t0
e−(t−s)A
g(z(s))ds.
18/55
29. Sketch of proof
For ρ 0, Z := {z : ∥z∥L∞
(J;H1
0 (Ω)) ≤ ρ} ⊂ L∞
(J; H1
0 (Ω)).
On the basis of Banach’s fixed-point theorem, we show a
sufficient condition of S having a fixed-point in Z.
S(Z) ⊂ Z Since the analytic semigroup e−tA
is bounded,
the first term of S(z) is estimated4
by
e−(t−t0)A
(ζ − ˆu0) H1
0
≤ µ−1
A e−(t−t0)A
(ζ − ˆu0) H−1
≤
M
µ
e−(t−t0)λmin
ε0.
Then
e−(t−t0)A
(ζ − ˆu0) L∞(J;H1
0 (Ω))
≤
M
µ
ε0.
4
µ∥u∥H1
0
≤ ∥Au∥H−1 ≤ M∥u∥H1
0
is used.
19/55
30. Sketch of proof
For ρ 0, Z := {z : ∥z∥L∞
(J;H1
0 (Ω)) ≤ ρ} ⊂ L∞
(J; H1
0 (Ω)).
On the basis of Banach’s fixed-point theorem, we show a
sufficient condition of S having a fixed-point in Z.
S(Z) ⊂ Z Since the analytic semigroup e−tA
is bounded,
the first term of S(z) is estimated4
by
e−(t−t0)A
(ζ − ˆu0) H1
0
≤ µ−1
A e−(t−t0)A
(ζ − ˆu0) H−1
≤
M
µ
e−(t−t0)λmin
ε0.
Then
e−(t−t0)A
(ζ − ˆu0) L∞(J;H1
0 (Ω))
≤
M
µ
ε0.
4
µ∥u∥H1
0
≤ ∥Au∥H−1 ≤ M∥u∥H1
0
is used.
19/55
31. Sketch of proof
For ρ 0, Z := {z : ∥z∥L∞
(J;H1
0 (Ω)) ≤ ρ} ⊂ L∞
(J; H1
0 (Ω)).
On the basis of Banach’s fixed-point theorem, we show a
sufficient condition of S having a fixed-point in Z.
S(Z) ⊂ Z Since the analytic semigroup e−tA
is bounded,
the first term of S(z) is estimated4
by
e−(t−t0)A
(ζ − ˆu0) H1
0
≤ µ−1
A e−(t−t0)A
(ζ − ˆu0) H−1
≤
M
µ
e−(t−t0)λmin
ε0.
Then
e−(t−t0)A
(ζ − ˆu0) L∞(J;H1
0 (Ω))
≤
M
µ
ε0.
4
µ∥u∥H1
0
≤ ∥Au∥H−1 ≤ M∥u∥H1
0
is used.
19/55
35. Sketch of proof
The term of g1(s):
∫ t
t0
e−(t−s)A
g1(s)ds
H1
0
=
∫ t
t0
e−(t−s)A
(f(ω(s) + z(s)) − f(ω(s)))ds
H1
0
≤ µ−1
∫ t
t0
A e−(t−s)A
(f(ω(s) + z(s)) − f(ω(s)))
H−1
ds
= µ−1
∫ t
t0
A
1
2 e−(t−s)A
A
1
2 (f(ω(s) + z(s)) − f(ω(s)))
H−1
ds
≤ µ−1
e− 1
2
∫ t
t0
(t − s)− 1
2 e− 1
2 (t−s)λmin
A
1
2 (f(ω(s) + z(s)) − f(ω(s)))
H−1
≤ µ−1
M
1
2 e− 1
2
∫ t
t0
(t − s)− 1
2 e− 1
2 (t−s)λmin
∥f(ω(s) + z(s)) − f(ω(s))∥L2 ds
≤ µ−1
M
1
2 e− 1
2 ν(t) ∥f(ω + z) − f(ω)∥L∞(J;L2(Ω)).
21/55
36. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
22/55
37. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
22/55
38. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
22/55
39. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
22/55
40. Sketch of proof
For any z1, z2 in Z,
S(z1) − S(z2) =
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
holds. We have
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
H1
0
≤ µ−1
M
1
2 e−1
2 ν(t)∥f(z1 + ω) − f(z2 + ω)∥L∞(J;L2(Ω)).
Here, zi + ω ∈ B(ω, ρ) (i = 1, 2) holds. Then
∥S(z1) − S(z2)∥L∞
(J;H1
0 (Ω)) ≤
2
µ
√
Mτ
e
Lρ∥z1 − z2∥L∞
(J;H1
0 (Ω)).
23/55
41. Sketch of proof
For any z1, z2 in Z,
S(z1) − S(z2) =
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
holds. We have
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
H1
0
≤ µ−1
M
1
2 e−1
2 ν(t)∥f(z1 + ω) − f(z2 + ω)∥L∞(J;L2(Ω)).
Here, zi + ω ∈ B(ω, ρ) (i = 1, 2) holds. Then
∥S(z1) − S(z2)∥L∞
(J;H1
0 (Ω)) ≤
2
µ
√
Mτ
e
Lρ∥z1 − z2∥L∞
(J;H1
0 (Ω)).
23/55
42. Sketch of proof
For any z1, z2 in Z,
S(z1) − S(z2) =
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
holds. We have
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
H1
0
≤ µ−1
M
1
2 e−1
2 ν(t)∥f(z1 + ω) − f(z2 + ω)∥L∞(J;L2(Ω)).
Here, zi + ω ∈ B(ω, ρ) (i = 1, 2) holds. Then
∥S(z1) − S(z2)∥L∞
(J;H1
0 (Ω)) ≤
2
µ
√
Mτ
e
Lρ∥z1 − z2∥L∞
(J;H1
0 (Ω)).
23/55
43. Sketch of proof
The condition of theorem also implies
2
µ
√
Mτ
e
L (ρ) 1.
Therefore, S is a contraction mapping. Banach’s fixed point
theorem yields that there uniquely exists a fixed-point in Z.
24/55
44. Theorem (A posteriori error estimate)
Assume that existence and local uniqueness of the weak
solution u(t), t ∈ J, is proved in BJ (ω, ρ). Assume also that
ω satisfies
∫ t1
t0
e−(t1−s)A
(∂tω(s) + Aω(s) − f(ω(s))) ds
H1
0
≤ ˜δ.
Then, the following a posteriori error estimate holds:
∥u(t1) − ˆu1∥H1
0
≤
M
µ
e−τλmin
ε0 +
2
µ
√
Mτ
e
Lρρ + ˜δ =: ε1.
25/55
45. On several intervals
For n ∈ N, 0 = t0 t1 · · · tn ∞.
Jk := (tk−1, tk], τk := tk − tk−1, and J =
∪
Jk. (k=1,2,...,n)
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
where u0 ∈ H1
0 (Ω) is a given initial function satisfies
∥u0 − ˆu0∥H1
0
≤ ε0.
26/55
46. Approximate solution (Backward Euler)
Find {uh
k}k≥0 ⊂ Vh such that
(
uh
k − uh
k−1
τ
, vh
)
L2
+ a(uh
k, vh)L2 = (f(uh
k), vh)L2
and
(
uh
0, vh
)
L2 = (u0, vh)L2 for ∀vh ∈ Vh. Numerically
compute each approximation ˆuk (≈ uh
k) ∈ Vh.
From the data ˆuk(≈ uh
k) ∈ Vh, we construct ω(t):
ω(t) :=
n∑
k=0
ˆukϕk(t), t ∈ T,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
27/55
47. Approximate solution (Backward Euler)
Find {uh
k}k≥0 ⊂ Vh such that
(
uh
k − uh
k−1
τ
, vh
)
L2
+ a(uh
k, vh)L2 = (f(uh
k), vh)L2
and
(
uh
0, vh
)
L2 = (u0, vh)L2 for ∀vh ∈ Vh. Numerically
compute each approximation ˆuk (≈ uh
k) ∈ Vh.
From the data ˆuk(≈ uh
k) ∈ Vh, we construct ω(t):
ω(t) :=
n∑
k=0
ˆukϕk(t), t ∈ T,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
27/55
67. 大域解の存在に関する先行研究
S. Cai,
“A computer-assisted proof for the pattern formation on
reaction-diffusion systems”, 学位論文, Graduate School of
Mathematics, Kyushu University (2012) 71 pages.
▶ 反応拡散方程式のあるクラスの定常解に対する精度保
証付き数値計算法を示している.
▶ (t′
, ∞) で定常解まわりに大域的に存在する範囲を
L∞
(Ω) × L∞
(Ω) 上で生成された解析半群を用いて,計
算している.
40/55
68. Considered problem
Let Ω be a bounded polygonal domain in R2
and J := (0, ∞).
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
where A = −∆, u0 ∈ H1
0 (Ω) is an initial function, and
f : R → R is a twice Fr´echet differentiable nonlinear mapping.
41/55
69. Considered problem
Let Ω be a bounded polygonal domain in R2
and J := (0, ∞).
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
where A = −∆, u0 ∈ H1
0 (Ω) is an initial function, and
f : R → R is a twice Fr´echet differentiable nonlinear mapping.
41/55
70. Aim of this part
Let Ω be a bounded polygonal domain in R2
.
(PG)
∂tu + Au = f(u) in (t′
, ∞) × Ω,
u(t, x) = 0 on (t′
, ∞) × ∂Ω,
u(t′
, x) = η in Ω,
where η ∈ H1
0 (Ω) satisfies ∥η − ˆun∥H1
0
≤ εn for a certain
εn 0.
We enclose a solution for t ∈ (t′
, ∞) in a neighborhood of a
stationary solution ϕ ∈ D(A) of (PJ ) such that
{
Aϕ = f(ϕ) in Ω,
ϕ = 0 on ∂Ω.
42/55
71. 記号
For ρ 0, v ∈ L∞
((t′
, ∞); H1
0 (Ω)), define a ball
B(v, ρ) :=
{
y ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥y − v∥L∞
((t′,∞);H1
0 (Ω)) ≤ ρ
}
.
The Fr´echet derivative of f at w is denoted by
f′
[w] : L∞
((t′
, ∞); H1
0 (Ω)) → L∞
((t′
, ∞); L2
(Ω)).
For y ∈ B(v, ρ), we assume that there exists a non-decreasing
function L : R → R such that
∥f′
[y]u∥L∞(J;L2(Ω)) ≤ L(ρ)∥u∥H1
0
, u ∈ H1
0 (Ω).
43/55
72. 記号
Define a function space Xλ: for a fixed λ 0,
Xλ :=
{
u ∈ L∞
((t′
, ∞); H1
0 (Ω)) : ess sup
t∈(t′,∞)
e(t−t′)λ
∥u(t)∥H1
0
∞
}
which becomes a Banach space with the norm
∥ · ∥Xλ
:= ess sup
t∈(t′,∞)
e(t−t′)λ
∥u(t)∥H1
0
.
44/55
73. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
74. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
75. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
76. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
77. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
78. Sketch of proof
Let z ∈ Uϕ. A nonlinear operator
S : L∞
((t′
, ∞); H1
0 (Ω)) → L∞
((t′
, ∞); H1
0 (Ω)) is defined by
S(z) := e−(t−t′)A
(η − ϕ) +
∫ t
t′
e−(t−s)A
(f(z(s)) − f(ϕ)) ds.
On the basis of Banach’s fixed-point theorem, we show a
condition of S having a fixed-point in Uϕ.
For s ∈ (t′
, ∞) and ψ1, ψ2 ∈ Uϕ, the mean-value theorem
states that there exists y ∈ Uϕ such that
∥f(ψ1(s)) − f(ψ2(s))∥L2 = ∥f′
[y(s)](ψ1(s) − ψ2(s))∥L2 .
Since y ∈ Uϕ ⊂ B(ϕ, ρ) holds, we obtain
∥f(ψ1(s)) − f(ψ2(s))∥L2 ≤ L(ρ)∥ψ1(s) − ψ2(s)∥H1
0
.
46/55
79. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
80. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
81. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
82. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
84. 藤田型方程式
Let Ω := (0, 1)2
be an unit square domain in R2
.
(F)
∂tu − ∆u = u2
in (0, ∞) × Ω,
u(t, x) = 0 on (0, ∞) × ∂Ω,
u(0, x) = u0(x) in Ω,
where u0(x) = γ sin(πx) sin(πy).
▶ Vh :=
{∑N
k,l=1 ak,l sin(kπx) sin(lπy) : ak,l ∈ R
}
;
▶ Crank-Nicolson scheme is employed;
▶ we fixed λ = 1/40 in the global existence theorem.
49/55