This document presents a new version of the quasi-boundary value method for solving the 1-D nonlinear ill-posed heat equation. The method approximates the solution of the nonlinear backward problem by solving a regularized problem. Error estimates between the exact and approximate solutions are provided. Under additional smoothness assumptions on the exact solution, an improved error estimate of O(ε(t+m)/(T+m)) is proved, where ε is a regularization parameter and t, T, m are time variables. This error estimate converges to zero faster than previous estimates as t approaches zero. A numerical experiment is also presented to demonstrate the effectiveness of the new method.
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.
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.
Geometric and viscosity solutions for the Cauchy problem of first orderJuliho Castillo
This document summarizes a doctoral dissertation on geometric and viscosity solutions to first order Cauchy problems. It introduces two types of solutions - viscosity solutions and minimax solutions - which are generally different. The aim is to show that iterating the minimax procedure over shorter time intervals approaches the viscosity solution. This extends previous work relating geometric and viscosity solutions in the symplectic case. The document outlines characteristics methods, generating families, Clarke calculus tools, and a proof constructing generating families to relate iterated minimax solutions to viscosity solutions.
Average Polynomial Time Complexity Of Some NP-Complete ProblemsAudrey Britton
This document discusses the average polynomial time complexity of some NP-complete problems. Specifically:
- It proves that when N(n) = [n^e] for 0 < e < 2, the CLIQUEN(n) problem (finding a clique of size k in a graph with n vertices and at most N(n) edges) is NP-complete, yet can be solved in average and almost everywhere polynomial time.
- It considers the CLIQUE problem under different non-uniform distributions on the input graphs, showing the problem can be solved in average polynomial time for certain values of the distribution parameter p.
- The main result shows that a family of subproblems of CLIQUE that remain
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
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.
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
An Approach For Solving Nonlinear Programming ProblemsMary Montoya
This document presents an approach for solving nonlinear programming problems using measure theory. It begins by transforming a nonlinear programming problem into an optimal control problem by treating the variables as time-varying and integrating the objective and constraint functions. It then solves the optimal control problem using measure theory by representing the control-trajectory pair as a positive Radon measure on the space of trajectories and controls. Finally, it shows that the optimal solution to the transformed optimal control problem provides an approximate optimal solution to the original nonlinear programming problem.
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.
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.
Geometric and viscosity solutions for the Cauchy problem of first orderJuliho Castillo
This document summarizes a doctoral dissertation on geometric and viscosity solutions to first order Cauchy problems. It introduces two types of solutions - viscosity solutions and minimax solutions - which are generally different. The aim is to show that iterating the minimax procedure over shorter time intervals approaches the viscosity solution. This extends previous work relating geometric and viscosity solutions in the symplectic case. The document outlines characteristics methods, generating families, Clarke calculus tools, and a proof constructing generating families to relate iterated minimax solutions to viscosity solutions.
Average Polynomial Time Complexity Of Some NP-Complete ProblemsAudrey Britton
This document discusses the average polynomial time complexity of some NP-complete problems. Specifically:
- It proves that when N(n) = [n^e] for 0 < e < 2, the CLIQUEN(n) problem (finding a clique of size k in a graph with n vertices and at most N(n) edges) is NP-complete, yet can be solved in average and almost everywhere polynomial time.
- It considers the CLIQUE problem under different non-uniform distributions on the input graphs, showing the problem can be solved in average polynomial time for certain values of the distribution parameter p.
- The main result shows that a family of subproblems of CLIQUE that remain
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
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.
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
An Approach For Solving Nonlinear Programming ProblemsMary Montoya
This document presents an approach for solving nonlinear programming problems using measure theory. It begins by transforming a nonlinear programming problem into an optimal control problem by treating the variables as time-varying and integrating the objective and constraint functions. It then solves the optimal control problem using measure theory by representing the control-trajectory pair as a positive Radon measure on the space of trajectories and controls. Finally, it shows that the optimal solution to the transformed optimal control problem provides an approximate optimal solution to the original nonlinear programming problem.
We understand that you're a college student and finances can be tight. That's why we offer affordable pricing for our online statistics homework help. Your future is important to us, and we want to make sure you can achieve your degree without added financial stress. Seeking assistance with statistics homework should be simple and stress-free, and that's why we provide solutions starting from a reasonable price.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
if you are struggling with your Multiple Linear Regression homework, do not hesitate to seek help from our statistics homework help experts. We are here to guide you through the process and ensure that you understand the concept and the steps involved in performing the analysis. Contact us today and let us help you ace your Multiple Linear Regression homework!
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
This summary provides an overview of numerical methods for solving initial value problems (IVPs) for ordinary differential equations:
1. Several common numerical methods for solving IVPs are presented, including explicit and implicit Euler methods, the trapezoidal (midpoint) rule, improved Euler (Runge-Kutta 2), and Runge-Kutta 4.
2. The concepts of consistency and convergence are introduced. A method is consistent if the local error decays to zero as the step size decreases, and convergent if the global error decreases with decreasing step size. Order refers to the rate of decay of local error.
3. Stability is also important, especially for moderate step sizes. Linear stability is introduced
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
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.
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant combination of the new integral transform “ELzaki Transform” and the homotopy perturbation method. The method is really capable of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed. Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas dynamics equation
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, we apply a new method called ELzaki transform homotopy perturbation method (ETHPM) to solve porous medium equation. This method is a combination of the new integral transform “ELzaki transform” and the homotopy perturbation method. The nonlinear term can be easily handled by homotopy perturbation method. The porous medium equations have importance in engineering and sciences and constitute a good model for many systems in various fields. Some cases of the porous medium equation are solved as examples to illustrate ability and reliability of mixture of ELzaki transform and homotopy perturbation method. The results reveal that the combination of ELzaki transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems and can be applied to other nonlinear problems. This method is seen as a better alternative method to some existing techniques for such realistic problems. Key words: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and porous medium equation
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
This document presents a modified q-homotopy analysis method (mq-HAM) for solving high-order nonlinear partial differential equations. The mq-HAM improves upon the standard q-HAM by avoiding repeated computations at each iteration step, making it more efficient. As an illustrative example, the method is applied to solve second- and third-order nonlinear cases. The key steps of the mq-HAM include transforming the high-order PDE into a system of first-order equations, then obtaining a series solution through a zero-order deformation equation and its derivatives with respect to an embedded parameter.
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.
This document discusses partial differential equations and the heat equation. It begins by defining partial differential equations and providing an example of the heat equation modeling heat transfer through a solid body. It then derives the one-dimensional heat equation to model heat flow through a uniform bar. The method of separation of variables is introduced to solve the heat equation under various boundary and initial conditions. Finally, examples are provided to demonstrate solving heat transfer problems using Fourier series expansions.
This document summarizes numerical studies of line soliton solutions to the Kadomtsev-Petviashvili (KP) equation. The KP equation models nonlinear shallow water waves and admits exact solitary wave solutions called line solitons. The document presents pseudospectral schemes for numerically solving the KP equation with initial conditions formed from pieces of exact one-soliton solutions. It demonstrates convergence of the numerical solutions to three types of exact two-soliton solutions: a (3142)-soliton, a Y-shaped soliton, and an O-shaped soliton. Parameters in the exact solutions are optimized to minimize the error between numerical and exact solutions.
Pierre-Simon Laplace made important contributions to mathematics, astronomy, physics, and statistics. He began working in calculus, which led to developing the Laplace transform. The Laplace transform is a linear operator that switches a function of time to a function of a complex variable, allowing differential equations to be solved more easily. It has various restrictions on the original function for the transform to be applied. The transform is widely used to solve ordinary and partial differential equations and has applications in areas like semiconductor mobility and modeling physical systems.
This document provides an overview of single step methods and Runge-Kutta methods for numerically solving ordinary and partial differential equations. It discusses:
1) Single step methods can be explicit, where the next value depends only on the current value, or implicit, where it depends on both the current and next values.
2) Runge-Kutta methods approximate the solution curve slope over an interval using a weighted average of slopes, rather than a single slope.
3) General Runge-Kutta methods use ν evaluations of the slope function f to compute ν intermediate slopes (stages). The weights are determined by comparing to Taylor series expansions.
We understand that you're a college student and finances can be tight. That's why we offer affordable pricing for our online statistics homework help. Your future is important to us, and we want to make sure you can achieve your degree without added financial stress. Seeking assistance with statistics homework should be simple and stress-free, and that's why we provide solutions starting from a reasonable price.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
if you are struggling with your Multiple Linear Regression homework, do not hesitate to seek help from our statistics homework help experts. We are here to guide you through the process and ensure that you understand the concept and the steps involved in performing the analysis. Contact us today and let us help you ace your Multiple Linear Regression homework!
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
This summary provides an overview of numerical methods for solving initial value problems (IVPs) for ordinary differential equations:
1. Several common numerical methods for solving IVPs are presented, including explicit and implicit Euler methods, the trapezoidal (midpoint) rule, improved Euler (Runge-Kutta 2), and Runge-Kutta 4.
2. The concepts of consistency and convergence are introduced. A method is consistent if the local error decays to zero as the step size decreases, and convergent if the global error decreases with decreasing step size. Order refers to the rate of decay of local error.
3. Stability is also important, especially for moderate step sizes. Linear stability is introduced
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
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.
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant combination of the new integral transform “ELzaki Transform” and the homotopy perturbation method. The method is really capable of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed. Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas dynamics equation
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, we apply a new method called ELzaki transform homotopy perturbation method (ETHPM) to solve porous medium equation. This method is a combination of the new integral transform “ELzaki transform” and the homotopy perturbation method. The nonlinear term can be easily handled by homotopy perturbation method. The porous medium equations have importance in engineering and sciences and constitute a good model for many systems in various fields. Some cases of the porous medium equation are solved as examples to illustrate ability and reliability of mixture of ELzaki transform and homotopy perturbation method. The results reveal that the combination of ELzaki transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems and can be applied to other nonlinear problems. This method is seen as a better alternative method to some existing techniques for such realistic problems. Key words: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and porous medium equation
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
This document presents a modified q-homotopy analysis method (mq-HAM) for solving high-order nonlinear partial differential equations. The mq-HAM improves upon the standard q-HAM by avoiding repeated computations at each iteration step, making it more efficient. As an illustrative example, the method is applied to solve second- and third-order nonlinear cases. The key steps of the mq-HAM include transforming the high-order PDE into a system of first-order equations, then obtaining a series solution through a zero-order deformation equation and its derivatives with respect to an embedded parameter.
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.
This document discusses partial differential equations and the heat equation. It begins by defining partial differential equations and providing an example of the heat equation modeling heat transfer through a solid body. It then derives the one-dimensional heat equation to model heat flow through a uniform bar. The method of separation of variables is introduced to solve the heat equation under various boundary and initial conditions. Finally, examples are provided to demonstrate solving heat transfer problems using Fourier series expansions.
This document summarizes numerical studies of line soliton solutions to the Kadomtsev-Petviashvili (KP) equation. The KP equation models nonlinear shallow water waves and admits exact solitary wave solutions called line solitons. The document presents pseudospectral schemes for numerically solving the KP equation with initial conditions formed from pieces of exact one-soliton solutions. It demonstrates convergence of the numerical solutions to three types of exact two-soliton solutions: a (3142)-soliton, a Y-shaped soliton, and an O-shaped soliton. Parameters in the exact solutions are optimized to minimize the error between numerical and exact solutions.
Pierre-Simon Laplace made important contributions to mathematics, astronomy, physics, and statistics. He began working in calculus, which led to developing the Laplace transform. The Laplace transform is a linear operator that switches a function of time to a function of a complex variable, allowing differential equations to be solved more easily. It has various restrictions on the original function for the transform to be applied. The transform is widely used to solve ordinary and partial differential equations and has applications in areas like semiconductor mobility and modeling physical systems.
This document provides an overview of single step methods and Runge-Kutta methods for numerically solving ordinary and partial differential equations. It discusses:
1) Single step methods can be explicit, where the next value depends only on the current value, or implicit, where it depends on both the current and next values.
2) Runge-Kutta methods approximate the solution curve slope over an interval using a weighted average of slopes, rather than a single slope.
3) General Runge-Kutta methods use ν evaluations of the slope function f to compute ν intermediate slopes (stages). The weights are determined by comparing to Taylor series expansions.
How Does CRISIL Evaluate Lenders in India for Credit RatingsShaheen Kumar
CRISIL evaluates lenders in India by analyzing financial performance, loan portfolio quality, risk management practices, capital adequacy, market position, and adherence to regulatory requirements. This comprehensive assessment ensures a thorough evaluation of creditworthiness and financial strength. Each criterion is meticulously examined to provide credible and reliable ratings.
Understanding how timely GST payments influence a lender's decision to approve loans, this topic explores the correlation between GST compliance and creditworthiness. It highlights how consistent GST payments can enhance a business's financial credibility, potentially leading to higher chances of loan approval.
Seminar: Gender Board Diversity through Ownership NetworksGRAPE
Seminar on gender diversity spillovers through ownership networks at FAME|GRAPE. Presenting novel research. Studies in economics and management using econometrics methods.
"Does Foreign Direct Investment Negatively Affect Preservation of Culture in the Global South? Case Studies in Thailand and Cambodia."
Do elements of globalization, such as Foreign Direct Investment (FDI), negatively affect the ability of countries in the Global South to preserve their culture? This research aims to answer this question by employing a cross-sectional comparative case study analysis utilizing methods of difference. Thailand and Cambodia are compared as they are in the same region and have a similar culture. The metric of difference between Thailand and Cambodia is their ability to preserve their culture. This ability is operationalized by their respective attitudes towards FDI; Thailand imposes stringent regulations and limitations on FDI while Cambodia does not hesitate to accept most FDI and imposes fewer limitations. The evidence from this study suggests that FDI from globally influential countries with high gross domestic products (GDPs) (e.g. China, U.S.) challenges the ability of countries with lower GDPs (e.g. Cambodia) to protect their culture. Furthermore, the ability, or lack thereof, of the receiving countries to protect their culture is amplified by the existence and implementation of restrictive FDI policies imposed by their governments.
My study abroad in Bali, Indonesia, inspired this research topic as I noticed how globalization is changing the culture of its people. I learned their language and way of life which helped me understand the beauty and importance of cultural preservation. I believe we could all benefit from learning new perspectives as they could help us ideate solutions to contemporary issues and empathize with others.
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quan2009.pdf
1. c
de Gruyter 2009
J. Inv. Ill-Posed Problems 17 (2009), 913–932 DOI 10.1515 / JIIP.2009.053
A new version of quasi-boundary value method
for a 1-D nonlinear ill-posed heat problem
P. H. Quan, D. D. Trong, and N. H. Tuan
Abstract. In this paper, a simple and convenient new regularization method which is called modified
quasi-boundary value method for solving nonlinear backward heat equation is given. Some new
quite sharp error estimates between the approximate solution are provided and generalize the results
in our paper [17, 19, 20]. The approximation solution is calculated by the contraction principle. A
numerical experiment is given.
Key words. Backward heat problem, nonlinearly ill-posed problem, quasi-boundary value methods,
quasi-reversibility methods, contraction principle.
AMS classification. 35K05, 35K99, 47J06, 47H10.
1. Introduction
Let T be a positive number. We consider the problem of finding the temperature u(x, t),
(x, t) ∈ (0, π) × [0, T] such that
ut − uxx = f(x, t, u(x, t)), (x, t) ∈ (0, π) × (0, T), (1.1)
u(0, t) = u(π, t) = 0, t ∈ (0, T), (1.2)
u(x, T) = g(x), x ∈ (0, π), (1.3)
where g(x), f(x, t, z) are given. The problem is called the backward heat problem, the
backward Cauchy problem or the final value problem.
As is known, the nonlinear problem is severely ill-posed, i.e., solutions do not al-
ways exist, and in the case of existence, these do not depend continuously on the given
data. In fact, from small noise contaminated physical measurements, the correspond-
ing solutions have large errors. This makes it difficult to find numerical calculations.
Hence, a regularization is in order. In the mathematical literature various methods
have been proposed for solving backward Cauchy problems. We can notably men-
tion the method of quasi-reversibility (QR method) of Lattes and Lions [2, 3, 7], the
quasi-boundary value method (Q.B.V method) [4] and the C-regularized semigroups
technique [11].
In the method of quasi-reversibility, the main idea is to replace A by Aε = gε(A). In
the original method in [7], Lattes and Lions have proposed gε(A) = A−εA2, to obtain
well-posed problem in the backward direction. Then, using the information from the
solution of the perturbed problem and solving the original problem, we get another
All authors were supported by the Council for Natural Sciences of Vietnam.
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2. 914 P. H. Quan, D. D. Trong, and N. H. Tuan
well-posed problem and this solution sometimes can be taken to be the approximate
solution of the ill-posed problem.
Difficulties may arise when using the method quasi-reversibility discussed above.
The essential difficulty is that the order of the operator is replaced by an operator of
second order, which produces serious difficulties on the numerical implementation,
in addition, the error c(ε) introduced by small change in the final value g is of the
order eT/ε
.
In 1983, Showalter, in [15, 16], presented a different method called the quasi-
boundary value (QBV) method to regularize that linear homogeneous problem which
gave a stability estimate better than the one of the disscused method. The main idea of
the method is to add an appropriate “corrector” into the final data. Using the method,
Clark and Oppenheimer, in [4], and Denche-Bessila, very recently in [5], regularized
the backward problem by replacing the final condition by
u(T) + εu(0) = g (1.4)
and
u(T) − εu0
(0) = g, (1.5)
respectively.
To the authors’ knowledge, so far there are many papers on the linear homogeneous
case of the backward problem, but we only find a few papers on the nonhomogeneous
case, and especially, the nonlinear case is very scarce.
For recent articles considering the nonlinear backward-parabolic heat, we refer the
reader to [14,17,20]. In [17], the authors used the quasi-boundary value method to reg-
ularize the latter problem. Thus, they have established, under the hypothesis that f is a
global Lipschitzian function, the existence of a unique solution for some approximated
well-posed problem as follows:
V ε
(x, t) =
∞
X
p=1
e−tp2
ε + e−T p2 ϕp −
Z T
t
e−tp2
εs/T + e−sp2 fp(V ε
)(s) ds
sin px, (1.6)
where ε is a positive parameter such that ε 0 and
fp(u)(t) =
2
π
hf(x, t, u(x, t)), sin (px)i =
2
π
Z π
0
f(x, t, u(x, t)) sin (px) dx, (1.7)
ϕp =
2
π
hϕ(x), sin (px)i =
2
π
Z π
0
ϕ(x) sin (px) dx (1.8)
and h · , · i is the inner product in L2(0, π).
Let ϕ and ϕε denote the exact and measured data at t = T, respectively, which
satisfy
kϕ − ϕεk ≤ ε.
Under a strong smoothness assumption on the original solution namely
Z T
0
∞
X
p=1
e2sp2
f2
p (u)(s) ds ∞, (1.9)
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3. Quasi-boundary value method 915
we obtain an error estimate
ku( · , t) − V ε
( · , t)k ≤
p
M1 exp
3k2T(T − t)
2
εt/T
, (1.10)
where
M1 = 3ku( · , 0)k2
+ 6π
Z T
0
∞
X
p=1
e2sp2
f2
p (u)(s) ds.
And in [19], the error is also given as the similar form
ku(t) − uε
(t)k ≤ Mβ(ε)t/T
.
It is easy to see that the two above errors do not tend to zero, if ε is fixed and t tends to
zero . Hence, the convergence of the approximate solution is very slow when t is in a
neighborhood of zero. Moreover, the error in case t = 0 is not given in here. These are
some disadvantages of the above methods.
In the present paper, we will improve the latter results by using the method given
in [17]. The nonlinear backward problem is approximated by the following one dimen-
sional problem:
uε
t − uε
xx =
∞
X
k=1
e−T k2
εk2 + e−T k2 fk(uε
)(t) sin (kx), (x, t) ∈ (0, π) × (0, T), (1.11)
uε
(0, t) = uε
(π, t) = 0, t ∈ [0, T], (1.12)
uε
(x, T) =
∞
X
k=1
e−T k2
εk2 + e−T k2 gk sin (kx), x ∈ [0, π], (1.13)
where ε ∈ (0, e T), and
gk =
2
π
hg(x), sin kxi =
2
π
Z π
0
g(x) sin (kx) dx,
fk(u)(t) =
2
π
hf(x, t, u(x, t)), sin kxi =
2
π
Z π
0
f(x, t, u(x, t)) sin kx dx
and h · , · i is the inner product in L2(0, π).
The paper is organized as follows. In Theorem 2.1 and 2.2, we will show that (1.11)–
(1.13) is well-posed and that the unique solution uε
(x, t) of it is given by
uε
(x, t) =
∞
X
k=1
εk2
+ e−T k2 −1
e−tk2
gk −
Z T
t
e(s−t−T )k2
fk(uε
)(s) ds
sin kx.
(1.14)
Then, in Theorem 2.4 and 2.6, we estimate the error between an exact solution u of
problem (1.1)–(1.3) and the approximation solution uε
of (1.14). Under some addi-
tional conditions on the smoothness of the exact solution u, we also show that
kuε
( · , t) − u( · , t)k ≤ Cε(t+m)/(T +m)
, (1.15)
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4. 916 P. H. Quan, D. D. Trong, and N. H. Tuan
where m 0, C depends on u and k · k is the norm of L2(0, π). Note that (1.15) is an
improved result of many recent papers, such as [17,19,20]. Oncemore, the convergence
of the approximate solution in t = 0 is also proved. The notations about the usefulness
and advantages of this method can be found in Remarks 2.3, 2.5. Finally, a numerical
experiment will be given in Section 4, which proves the efficiency of our method.
2. The main results
From now on, for clarity, we denote the solution of (1.1)–(1.3) by u(x, t), and the
solution of the problem (1.11)–(1.13) by uε
(x, t). Let ε be a positive number such that
0 ε e T.
The function f is called global Lipchitz function if f ∈ L∞
([0, π] × [0, T] × R)
satisfies
|f(x, y, w) − f(x, y, v)| ≤ L|w − v| (H1)
for a constant L 0 independent of x, y, w, v.
Theorem 2.1 (Existence and uniqueness of the regularized problem). Let (H1) hold.
Then problem (1.11)–(1.13) has a unique weak solution
uε
∈ W = C([0, T]; L2
(0, π)) ∩ L2
(0, T; H1
0 (0, π)) ∩ C1
(0, T; H1
0 (0, π)) (2.1)
satisfying (1.14).
Theorem 2.2 (Stability of the regularized solution). Let u and v be two solutions of
(1.11)–(1.13) corresponding to the final values g and h in L2(0, π). Then we have the
inequality
ku( · , t) − v( · , t)k ≤ εt/T −1
T
1 + ln (T/ε)
1−t/T
exp (L2
(T − t)2
) kg − hk.
Remark 2.3. In [18], the stability of magnitude is eT/ε
. And in [4, 5, 14, 17], the
stability estimate is of order εt/T −1, which is better than the previous results.
In this paper, we give different estimation of the stability is order of
Cεt/T −1
T
1 + ln (T/ε)
1−t/T
. (2.2)
It is clear to see that the order of stability (2.2) is less than the orders given in the above
results. This is one of the advantages of our method.
Despite the uniqueness, problem (1.1)–(1.3) is still ill-posed. Hence, we have to
resort to a regularization. We have the following result.
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5. Quasi-boundary value method 917
Theorem 2.4. Let (H1) hold.
a) If u(x, t) ∈ W is the solution of the problem (1.1)–(1.3) such that
Z T
0
∞
X
k=1
k4
e2sk2
f2
k(u)(s) ds ∞ (2.3)
and kuxx( · , 0)k ∞, then
ku( · , t) − uε
( · , t)k ≤ CM εt/T
T
1 + ln (T/ε)
1−t/T
. (2.4)
b) If u(x, t) satisfies
Q = sup
0≤t≤T
∞
X
k=1
k4
e2tk2
|hu(x, t), sin kxi|2
∞, (2.5)
then
ku( · , t) − uε
( · , t)k ≤ CQεt/T
T
1 + ln (T/ε)
1−t/T
,
for every t ∈ [0, T], where
M = 3kuxx(0)k2
+
3π
2
T
Z T
0
∞
X
k=1
k4
e2sk2
f2
k(u)(s) ds, (2.6)
CM =
p
πM e3L2T (T −t), (2.7)
CQ =
q
πQ e3L2T (T −t). (2.8)
c) If there exists a positive number m 0 such that
Qm = sup
0≤t≤T
∞
X
k=1
k4
e(2t+2m)k2
|hu(x, t), sin kxi|2
∞, (2.9)
then
ku( · , t) − Uε
( · , t)k ≤ ε(t+m)/(T +m)
p
Qm eL2
(T +1)(T −t)
(2.10)
for every t ∈ [0, T], where Uε
is the unique solution of the problem
Uε
(x, t) =
∞
X
k=1
εk2
+ e−(T +m)k2 −1
×
e−(t+m)k2
gk −
Z T
t
e(s−t−T −m)k2
fk(Uε
)(s) ds
sin kx. (2.11)
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6. 918 P. H. Quan, D. D. Trong, and N. H. Tuan
Remark 2.5. 1. In [17, p. 241] and [20] the error estimate between the exact solution
and the approximate solution is U(ε, t) = Cεt/T
. So, if the time t is close to the
original time t = 0, the convergence rates here are very slow. This implies that the
methods studied in [31, 33] are not useful to derive the error estimations in the case t
is near zero. To improve this, we give the convergence rate in the present theorem in
slightly different form, defined by V (ε, t) = Dεt/T
[T/(1 + ln (T/ε)]1−t/T
. We note
that limε→0 V (ε, t)/U(ε, t) = 0. Moreover, we also have limε→0(limt→0 U(ε, t)) = C
and limε→0(limt→0 V (ε, t)) = limε→0[DT/(1 + ln (T/ε))] = 0. This also proves
that our method gives a better approximation than the previous case which we know.
Comparing (1.12) with the previous results obtained in [17, 20], we realize that this
estimate is sharp and the best known estimate.
2. One superficial advantage of this method is that there is an error estimation for
the time t = 0. We have the following logarithmic estimate:
ku( · , 0) − uε
( · , 0)k ≤
p
M e3L2T 2 T
1 + ln (T/ε)
. (2.12)
These estimates, as noted above, are very seldom in the theory of ill-posed problems.
3. In the linear nonhomogeneous case f(x, t, u) = f(x, t), the error estimates were
given in [19]. And the assumption of f in (2.3) is not used, only in L2(0, T; L2(0, π)).
4. In Theorem 2.4a) we ask for a condition on the expansion coefficient fk. This
condition on u is severe and requires more than analiticy of x → f(x, t, u(x, t)) for all
t ∈ [0, T]. We note that the solution u depends on the nonlinear term f and therefore fk
and fk(u) are very difficult to be valued. Such an obscurity makes this theorem hard
to be used for numerical computations. To improve this, in Theorem 2.6b) we only
require the assumption on u, depending not on the function f(u). Once more, we note
that in the simple case of the right hand side f(u) = 0, the term Q becomes
∞
X
k=1
k4
e2tk2
|hu(x, t), sin kxi|2
= kuxx( · , 0)k.
So, the condition (2.5) is nature and acceptable.
5. In Theorem 2.4c) we give the error which is the Hölder form on the whole t ∈
[0, T]. Comparing (2.10) with (2.12), we can see that (2.10) is the optimal order.
In the case of nonexact data, we have the following theorem.
Theorem 2.6. Let the exact solution u of (1.1)–(1.3) correspond to g. Let gε be a
measured data such that
kgε − gk ≤ ε.
Then there exists a function wε
satisfying
a) kwε
( · , t) − u( · , t)k ≤ (1 +
√
M ) exp
3L2
T (T −t)
2
εt/T
T
1+ln (T/ε)
1−t/T
,
for every t ∈ [0, T], where u is defined in Theorem 2.4a).
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7. Quasi-boundary value method 919
b) kwε
( · , t) − u( · , t)k ≤ (1 +
√
πQ ) exp
3L2
T (T −t)
2
εt/T
T
1+ln (T/ε)
1−t/T
,
for every t ∈ [0, T], where u is defined in Theorem 2.4a), and M, Q are defined in
Theorem 2.4b).
3. Proofs of the main theorems
We give some lemmas which will be useful for proving the main theorems.
Lemma 3.1. Denote h(x) = 1/(εx + e−xT
) and Mε = T(ε + ε ln (T/ε))−1 then
h(x) ≤ Mε =
T
ε(1 + ln (T/ε))
. (3.1)
The proof of this lemma can be found in [19].
For 0 ≤ t ≤ s ≤ T, denote
Gε(s, t, k) =
e(s−t−T )k2
εk2 + e−T k2 . (3.2)
It is easy to see that
Gε(T, t, k) =
e−tk2
εk2 + e−T k2 . (3.3)
Lemma 3.2.
Gε(s, t, k) ≤ M(s−t)/T
ε . (3.4)
Proof. We have
Gε(s, t, k) =
e(s−t−T )k2
εk2 + e−T k2 =
e(s−t−T )k2
(εk2 + e−T k2
)(s−t)/T (εk2 + e−T k2
)(T +t−s)/T
≤
e(s−t−T )k2
(e−T k2
)(T +t−s)/T
1
(εk2 + e−T k2
)s/T −t/T
≤
T
ε(1 + ln (T/ε))
s/T −t/T
= εt/T −s/T
T
1 + ln (T/ε)
s/T −t/T
= M(s−t)/T
ε . (3.5)
2
Lemma 3.3.
Gε(T, t, k) ≤ M(T −t)/T
ε . (3.6)
Proof. Let s = T in Lemma 3.2. Then we obtain Lemma 3.3. 2
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8. 920 P. H. Quan, D. D. Trong, and N. H. Tuan
Proof of Theorem 2.1. The proof consists of Steps 1–3.
Step 1. Existence and uniqueness of the solution of the integral equation (1.14).
Put
F(w)(x, t) = P(x, t) −
∞
X
k=1
Z T
t
Gε(s, t, k)fk(w)(s) ds sin (kx)
for w ∈ C([0, T]; L2(0, π)), where
P(x, t) =
∞
X
k=1
Gε(T, t, k) hg(x), sin kxi sin kx.
We claim that, for every w, v ∈ C([0, T]; L2(0, π)), p ≥ 1, we have
kFp
(w)( · , t) − Fp
(v)( · , t)k2
≤
LT
ε(1 + ln (T/ε))
2p (T − t)p
Cp
p!
k|w − v|k2
, (3.7)
where C = max {T, 1}, and k| · |k is the sup norm on C([0, T]; L2(0, π)).
We shall prove the latter inequality by induction. For p = 1, using (3.2) and
Lemma 3.2, we find that
kF(w)( · , t) − F(v)( · , t)k2
=
π
2
∞
X
k=1
h Z T
t
Gε(s, t, k)(fk(w)(s) − fk(v)(s)) ds
i2
≤
π
2
∞
X
k=1
Z T
t
e(s−t−T )k2
εk2 + e−T k2
2
ds
Z T
t
(fk(w)(s) − fk(v)(s))2
ds
≤
π
2
∞
X
k=1
M2
ε (T − t)
Z T
t
(fk(w)(s) − fk(v)(s))2
ds
=
π
2
M2
ε (T − t)
Z T
t
∞
X
k=1
(fk(w)(s) − fk(v)(s))2
ds
= M2
ε (T − t)
Z T
t
Z π
0
(f(x, s, w(x, s)) − f(x, s, v(x, s)))2
dx ds
≤ L2
M2
ε (T − t)
Z T
t
Z π
0
|w(x, s) − v(x, s)|2
dx ds
= CL2
M2
ε (T − t) k|w − v|k2
.
Thus (3.7) holds.
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9. Quasi-boundary value method 921
Suppose that (3.7) holds for p = m. We prove that (3.7) holds for p = m + 1. We
have
kFm+1
(w)( · , t) − Fm+1
(v)( · , t)k2
=
π
2
∞
X
k=1
h Z T
t
Gε(s, t, k)(fk(Fm
(w))(s) − fk(Fm
(v))(s)) ds
i2
≤
π
2
M2
ε
∞
X
k=1
h Z T
t
|fk(Fm
(w))(s) − fk(Fm
(v))(s)| ds
i2
≤
π
2
M2
ε (T − t)
Z T
t
∞
X
k=1
|fk(Fm
(w))(s) − fk(Fm
(v))(s)|2
ds
≤ M2
ε (T − t)
Z T
t
kf( · , s, Fm
(w)( · , s)) − f( · , s, Fm
(v)( · , s))k2
ds
≤ M2
ε (T − t)L2
Z T
t
kFm
(w)( · , s) − Fm
(v)( · , s)k2
ds
≤ M2
ε (T − t)L2m+2
M2m
ε
Z T
t
(T − s)m
m!
dsCm
k|w − v|k2
≤ (LMε)2m+2 (T − t)m+1
(m + 1)!
Cm+1
k|w − v|k2
.
Therefore, by the induction principle, we have
k|Fp
(w) − Fp
(v)|k ≤
LT
ε(1 + ln (T/ε))
p Tp/2
√
p!
Cp/2
k|w − v|k
for all w, v ∈ C([0, T]; L2(0, π)).
We consider F : C([0, T]; L2(0, π)) → C([0, T]; L2(0, π)). Since
lim
p→∞
LT
ε(1 + ln (T/ε))
p Tp/2Cp/2
√
p!
= 0,
there exists a positive integer p0 such that
LT
ε(1 + ln (T/ε))
p0 Tp0/2Cp0/2
p
(p0)!
1
and Fp0 is a contraction. It follows that the equation Fp0 (w) = w has a unique solution
uε
∈ C([0, T]; L2(0, π)).
We claim that F(uε
) = uε
. In fact, we have F(Fp0 (uε
)) = F(uε
). Hence it
follows Fp0 (F(uε
)) = F(uε
). By the uniqueness of the fixed point of Fp0 , we
get that F(uε
) = uε
, that is, the equation F(w) = w has a unique solution uε
∈
C([0, T]; L2(0, π)).
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10. 922 P. H. Quan, D. D. Trong, and N. H. Tuan
Step 2. If uε
∈ W satisfies (1.14) then uε
is solution of (1.11)–(1.13). We have
uε
(x, t) =
∞
X
k=1
(εk2
+ e−T k2
)−1
e−tk2
gk −
Z T
t
e(s−t−T )k2
fk(uε
)(s) ds
sin kx,
0 ≤ t ≤ T. (3.8)
We can verify directly that uε
∈ C([0, T]; L2(0, π) ∩ C1((0, T); H1
0 (0, π)) ∩
L2(0, T; H1
0 (0, π))). In fact, uε
∈ C∞
((0, T]; H1
0 (0, π))). Moreover, by direct compu-
tation, one has
uε
t (x, t) =
∞
X
k=1
−k2
(εk2
+ e−T k2
)−1
e−tk2
gk −
Z T
t
e(s−t−T )k2
fk(uε
)(s) ds
sin kx
+
∞
X
k=1
e−T k2
(εk2
+ e−T k2
)−1
fk(uε
)(t) sin kx
= −
2
π
∞
X
k=1
k2
huε
(x, t), sin kxi sin kx
+
∞
X
k=1
e−T k2
(εk2
+ e−T k2
)−1
fk(uε
)(t) sin kx
= uε
xx(x, t) +
∞
X
k=1
e−T k2
(εk2
+ e−T k2
)−1
fk(uε
)(t) sin kx (3.9)
and
uε
(x, T) =
∞
X
k=1
e−T k2
(εk2
+ e−T k2
)−1
gk sin (kx). (3.10)
So uε
is the solution of (1.11)–(1.13).
Step 3. The problem (1.11)–(1.13) has at most one (weak) solution uε
∈ W.
Let uε
and vε
be two solutions of problem (1.11)–(1.13) such that uε
, vε
∈ W.
Putting wε
(x, t) = uε
(x, t) − vε
(x, t), then wε
satisfies the equation
wε
t − wε
xx =
∞
X
k=1
e−T k2
(εk2
+ e−T k2
)−1
(fk(uε
)(t) − fk(vε
)(t)) sin (kx). (3.11)
Noticing that e−T k2
(εk2 + e−T k2
)−1 ≤ 1/ε, we have
kwε
t − wε
xxk2
≤
1
ε2
∞
X
k=1
(fk(uε
)(t) − fk(vε
)(t))2
≤ kf( · , t, uε
( · , t)) − f( · , t, vε
( · , t))k2
/ε2
≤ M2
kuε
( · , t) − vε
( · , t)k2
/ε2
= M2
kwε
( · , t)k2
/ε2
.
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11. Quasi-boundary value method 923
Using the Lees–Protter in [29], we get wε
( · , t) = 0. This completes the proof of
Step 3.
The three steps complete the proof of the theorem. 2
Proof of Theorem 2.2. Using (1.14) and the inequality (a+b)2 ≤ 2(a2 +b2), we obtain
ku( · , t) − v( · , t)k2
=
π
2
∞
X
k=1
12.
13.
14. Gε(T, t, k)(gk − hk) −
Z T
t
Gε(s, t, k)(fk(u)(s) − fk(v)(s)) ds
15.
16.
17. 2
≤ π
∞
X
k=1
(Gε(T, t, k) |gk − hk|)2
+ π
∞
X
k=1
Z T
t
Gε(s, t, k) |fk(u)(s) − fk(v)(s)| ds
2
. (3.12)
Using Lemma 3.2, Lemma 3.3 and (3.12), we get
ku( · , t) − v( · , t)k2
≤ M2−2t/T
ε kg − hk2
+ 2L2
(T − t)M−2t/T
ε
Z T
t
M2s/T
ε ku( · , s) − v( · , s)k2
ds.
(3.13)
It follows from (3.13) that
M2t/T
ε ku( · , t) − v( · , t)k2
≤ M−2
ε kg − hk2
+ 2L2
(T − t)
Z T
t
M2s/T
ε ku( · , s) − v( · , s)k2
ds.
Using Gronwall’s inequality we have
M2t/T
ε ku( · , t) − v( · , t)k2
≤ M−2
ε exp (2L2
(T − t)2
) kg − hk2
.
Thus
ku( · , t) − v( · , t)k ≤ εt/T −1
T
1 + ln (T/ε)
1−t/T
exp (L2
(T − t)2
) kg − hk.
This completes the proof of the theorem. 2
Proof of Theorem 2.4a). Suppose the problem (1.1)–(1.3) has an exact solution u,
then u is given by
u(x, t) =
∞
X
k=1
e−(t−T )k2
gk −
Z T
t
e−(t−s)k2
fk(u)(s) ds
sin kx. (3.14)
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18. 924 P. H. Quan, D. D. Trong, and N. H. Tuan
Since
uk(0) = eT k2
gk −
Z T
0
esk2
fk(u)(s) ds
implies
gk = e−T k2
uk(0) +
Z T
0
e(s−T )k2
fk(u)(s) ds,
we get
u(x, T) =
∞
X
k=1
gk sin kx =
∞
X
k=1
e−T k2
uk(0) +
Z T
0
e−(T −s)k2
fk(u)(s) ds
sin kx.
Since (1.15) and (3.14), we have
uε
k(t) = (εk2
+ e−T k2
)−1
e−tk2
gk −
Z T
t
e(s−t−T )k2
fk(uε
)(s) ds
, (3.15)
uk(t) = eT k2
e−tk2
gk −
Z T
t
e(s−t−T )k2
fk(u)(s) ds
. (3.16)
With (3.3), (3.15) and (3.16) we have
uk(t) − uε
k(t) =
eT k2
−
1
εk2 + e−T k2
e−tk2
gk −
Z T
t
e(s−t−T )k2
fk(u)(s) ds
+
Z T
t
e(s−t−T )k2
εk2 + e−T k2 (fk(uε
)(s) − fk(u)(s)) ds
=
εk2 e−tk2
εk2 + e−T k2
eT k2
gk −
Z T
0
esk2
fk(u)(s) ds +
Z t
0
esk2
fk(u)(s) ds
+
Z T
t
Gε(s, t, k)(fk(uε
)(s) − fk(u)(s)) ds. (3.17)
Since (3.6) and
eT k2
gk −
Z T
0
esk2
fk(u)(s) ds = uk(0),
we have
|uk(t) − uε
k(t)| ≤ εGε(T, t, k)
30. +
Z T
t
Gε(s, t, k)|fk(u)(s) − fk(uε
)(s)| ds
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31. Quasi-boundary value method 925
≤ ε · M(T −t)/T
ε
|k2
uk(0)| +
Z t
0
|k2
esk2
fk(u)(s)| ds
+
Z T
t
M(s−t)/T
ε |fk(u)(s) − fk(uε
)(s)| ds
= ε · M(T −t)/T
ε
|k2
uk(0)| +
Z T
0
|k2
esk2
fk(u)(s)| ds
+
Z T
t
1
ε
Ms/T −1
ε |fk(u)(s) − fk(uε
)(s)| ds
.
Applying the inequality (a + b + c)2 ≤ 3(a2 + b2 + c2), we get
ku( · , t) − uε
( · , t)k2
=
π
2
∞
X
k=1
|uk(t) − uε
k(t)|2
≤
3π
2
ε2
· M(2T −2t)/T
ε
∞
X
k=1
h
|k2
uk(0)|2
+
Z T
0
|k2
esk2
fk(u)(s)| ds
2
+
Z T
t
1
ε
Ms/T −1
ε |fk(u)(s) − fk(uε
)(s)| ds
2i
= ε2
· M(2T −2t)/T
ε (I1 + I2 + I3),
where
I1 =
3π
2
∞
X
k=1
|k2
uk(0)|2
, I2 =
3π
2
∞
X
k=1
Z T
0
|k2
esk2
fk(u)(s)| ds
2
,
I3 =
3π
2
∞
X
k=1
Z T
t
1
ε
Ms/T −1
ε |fk(u)(s) − fk(uε
)(s)| ds
2
.
The terms I1, I2, I3 can be estimated as
I1 = 3ε2
M(2T −2t)/T
ε kuxx(0)k2
.
Using the Hölder inequality, we estimate the term I2 as follows:
I2 ≤
3π
2
T
Z T
0
∞
X
k=1
(k2
esk2
fk(u)(s))2
ds ≤
3π
2
T
Z T
0
∞
X
k=1
k4
e2sk2
f2
k(u)(s) ds
≤
3π
2
T
Z T
0
∞
X
k=1
k4
e2sk2
f2
k(u)(s) ds. (3.18)
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32. 926 P. H. Quan, D. D. Trong, and N. H. Tuan
Using condition (H1) from page 916, we get
I3 ≤
3π
2
(T − t)
Z T
t
1
ε2
M2s/T −2
ε
∞
X
k=1
(fk(u)(s) − fk(uε
)(s))2
ds
≤ 3(T − t)
Z T
t
1
ε2
M2s/T −2
ε kf( · , s, u( · , s)) − f( · , s, uε
( · , s))k2
ds
≤ 3L2
T
Z T
t
1
ε2
M2s/T −2
ε ku( · , s) − uε
( · , s)k2
ds. (3.19)
Combining (3.18), (3.19), we obtain
ku( · , t) − uε
( · , t)k2
≤ ε2
· M(2T −2t)/T
ε
h
3kuxx(0)k2
+
3π
2
T
Z T
0
∞
X
k=1
k4
e2sk2
f2
k(u)(s) ds
+ 3L2
T
Z T
t
1
ε2
M2s/T −2
ε ku( · , s) − uε
( · , s)k2
ds
i
.
It follows that
1
ε2
· M(2t−2T )/T
ε ku( · , t) − uε
( · , t)k2
≤ M + 3L2
T
Z T
t
1
ε2
M2s/T −2
ε ku( · , s) − uε
( · , s)k2
ds.
Using Gronwall’s inequality, we get
ε−2t/T
T
1 + ln (T/ε)
2t/T −2
ku( · , t) − uε
( · , t)k2
≤ M e3L2
T (T −t)
.
Hence
ku( · , t) − uε
( · , t)k2
≤ M e3L2
T (T −t)
ε2t/T
T
1 + ln (T/ε)
2−2t/T
.
This completes the proof of Theorem 2.4a). 2
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33. Quasi-boundary value method 927
Proof of Theorem 2.4b). Since (3.17), we prove in a similar manner to that in the above
part, that
|uk(t) − uε
k(t)|
≤
34.
35.
36. eT k2
−
1
εk2 + e−T k2
e−tk2
gk −
Z T
t
e(s−t−T )k2
fk(u)(s) ds
51. +
Z T
t
Gε(s, t, k)|fk(uε
)(s) − fk(u)(s)| ds
≤
ε e−tk2
εk2 + e−T k2 k2
etk2
|uk(t)| +
Z T
t
Gε(s, t, k)|fk(u)(s) − fk(uε
)(s)| ds (3.20)
≤ ε · M(T −t)/T
ε |k2
etk2
uk(t)| +
Z T
t
M(s−t)/T
ε |fk(u)(s) − fk(uε
)(s)| ds. (3.21)
This implies that
ku( · , t) − uε
( · , t)k2
=
π
2
∞
X
k=1
|uk(t) − uε
k(t)|2
≤ π
∞
X
k=1
ε2
· M(2T −2t)/T
ε |k2
etk2
uk(t)|2
+ π
∞
X
k=1
ε2
· M(2T −2t)/T
ε
×
Z T
t
ε−s/T
T
1 + ln (T/ε)
s/T −1
|fk(u)(s) − fk(uε
)(s)| ds
2
.
Since Mε = T(ε + ε ln (T/ε))−1 implies that
ε2
· M(2T −2t)/T
ε = ε2t/T
T
1 + ln (T/ε)
2−2t/T
,
we obtain
ku( · , t) − uε
( · , t)k2
≤ ε2t/T
T
1 + ln (T/ε)
2−2t/T
π
∞
X
k=1
k4
e2tk2
u2
k(t)
+ 2L2
Tε2t/T
T
1 + ln (T/ε)
2−2t/T
Z T
t
ε−2s/T
T
1 + ln (T/ε)
2s/T −2
× ku( · , s) − uε
( · , s)k2
ds.
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52. 928 P. H. Quan, D. D. Trong, and N. H. Tuan
Using again Gronwall’s inequality, we get
ε−2t/T
T
1 + ln (T/ε)
2t/T −2
ku( · , t) − uε
( · , t)k2
≤ Q e2L2
T (T −t)
.
This completes the proof of Theorem 2.4b. 2
Proof of Theorem 2.4c). We have
|uk(t) − Uε
k (t)|
≤
67. εk2 e−tk2
εk2 + e−(T +m)k2
eT k2
gk −
Z T
t
esk2
fk(u)(s) ds
68.
69.
70. +
Z T
t
e(s−t−T −m)k2
εk2 + e−(T +m)k2 |fk(Uε
)(s) − fk(u)(s)| ds
≤
ε e−(t+m)k2
εk2 + e−(T +m)k2 k2
e(t+m)k2
|uk(t)| +
Z T
t
Gε(s, t, k)|fk(u)(s) − fk(uε
)(s)| ds
≤ ε(t+m)/(T +m)
k2
e(t+m)k2
|uk(t)| +
Z T
t
M(s−t)/(T +m)
ε |fk(u)(s) − fk(Uε
)(s)| ds
≤ ε(t+m)/(T +m)
k2
e(t+m)k2
|uk(t)| +
Z T
t
ε
T
(s−t)/(T +m)
|fk(u)(s) − fk(Uε
)(s)| ds.
(3.22)
Then, we obtain
ku( · , t) − Uε
( · , t)k2
=
π
2
∞
X
k=1
|uk(t) − Uε
k (t)|2
≤ ε2(t+m)/(T +m)
∞
X
k=1
k4
e2(t+m)k2
u2
k(t)
+ 2L2
(T + 1)ε2t/(T +m)
Z T
t
ε−2s/(T +m)
ku( · , s) − Uε
( · , s)k2
ds.
Using again Gronwall’s inequality, we get
ε−2t/T +m
ku( · , t) − Uε
( · , t)k2
≤ ε2m/(T +m)
Qm e2L2
(T +1)(T −t)
.
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71. Quasi-boundary value method 929
Hence, we obtain
ku( · , t) − Uε
( · , t)k ≤ ε(t+m)/(T +m)
p
Qm eL2
(T +1)(T −t)
. (3.23)
2
Proof of Theorem 2.6. Let uε
be the solution of problem (1.11)–(1.13) corresponding
to g and let wε
be the solution of problem (1.11)–(1.13) corresponding to gε.
Using Theorem 2.2 and 2.4a), we get
kwε
( · , t) − u( · , t)k ≤ kwε
( · , t) − uε
( · , t)k + kuε
( · , t) − u( · , t)k
≤ εt/T −1
T
1 + ln (T/ε)
1−t/T
exp (L2
(T − t)2
)kgε − gk
+
p
M e3L2T (T −t) εt/T
T
1 + ln (T/ε)
1−t/T
≤ (1 +
√
M ) exp
3L2T(T − t)
2
εt/T
T
1 + ln (T/ε)
1−t/T
for every t ∈ [0, T].
Theorem 2.6b) is similarly proved as Theorem 2.6a). 2
4. A numerical experiment
We consider the equation
−uxx + ut = f(u) + g(x, t),
where
f(u) = u4
, g(x, t) = 2 et
sin x − e4t
sin4
x,
and
u(x, 1) = ϕ0(x) ≡ e sin x.
The exact solution of the equation is
u(x, t) = et
sin x.
Especially
u(x, 99/100) ≡ u(x) = exp (99/100) sin x.
Let ϕε(x) ≡ ϕ(x) = (ε + 1) e sin x. We have
kϕε − ϕk2 =
sZ π
0
ε2 e2 sin2
x dx = ε e
r
π
2
.
We find the regularized solution uε(x, 99/100) ≡ uε(x) having the form
uε(x) = vm(x) = w1,m sin x + w2,m sin 2x + w3,m sin 3x,
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72. 930 P. H. Quan, D. D. Trong, and N. H. Tuan
where
v1(x) = (ε + 1) e sin x, w1,1 = (ε + 1) e, w2,1 = 0, w3,1 = 0
and
a = 1/10000, tm = 1 − am, m = 1, 2, . . . , 100,
wi,m+1 =
e−tm+1i2
εi2 + e−tmi2 wi,m
−
2
π
Z tm
tm+1
e−tm+1i2
εi2 + e−tmi2 e(s−tm)i2
Z π
0
(v4
m(x) + g(x, s)) sin ix dx
ds,
i = 1, 2, 3.
Let aε = kuε − uk be the error between the regularization solution uε and the exact
solution u. Letting ε = ε1 = 10−5, ε = ε2 = 10−7, ε = ε3 = 10−11, we have
ε uε aε
ε1 = 10−5
2.685490624 sin (x) − 0.00009487155350 sin (3x) 0.005744631447
ε2 = 10−7
2.691122866 sin (x) + 0.00001413193606 sin (3x) 0.0001124971593
ε3 = 10−11
2.691180223 sin (x) + 0.00002138991088 sin (3x) 0.00005831365439
We have, in view of the error table in [17, p. 214],
ε uε aε
ε1 = 10−5
2.430605996 sin (x) − 0.0001718460902 sin (3x) 0.3266494251
ε2 = 10−7
2.646937077 sin (x) − 0.002178680692 sin (3x) 0.05558566020
ε3 = 10−11
2.649052245 sin (x) − 0.004495263004 sin (3x) 0.05316693437
By applying the stabilized quasi-reversibility method from [20], we have the ap-
proximate solution uε(x, 99/100) ≡ uε(x) having the form
uε(x) = vm(x) = w1,m sin x + w6,m sin 6x,
where v1(x) = (ε + 1) e sin x, w1,1 = (ε + 1) e, w6,1 = 0, and a = 1/10000, tm =
1 − am for m = 1, 2, . . . , 100, and
wi,m+1 = (ε + e−tmi2
)(tm+1−tm)/tm
wi,m
−
2
π
Z tm
tm+1
e(s−tm+1)i2
Z π
0
(v4
m(x) + g(x, s)) sin ix dx
ds
for i = 1, 6. The following table shows the approximation error in this case.
ε uε kuε − uk
ε1 = 10−5
2.690989330 sin(x) − 0.06078794774 sin(6x) 0.003940316590
ε2 = 10−7
2.691002638 sin(x) − 0.05797060493 sin(6x) 0.003592425036
ε3 = 10−11
2.691023938 sin(x) − 0.05663820292 sin(6x) 0.003418420030
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73. Quasi-boundary value method 931
Looking at three above tables with comparison between other methods, we can see
that the error results of the second and third tables are smaller than those in the first
table. This shows that our approach has a nice regularizing effect and gives a better
approximation compared to the previous method in [17,20].
Acknowledgments. The authors would like to thank the anonymous referees for their
careful reading and valuable comments and suggestions.
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74. 932 P. H. Quan, D. D. Trong, and N. H. Tuan
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Received April 14, 2009
Author information
P. H. Quan, Department of Mathematics, Sai Gon University, 273 An Duong Vuong, Q.5, Ho Chi
Minh city, Vietnam.
D. D. Trong, Department of Mathematics, HoChiMinh City National University, 227 Nguyen Van
Cu, Q. 5, Ho Chi Minh city, Vietnam.
N. H. Tuan, Department of Mathematics, Sai Gon University, 273 An Duong Vuong, Q.5, Ho Chi
Minh city, Vietnam.
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