The document analyzes the analytic solution of Burger's equations using the variational iteration method. It begins by introducing the variational iteration method and how it can be used to solve differential equations. It then applies the method to obtain exact solutions for the (1+1), (1+2), and (1+3) dimensional Burger equations. Lengthy iterative solutions are presented for each case. The variational iteration method is shown to provide exact solutions to these Burger equations without requiring linearization.
Derivation and Application of Multistep Methods to a Class of First-order Ord...AI Publications
Of concern in this work is the derivation and implementation of the multistep methods through Taylor’s expansion and numerical integration. For the Taylor’s expansion method, the series is truncated after some terms to give the needed approximations which allows for the necessary substitutions for the derivatives to be evaluated on the differential equations. For the numerical integration technique, an interpolating polynomial that is determined by some data points replaces the differential equation function and it is integrated over a specified interval. The methods show that they are only convergent if and only if they are consistent and stable. In our numerical examples, the methods are applied on non-stiff initial value problems of first-order ordinary differential equations, where it is established that the multistep methods show superiority over the single-step methods in terms of robustness, efficiency, stability and accuracy, the only setback being that the multi-step methods require more computational effort than the single-step methods.
This document presents an internship project report on multistep methods for solving initial value problems of ordinary differential equations. It introduces the basic problem of finding the function y(t) that satisfies a given differential equation and initial condition. It discusses existence and uniqueness theorems, Picard's method of successive approximations, and approaches for approximating the required integrations, including the derivative, Taylor series, and Euler's methods. The report appears to evaluate various one-step and multistep numerical methods for solving initial value problems, including Runge-Kutta, Adams-Bashforth, and Adams-Moulton methods.
Adomian decomposition method for solving higher order boundary value problemsAlexander Decker
This document presents an application of the Adomian decomposition method to solve higher order boundary value problems. The Adomian decomposition method assumes a series solution for the unknown quantity and decomposes the nonlinear operator into a series of polynomials. The document outlines how the method can be applied to solve third order singular boundary value problems and fourth order beam bending problems. It describes the key steps of the Adomian decomposition method and introduces the inverse operators used to obtain approximate solutions to the boundary value problems.
11.solution of a singular class of boundary value problems by variation itera...Alexander Decker
1) The document proposes an effective methodology called the variation iteration method to find solutions to singular second order linear and nonlinear boundary value problems.
2) The variation iteration method constructs a sequence of correction functionals to iteratively solve the boundary value problem. It is shown that the limit of the convergent iterative sequence obtained from this method is the exact solution.
3) The convergence of the iterative sequence generated by the variation iteration method is analyzed. It is established that the sequence converges to the exact solution under certain continuity conditions on the problem functions.
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.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
11.solution of a subclass of singular second orderAlexander Decker
This document presents a method for solving a subclass of singular second order differential equations of Lane-Emden type using Taylor series. The method identifies conditions where the equations can be solved analytically as a polynomial function. Several examples from literature are provided and solved using the proposed Taylor series method to demonstrate its effectiveness. The method produces exact solutions in polynomial form and is shown to perform well on both linear and nonlinear boundary value problems.
Derivation and Application of Multistep Methods to a Class of First-order Ord...AI Publications
Of concern in this work is the derivation and implementation of the multistep methods through Taylor’s expansion and numerical integration. For the Taylor’s expansion method, the series is truncated after some terms to give the needed approximations which allows for the necessary substitutions for the derivatives to be evaluated on the differential equations. For the numerical integration technique, an interpolating polynomial that is determined by some data points replaces the differential equation function and it is integrated over a specified interval. The methods show that they are only convergent if and only if they are consistent and stable. In our numerical examples, the methods are applied on non-stiff initial value problems of first-order ordinary differential equations, where it is established that the multistep methods show superiority over the single-step methods in terms of robustness, efficiency, stability and accuracy, the only setback being that the multi-step methods require more computational effort than the single-step methods.
This document presents an internship project report on multistep methods for solving initial value problems of ordinary differential equations. It introduces the basic problem of finding the function y(t) that satisfies a given differential equation and initial condition. It discusses existence and uniqueness theorems, Picard's method of successive approximations, and approaches for approximating the required integrations, including the derivative, Taylor series, and Euler's methods. The report appears to evaluate various one-step and multistep numerical methods for solving initial value problems, including Runge-Kutta, Adams-Bashforth, and Adams-Moulton methods.
Adomian decomposition method for solving higher order boundary value problemsAlexander Decker
This document presents an application of the Adomian decomposition method to solve higher order boundary value problems. The Adomian decomposition method assumes a series solution for the unknown quantity and decomposes the nonlinear operator into a series of polynomials. The document outlines how the method can be applied to solve third order singular boundary value problems and fourth order beam bending problems. It describes the key steps of the Adomian decomposition method and introduces the inverse operators used to obtain approximate solutions to the boundary value problems.
11.solution of a singular class of boundary value problems by variation itera...Alexander Decker
1) The document proposes an effective methodology called the variation iteration method to find solutions to singular second order linear and nonlinear boundary value problems.
2) The variation iteration method constructs a sequence of correction functionals to iteratively solve the boundary value problem. It is shown that the limit of the convergent iterative sequence obtained from this method is the exact solution.
3) The convergence of the iterative sequence generated by the variation iteration method is analyzed. It is established that the sequence converges to the exact solution under certain continuity conditions on the problem functions.
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.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
11.solution of a subclass of singular second orderAlexander Decker
This document presents a method for solving a subclass of singular second order differential equations of Lane-Emden type using Taylor series. The method identifies conditions where the equations can be solved analytically as a polynomial function. Several examples from literature are provided and solved using the proposed Taylor series method to demonstrate its effectiveness. The method produces exact solutions in polynomial form and is shown to perform well on both linear and nonlinear boundary value problems.
This document discusses using the Taylor series method to solve a subclass of singular second order differential equations of Lane-Emden type.
The method is applied to solve boundary value problems that arise from modeling physical phenomena. The Taylor series method produces an analytic solution in the form of a polynomial function.
Several examples from literature are considered to demonstrate the suitability and viability of the Taylor series method for solving these types of differential equations.
Solution of a singular class of boundary value problems by variation iteratio...Alexander Decker
1. The document proposes an effective methodology called the variation iteration method to find solutions to a general class of singular second-order linear and nonlinear boundary value problems.
2. The variation iteration method generates a sequence of correction functionals that converges to the exact solution of the boundary value problem.
3. The author applies the variation iteration method to solve a specific class of boundary value problems and derives the sequence of correction functionals. Convergence of the iterative sequence is also analyzed.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
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.
On an Optimal control Problem for Parabolic Equationsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document discusses stochastic partial differential equations (SPDEs). It outlines several approaches that have been used to solve SPDEs, including methods based on diffusion processes, stochastic characteristic systems, direct methods from mathematical physics, and substitution of integral equations. It also discusses using backward stochastic differential equations to study SPDEs and introduces notation for the analysis of an Ito SDE with inverse time. The document is technical in nature and outlines the mathematical frameworks and equations involved in solving SPDEs through various probabilistic methods.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
The document summarizes key concepts from elementary quantum physics that will be built upon in the text, including:
1) The time-dependent and time-independent Schrodinger equations, which describe the wave function and energy levels of quantum systems.
2) Observables in quantum physics are represented by operators, and the measurement of an observable leaves the system in an eigenstate of that operator.
3) The Heisenberg uncertainty principle limits the precision with which conjugate variables like position and momentum can be known simultaneously.
4) Angular momentum is quantized and can be decomposed into orbital and spin components, with associated quantum numbers and eigenstates. Operators for total and z-component angular
IRJET- Wavelet based Galerkin Method for the Numerical Solution of One Dimens...IRJET Journal
This document presents a wavelet-based Galerkin method for numerically solving one-dimensional partial differential equations using Hermite wavelets. Hermite wavelets are used as the basis functions in the Galerkin method. The method is demonstrated on some test problems, and the numerical results obtained from the proposed method are compared to exact solutions and a finite difference method to evaluate the accuracy and efficiency of the proposed wavelet Galerkin approach.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
This document discusses partial differential equations (PDEs). It begins by defining PDEs as equations that involve partial derivatives with respect to two or more independent variables. Next, it provides examples of how PDEs can be formed and classified based on characteristics like order, degree, whether they are linear or nonlinear. Then, it discusses methods for solving common types of PDEs like linear PDEs. Finally, it derives the one-dimensional wave equation and shows its solution as a product of functions involving the independent variables x and t.
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
The document discusses a new type of fractional differential equation that combines a multi-point boundary condition and an integral boundary condition. It presents existence results for solutions to this type of equation. The paper defines relevant fractional calculus concepts and establishes Green's functions for the boundary value problem. It then applies fixed point theorems to show the existence of multiple positive solutions under certain conditions on the continuous function f in the equation. Examples are also provided to illustrate the results.
This document is a master's thesis written in Chinese that investigates the existence and uniqueness of solutions to stochastic differential equations (SDEs) with Lévy noise and non-Lipschitz coefficients. It introduces Lévy processes and their properties, including the Lévy-Itô decomposition. It defines stochastic integration with respect to compensated Poisson processes and provides Itô's formula for Lévy diffusions. The thesis proves that if weak existence and pathwise uniqueness hold for an SDE with Lévy noise, then it has a unique strong solution. It establishes conditions on the coefficients that ensure infinite lifetime and pathwise uniqueness of the solution.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
Generalization of Tensor Factorization and ApplicationsKohei Hayashi
This document presents two tensor factorization methods: Exponential Family Tensor Factorization (ETF) and Full-Rank Tensor Completion (FTC). ETF generalizes Tucker decomposition by allowing for different noise distributions in the tensor and handles mixed discrete and continuous values. FTC completes missing tensor values without reducing dimensionality by kernelizing Tucker decomposition. The document outlines these methods and their motivations, discusses Tucker decomposition, and provides an example applying ETF to anomaly detection in time series sensor data.
Conditional trees use permutation tests and conditional inference at each step of recursive partitioning to overcome problems with traditional CART trees, such as selection bias and overfitting. The algorithm selects the variable with the strongest association using permutation tests, then searches for the best split point using a test statistic and permutation tests. It repeats this process recursively on the partitions until a stopping criterion is met where permutation tests show no variable has significant influence on the response. This approach aims to provide an unbiased and interpretable tree structure.
This document describes the design and implementation of a fingerprint-based identity authentication system. The system uses an improved algorithm to extract minutiae features from fingerprints faster and more accurately than previous methods. It then employs an alignment-based matching algorithm to find correspondences between input and stored fingerprint templates without exhaustive search. Experimental results on standard fingerprint databases show the system can achieve good performance and satisfy response time requirements for authentication, taking about 1.4 seconds on average. The system provides a means of positive identity verification through fingerprint biometrics with a very high level of accuracy.
This document summarizes a study on designing an optical beam filter and detector for laser range finding using photonic crystals. The study proposes using a one-dimensional photonic crystal structure to replace the mid-pass filter and beam detector in a laser range finder system that uses frequency modulation of continuous waves (FMCW). Through simulation, the photonic crystal structure is designed to have a narrow bandwidth of 5 nanometers around 630-645 nm to detect the transmitted laser beam. Defects and apodization of the photonic crystal refractive index profile allow controlling the wavelength passband. This detector design could enable measuring distances depending on the transmitted laser wavelength and power, reducing measurement errors.
The document describes a method for detecting abandoned objects using a moving camera. It proposes matching a reference video without objects to a target video that may contain abandoned objects. The method uses geometric alignment between sequences and within sequences, local appearance comparison, and temporal filtering to identify differences indicating abandoned objects. SIFT features are extracted and matched between videos, and RANSAC is used to estimate homographies for alignment. False alarms from shadows, reflections, and high objects are removed. The approach is validated on 15 test videos.
This document discusses using the Taylor series method to solve a subclass of singular second order differential equations of Lane-Emden type.
The method is applied to solve boundary value problems that arise from modeling physical phenomena. The Taylor series method produces an analytic solution in the form of a polynomial function.
Several examples from literature are considered to demonstrate the suitability and viability of the Taylor series method for solving these types of differential equations.
Solution of a singular class of boundary value problems by variation iteratio...Alexander Decker
1. The document proposes an effective methodology called the variation iteration method to find solutions to a general class of singular second-order linear and nonlinear boundary value problems.
2. The variation iteration method generates a sequence of correction functionals that converges to the exact solution of the boundary value problem.
3. The author applies the variation iteration method to solve a specific class of boundary value problems and derives the sequence of correction functionals. Convergence of the iterative sequence is also analyzed.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
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.
On an Optimal control Problem for Parabolic Equationsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document discusses stochastic partial differential equations (SPDEs). It outlines several approaches that have been used to solve SPDEs, including methods based on diffusion processes, stochastic characteristic systems, direct methods from mathematical physics, and substitution of integral equations. It also discusses using backward stochastic differential equations to study SPDEs and introduces notation for the analysis of an Ito SDE with inverse time. The document is technical in nature and outlines the mathematical frameworks and equations involved in solving SPDEs through various probabilistic methods.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
The document summarizes key concepts from elementary quantum physics that will be built upon in the text, including:
1) The time-dependent and time-independent Schrodinger equations, which describe the wave function and energy levels of quantum systems.
2) Observables in quantum physics are represented by operators, and the measurement of an observable leaves the system in an eigenstate of that operator.
3) The Heisenberg uncertainty principle limits the precision with which conjugate variables like position and momentum can be known simultaneously.
4) Angular momentum is quantized and can be decomposed into orbital and spin components, with associated quantum numbers and eigenstates. Operators for total and z-component angular
IRJET- Wavelet based Galerkin Method for the Numerical Solution of One Dimens...IRJET Journal
This document presents a wavelet-based Galerkin method for numerically solving one-dimensional partial differential equations using Hermite wavelets. Hermite wavelets are used as the basis functions in the Galerkin method. The method is demonstrated on some test problems, and the numerical results obtained from the proposed method are compared to exact solutions and a finite difference method to evaluate the accuracy and efficiency of the proposed wavelet Galerkin approach.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
This document discusses partial differential equations (PDEs). It begins by defining PDEs as equations that involve partial derivatives with respect to two or more independent variables. Next, it provides examples of how PDEs can be formed and classified based on characteristics like order, degree, whether they are linear or nonlinear. Then, it discusses methods for solving common types of PDEs like linear PDEs. Finally, it derives the one-dimensional wave equation and shows its solution as a product of functions involving the independent variables x and t.
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
The document discusses a new type of fractional differential equation that combines a multi-point boundary condition and an integral boundary condition. It presents existence results for solutions to this type of equation. The paper defines relevant fractional calculus concepts and establishes Green's functions for the boundary value problem. It then applies fixed point theorems to show the existence of multiple positive solutions under certain conditions on the continuous function f in the equation. Examples are also provided to illustrate the results.
This document is a master's thesis written in Chinese that investigates the existence and uniqueness of solutions to stochastic differential equations (SDEs) with Lévy noise and non-Lipschitz coefficients. It introduces Lévy processes and their properties, including the Lévy-Itô decomposition. It defines stochastic integration with respect to compensated Poisson processes and provides Itô's formula for Lévy diffusions. The thesis proves that if weak existence and pathwise uniqueness hold for an SDE with Lévy noise, then it has a unique strong solution. It establishes conditions on the coefficients that ensure infinite lifetime and pathwise uniqueness of the solution.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
Generalization of Tensor Factorization and ApplicationsKohei Hayashi
This document presents two tensor factorization methods: Exponential Family Tensor Factorization (ETF) and Full-Rank Tensor Completion (FTC). ETF generalizes Tucker decomposition by allowing for different noise distributions in the tensor and handles mixed discrete and continuous values. FTC completes missing tensor values without reducing dimensionality by kernelizing Tucker decomposition. The document outlines these methods and their motivations, discusses Tucker decomposition, and provides an example applying ETF to anomaly detection in time series sensor data.
Conditional trees use permutation tests and conditional inference at each step of recursive partitioning to overcome problems with traditional CART trees, such as selection bias and overfitting. The algorithm selects the variable with the strongest association using permutation tests, then searches for the best split point using a test statistic and permutation tests. It repeats this process recursively on the partitions until a stopping criterion is met where permutation tests show no variable has significant influence on the response. This approach aims to provide an unbiased and interpretable tree structure.
This document describes the design and implementation of a fingerprint-based identity authentication system. The system uses an improved algorithm to extract minutiae features from fingerprints faster and more accurately than previous methods. It then employs an alignment-based matching algorithm to find correspondences between input and stored fingerprint templates without exhaustive search. Experimental results on standard fingerprint databases show the system can achieve good performance and satisfy response time requirements for authentication, taking about 1.4 seconds on average. The system provides a means of positive identity verification through fingerprint biometrics with a very high level of accuracy.
This document summarizes a study on designing an optical beam filter and detector for laser range finding using photonic crystals. The study proposes using a one-dimensional photonic crystal structure to replace the mid-pass filter and beam detector in a laser range finder system that uses frequency modulation of continuous waves (FMCW). Through simulation, the photonic crystal structure is designed to have a narrow bandwidth of 5 nanometers around 630-645 nm to detect the transmitted laser beam. Defects and apodization of the photonic crystal refractive index profile allow controlling the wavelength passband. This detector design could enable measuring distances depending on the transmitted laser wavelength and power, reducing measurement errors.
The document describes a method for detecting abandoned objects using a moving camera. It proposes matching a reference video without objects to a target video that may contain abandoned objects. The method uses geometric alignment between sequences and within sequences, local appearance comparison, and temporal filtering to identify differences indicating abandoned objects. SIFT features are extracted and matched between videos, and RANSAC is used to estimate homographies for alignment. False alarms from shadows, reflections, and high objects are removed. The approach is validated on 15 test videos.
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 document discusses power quality issues and techniques for detecting and mitigating power quality disturbances. It introduces the concepts of power quality and various power quality disturbances. It then discusses techniques for detecting disturbances such as wavelet transform and fuzzy classifiers. Finally, it proposes using a dynamic voltage restorer (DVR) to mitigate detected power quality problems by injecting voltage and regulating the load side voltage. The DVR is presented as an efficient custom power device that can compensate for issues like voltage sags and swells as well as reduce transients and harmonics.
This document summarizes vulnerabilities in web applications and methods to protect against them. It discusses how vulnerabilities can occur from issues like format string exploits, SQL injection, and cross-site scripting. The document also describes different approaches to testing for vulnerabilities, including white-box and black-box testing. Additionally, it analyzes vulnerability information from various organization's lists of top vulnerability categories to provide a comparative overview. The goal is to help organizations identify and address vulnerabilities in their web applications.
This document summarizes quantum cryptography and the BB84 protocol. It discusses how quantum cryptography can securely transmit encryption keys using quantum properties like photon polarization. The BB84 protocol, developed in 1984 by Bennett and Brassard, was the first quantum cryptographic protocol and is based on generating secret keys for encryption via single photon polarization or entanglement. It ensures detection of eavesdropping during key distribution through disturbance of quantum data from listening.
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 document summarizes a research paper that evaluates the performance of the Dynamic Source Routing (DSR) routing protocol for Variable Bit Rate (VBR) multimedia traffic in mobile ad hoc networks (MANETs) using the Network Simulator 2 (NS-2). The paper conducts simulations of DSR under different scenarios by varying the terrain size, connection rate, and data send rate. It compares the performance of DSR and an enhanced version of DSR based on packet delivery ratio, average end-to-end delay, and optimal path length. The results show that the enhanced DSR performs better in terms of packet delivery but worse in terms of delay and path length compared to the original DSR protocol.
Este documento presenta el proyecto TIC de un centro educativo. El proyecto tiene como objetivos mejorar el aprendizaje de los estudiantes utilizando la tecnología, desarrollar habilidades del pensamiento y valores. También busca capacitar a los profesores para que incorporen las TIC en la enseñanza. El proyecto incluye líneas de acción como alfabetización digital, biblioteca digital, formación de profesores y evaluación del impacto de las TIC en los estudiantes y profesores.
El documento presenta un resumen de la Ley 1014 de fomento a la cultura del emprendimiento en Colombia. La ley establece principios como la formación integral, el trabajo en equipo y el reconocimiento de la responsabilidad. Su objetivo es promover el espíritu emprendedor en el sistema educativo y crear un marco jurídico e interinstitucional para fomentar la cultura empresarial. También busca vincular el sistema educativo con el productivo y fortalecer procesos empresariales que contribuyan al desarrollo del país.
Shadi k waza'if by muhammad ahsan raza qadri ghaza liHaadi11
The document discusses the benefits of meditation for reducing stress and anxiety. Regular meditation practice can help calm the mind and body by lowering heart rate and blood pressure. Studies have shown that meditating for just 10-20 minutes per day can have significant positive impacts on both mental and physical health over time.
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APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...
Am26242246
1. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
Analytic Solution Of Burger’s Equations By Variational Iteration
Method
P. R. Mistry* and V. H. Pradhan**
*
(Department of Applied Mathematics & Humanities, S.V.N.I.T., Surat-395007, India)
**
(Departments of Applied Mathematics & Humanities, S.V.N.I.T., Surat-395007, India)
Abstract
By means of variational iteration method the 2 VARIATIONAL ITERATION METHOD:
solutions of (1+1), (1+2) and (1+3) dimensional Consider the following differential equation:
Burger equations are exactly obtained. In this Lu Nu h ( x, t ) (2.1)
paper, He's variational iteration method is where L is a linear operator, N a nonlinear
introduced to overcome the difficulty arising in
calculating Adomian polynomials. operator, and an h ( x, t ) is the source
inhomogeneous term. The VIM was proposed by
Key words: Burger’s equation, Nonlinear time “He”, where a correctional functional for equation
dependent partial differential equations, Variational (1.4.1) can be written as
iteration method and Lagrange multiplier. un 1 (t ) un (t )
t
, n 0 (2.2)
1 Introduction
We often come with non linear partial
( Lu ( ) Nu ( ) h ( )) d
0
n
n
differential equations obtained through
Where is a general Lagrange multiplier
mathematical models of scientific phenomena.
which can be identified optimally via the variational
There are some methods to obtain approximate theory. The subscript n indicates the n th
solution of this kind of equation. Some of them are approximation and
u n is considered as a restricted
numerical methods, homotopy analysis, Exp-
variation, i.e. un 0 . It is clear that the
function method, and linearization of the equation
successive approximations un , n 0 can be
[1, 6, 7]. In 1999, the variational iteration method
established by determining , so we first
was developed by mathematician “He”. This method
determine the Lagrange multiplier that will be
is used for solving linear and non linear differential identified optimally via integration by parts. The
equations. The method introduces a reliable and successive approximation un ( x, t ), n 0 of the
efficient process for a wide variety of scientific and solution u( x, t ) will be readily obtained using the
engineering applications [1, 2, 4, 5,]. It is based on Lagrange multiplier obtained and by using any
Lagrange multiplier and it has the merits of selective function u 0 . The initial values u( x,0)
simplicity and easy execution. This method avoids and ut ( x,0) are usually used for selecting the
linearization of the problem. zeroth approximation u 0 . With determined, then
In this paper exact solution of (1+1), (1+2) several approximation un ( x, t ), n 0 , follow
and (1+3) dimensional Burger equation [3] has been immediately. Consequently the exact solution may
obtained by variational iteration method. The be obtained by using
mentioned problem has been solved by u lim un (2.3)
n
N.Taghizadeh etc. [3] by Homotopy Perturbation
Method and Reduced Differential Transformation 3. APPLICATION OF VIM FOR
BURGER’S EQUATION:
Method here in the present paper we have solved the 3.1 (1+1)-Dimensional Burgers equation
same problem by variational iteration mehod.. u u 2u
u 2 0 (3.1.1)
t x x
242 | P a g e
2. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
u4 {{{{x(1 t t 2 2 t 3 3 t 4 4
I.C.: u x,0 x (3.1.2)
13t 5 5 2t 6 6 29t 7 7 71t 8 8
Now following the variational iteration 15 3 63 252
method given in the above section we get the 86t
9 9
22t
10 10
5t
11 11
t12 12
following functional
567 315 189 126
un 1 ( x, t ) un ( x, t ) t
13 13
t
14 14
t
15 15
)}}}} (3.1.9)
un 2u 567 3969 59535
u
t
(3.1.3)
0 t
un n
x
2n d
x
u5 {{{{{x(1 t t 2 2 t 3 3 t 4 4
43t 6 6 13t 7 7 943t 8 8
t 5 5
Stationary conditions can be obtained as follows: 45 15 1260
3497t 9 9 27523t10 10 1477t11 11
' 0
5670 56700 4050
(3.1.4)
1 17779t 12 12
13141t 13 13
1019t14 14
t
68040 73710 8820
The Lagrange multiplier can therefore be 63283t 15 15
43363t 16 16
1080013t17 17
simply identified as 1 , and substituting this
893025 1058400 48580560
value of Lagrange multiplier into the functional
(3.1.3) gives the following iteration equation. 2588t 18 18
162179t 19 19
16511t 20 20
229635 30541455 7144200
un 1 ( x, t ) un ( x, t ) 207509t 21 21
557t 22 22
2447t 23 23
t
un u 2 un (3.1.5) 225042300 1666980 22504230
t un xn
0
2
x
d
16927t 24 24 5309t 25 25 t 26 26
540101520 675126900 595350
As stated before, we can select Initial 2t 27 27 13t 28 28 t 29 29
condition given in the equation (3.1.2) and using
this selection in (3.1.5) we obtain the following 6751269 315059220 236294415
successive approximations: t 30 30 t 31 31
)}}}}}(3.1.10)
3544416225 109876902975
u1 {x tx } (3.1.6)
1 u x, t u1 u2 u3 .... (3.1.11)
u2 {{x (tx t 2 x t 3 x 2 )}} (3.1.7)
3 x
u x, t (3.1.12)
1
u3 {{{x tx t 2 x 2 t 3 x 3 1 t
3
3.2 (2+1)-Dimensional Burger’s equation
1 3 2
(tx t 2 x t x )
3 u u u
(3.1.8)
2 4 3 u u
(tx t x t x t x
2 3 2
t x y
3 (3.2.1)
2u 2u
1 1 1
t 5 x 4 t 6 x 5 t 7 x 6 )}}} 2 2 0
3 9 63 x y
I.C.: u x, y,0 x y (3.2.2)
Now following the variational iteration method, we
get the following functional
243 | P a g e
3. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
un 1 ( x, t ) un ( x, t ) u4 {{{{x y 2tx
un u u 2ty 4t 2 x 2
un n un n
t
t x y (3.2.3) 4t 2 y 2 8t 3 x 3
2u 2u d
8t 3 y 3 16t 4 x 4
0
2n 2n
x y
16t 4 y 4
416 5 5
t x
15
Stationary conditions can be obtained as follows: 416 5 5 128 6 6
t y t x
' 0
15 3
128 6 6 3712 7 7
1
(3.2.4)
t y t x
t 3 63
3712 7 7 4544 8 8
The Lagrange multiplier can therefore be t y t x
simply identified as 1 , and substituting this 63 63
value of Lagrange multiplier into the functional 4544 8 8 44032 9 9
(3.2.3) gives the following iteration equation. t y t x
63 567
un 1 ( x, t ) un ( x, t ) 44032 9 9 22528 10 10
t y t x
567 315
un u u
un n un n
22528 10 10 10240 11 11
t y t x
t x y
t
(3.2.5)
2u 2u d
315 189
10240 11 11 2048 12 12
0
2n 2n t y t x
x y
189 63
2048 12 12 8192 13 13
As stated before, we can select Initial t y t x
condition given in the equation (3.2.2) and using 63 567
this selection in (3.2.5) we obtain the following
8192 13 13 16384t14 x 14
successive approximations: t y
567 3969
u1 {x y 2t ( x y ) } (3.2.6) 16384t y
14 14
32768t15 x 15
u2 {{x y 2tx 2ty 3969 59535
8 32768t y
15 15
4t 2 x 2 4t 2 y 2 t 3 x 3 (3.2.7) }}}} (3.2.9)
3 59535
8
t 3 y 3}}
3
u3 {{{x y 2tx 2ty
4t 2 x 2 4t 2 y 2 8t 3 x 3 8t 3 y 3
32 32 32
t 4 x 4 t 4 y 4 t 5 x 5 (3.2.8)
3 3 3
32 64 64
t 5 y 5 t 6 x 6 t 6 y 6
3 9 9
128 7 7 128 7 7
t x t y }}}
63 63
244 | P a g e
4. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
u5 {{{{{x y 2tx 2ty un 1 ( x, t ) un ( x, t )
4t 2 x 2 4t 2 y 2 8t 3 x 3 un u u u
un n un n un n
8t y 16t x 16t y
3 3 4 4 4 4 t
t x y z
(3.3.3)
2752 6 6 2u 2u 2u d
32t 5 x 5 32t 5 y 5 t x 0
2 2 2
n n n
45 x y z
2752 6 6 1664 7 7
t y t x
45 15 Stationary conditions can be obtained as follows:
1664 7 7 60352 8 8
t y t x ' 0
15 315 (3.3.4)
60352 8 8 895232t 9 x 9 1 t
t y
315 2835
The Lagrange multiplier can therefore be
895232t y 9 9
7045888t10 x 10 simply identified as 1 , and substituting this
2835 14175 value of Lagrange multiplier into the functional
(3.3.3) gives the following iteration equation.
7045888t y 10 10
1512448t11 x 11
14175 2025 un 1 ( x, t ) un ( x, t )
1512448t y 11 11
9102848t12 x 12 un
u u u
2025 8505 un n un n un n
t x y z
t
(3.3.5)
9102848t y 53825536t13 x 13 2u 2u 2u d
12 12
8505 36855 0
2 2 2
n n n
x y z
53825536t y 13 13
4173824t14 x 14
36855 2205 As stated before, we can select Initial condition
4173824t y 14 14
2073657344t15 x 15 given in the equation (3.3.2) and using this selection
in (3.3.5) we obtain the following successive
2205 893025 approximations:
2073657344t y 15 15
893025 u1 {x y z 3t ( x y z ) } (3.3.6)
88807424t16 x 16 u2 {{x y z 3tx 3ty
.......}}}}} (3.2.10)
33075 3tz 9t 2 x 2 9t 2 y 2
(3.3.7)
9t 2 z 2 9t 3 x 3
u x, t u1 u2 u3 .... (3.2.11)
x y 9t 3 y 3 9t 3 z 3}}
u x, y , t (3.2.12) u3 {{{x y z 3tx 3ty
1 2 t
3tz 9t 2 x 2 9t 2 y 2
3.3 (3+1)-Dimensional Burger’s equation
u u u u 9t 2 z 2 27t 3 x 3 27t 3 y 3
u u u
t x y z 27t 3 z 3 54t 4 x 4 54t 4 y 4
(3.3.1)
2u 2u 2u 54t 4 z 4 81t 5 x 5 81t 5 y 5 (3.3.8)
2 2 2 0
x y z 81t z 81t x 81t y
5 5 6 6 6 6
I.C.: u x, y, z,0 x yz (3.3.2) 81t 6 z 6
243 7 7 243 7 7
t x t y
Now following the variational iteration method, we 7 7
get the following functional 243 7 7
t z }}}
7
245 | P a g e
5. P. R. Mistry, V. H. Pradhan / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.242-246
u4 {{{{x y z 3tx 3ty iteration method is a powerful mathematical tool to
solving Burger’s equations. In our work, we use the
3tz 9t 2 x 2 9t 2 y 2 9t 2 z 2 Mathematica Package to calculate the series
obtained from the variational iteration method.
27t 3 x 3 27t 3 y 3 27t 3 z 3
81t 4 x 4 81t 4 y 4 81t 4 z 4 References
1. Sadighi A and Ganji D, Exact solution of
1053 5 5 1053 5 5
t x t y nonlinear diffusion equations by variational
5 5 iteration method, Commuters and
Mathematics with Applications, 54, (2007),
1053 5 5
t z 486t 6 x 6 pp. 1112-1121.
5 2. He.J.H. Variational iteration method -- a
1053 5 5 kind of non-linear analytical technique:
t z 486t 6 x 6 some examples, Internat. J. Nonlinear
5 Mech. 34, (1999).,pp. 699–708
486t 6 y 6 486t 6 z 6 3. N.Taghizadeh, M.Akbari and
A.Ghelichzadeh, Exact Solution of Burgers
7047 7 7 7047 7 7
t x t y Equations by Homotopy Perturbation
7 7 Method and Reduced Differential
Transformation Method, Australian Journal
7047 7 7 51759 8 8
t z t x of Basic and Applied Sciences, 5(5):
7 28 (2011),pp. 580-589,
...........}}}} (3.3.9) 4. Salehpoor E. and Jafari H., Variational
iteration method: A tools for solving partial
u x, y, z, t u1 u2 u3 ... (3.3.10) differential equations, The journal of
Mathematics and Computer Science, 2,
x yz
u x, y , z , t (3.3.11) (2011)pp. 388-393.
1 3 t 5. Wazwaz A. M.., The variational iteration
method: A Powerful scheme for handling
3.4 (n+1)-Dimensional Burger’s equation linear and nonlinear diffusion equations.
Computer and Mathematics with
u u u u u Applications 54, (2007) pp.933-939.
u u u ...... u 6. M.A. Abdou and A.A. Soliman,
t x1 x2 x3 xn Variational iteration method for solving
Burger’s and coupled Burger’s equations
2u 2u 2u 2u Journal of Computational and Applied
2 2 2 .... 0 Mathematics 181 (2005), pp. 245 – 251.
x1 x2 x3 xn 2
7. Junfeng Lu, Variational iteration method
(3.4.1) for solving a nonlinear system of second-
u x1 , x2 , x3 ,.........., xn , 0 order boundary value problems, Computers
I.C.: (3.4.2) and Mathematics with Applications 54
x1 x2 ...... xn (2007), pp.1133–1138.
Similarly, we apply VIM on the equation (3.4.1) we
get the following exact solution.
u x1 , x2 , x3 ,.............., xn , t
x1 x2 x3 x4 ............ xn (3.4.3)
1 n t
4. Conclusion
In this paper, the variational iteration
method has been successfully applied to finding the
solution of (1+1), (1+2) and (1+3) dimensional
Burger’s equations. The solution obtained by the
variational iteration method is an infinite power
series for appropriate initial condition, which can, in
turn, be expressed in a closed form, the exact
solution. The results show that the variational
246 | P a g e