This document presents a modification of the Adomian decomposition method to solve nonlinear partial differential equations. The modification involves using Padé approximation and Laplace transforms on the series solution obtained from the Adomian decomposition method. This is done to obtain a closed-form solution. The document provides details on the Adomian decomposition method, Padé approximation, and the proposed modified algorithm. It then applies the modified algorithm to some illustrative examples, including solving a differential equation modeling oscillatory behavior and obtaining the closed-form solution.
This document discusses regularization methods for solving inverse problems. It begins with an introduction to inverse problems, explaining that they are typically ill-posed and lack stability. Regularization is introduced as a way to approximate an ill-posed inverse problem with a family of nearby well-posed problems. Two main categories of regularization methods are described: classical regularization methods, which include singular value decomposition, Tikhonov regularization, and truncated iterative methods; and local regularization methods, designed for Volterra integral equations of the first kind. The document provides mathematical definitions and explanations of these various regularization techniques.
Numerical Solution of Third Order Time-Invariant Linear Differential Equation...theijes
In this paper, we state the Adomian Decomposition Method (ADM) for third order time-invariant linear homogeneous differential equations. And we applied it to find solutions to the same class of equations. Three test problems were used as concrete examples to validate the reliability of the method, and the result shows remarkable solutions as those that are obtained by any knows analytical method(s).
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.
He laplace method for special nonlinear partial differential equationsAlexander Decker
This document presents the He-Laplace method for solving nonlinear partial differential equations. The method combines Laplace transforms, homotopy perturbation method, and He's polynomials. It is shown that the He-Laplace method can easily handle nonlinear terms through the use of He's polynomials and provides better results than traditional methods. An example demonstrates the application of the He-Laplace method to solve a nonlinear parabolic-hyperbolic partial differential equation.
This document describes an experimental evaluation of combinatorial preconditioners for solving linear systems. It compares Vaidya's algorithm for constructing combinatorial preconditioners to newer algorithms presented by Spielman, including a low-stretch spanning tree constructor and tree augmentation approach. The algorithms were implemented in Java and experimentally evaluated using a test framework on various matrices. The main results found that the new augmentation algorithm did not consistently outperform Vaidya's algorithm, though it did sometimes have significantly better performance. Using low-stretch trees as a basis for augmentation provided a consistent but modest improvement over Vaidya.
This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
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.
This document discusses regularization methods for solving inverse problems. It begins with an introduction to inverse problems, explaining that they are typically ill-posed and lack stability. Regularization is introduced as a way to approximate an ill-posed inverse problem with a family of nearby well-posed problems. Two main categories of regularization methods are described: classical regularization methods, which include singular value decomposition, Tikhonov regularization, and truncated iterative methods; and local regularization methods, designed for Volterra integral equations of the first kind. The document provides mathematical definitions and explanations of these various regularization techniques.
Numerical Solution of Third Order Time-Invariant Linear Differential Equation...theijes
In this paper, we state the Adomian Decomposition Method (ADM) for third order time-invariant linear homogeneous differential equations. And we applied it to find solutions to the same class of equations. Three test problems were used as concrete examples to validate the reliability of the method, and the result shows remarkable solutions as those that are obtained by any knows analytical method(s).
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.
He laplace method for special nonlinear partial differential equationsAlexander Decker
This document presents the He-Laplace method for solving nonlinear partial differential equations. The method combines Laplace transforms, homotopy perturbation method, and He's polynomials. It is shown that the He-Laplace method can easily handle nonlinear terms through the use of He's polynomials and provides better results than traditional methods. An example demonstrates the application of the He-Laplace method to solve a nonlinear parabolic-hyperbolic partial differential equation.
This document describes an experimental evaluation of combinatorial preconditioners for solving linear systems. It compares Vaidya's algorithm for constructing combinatorial preconditioners to newer algorithms presented by Spielman, including a low-stretch spanning tree constructor and tree augmentation approach. The algorithms were implemented in Java and experimentally evaluated using a test framework on various matrices. The main results found that the new augmentation algorithm did not consistently outperform Vaidya's algorithm, though it did sometimes have significantly better performance. Using low-stretch trees as a basis for augmentation provided a consistent but modest improvement over Vaidya.
This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
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.
The Engineer of Industrial Universtiy of Santander, Elkin Santafe, give us a little summary about direct methods for the solution of systems of equations
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
This document provides an introduction and outline for a discussion of orthonormal bases and eigenvectors. It begins with an overview of orthonormal bases, including definitions of the dot product, norm, orthogonal vectors and subspaces, and orthogonal complements. It also discusses the relationship between the null space and row space of a matrix. The document then provides an introduction to eigenvectors and outlines topics that will be covered, including what eigenvectors are useful for and how to find and use them.
The document describes the (G'/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations (PDEs) arising in mathematical physics. The method involves expressing the solution as a polynomial in (G'/G), where G satisfies a second order linear ordinary differential equation. The method is demonstrated by using it to find traveling wave solutions of the variable coefficients KdV (vcKdV) equation, the modified dispersive water wave (MDWW) equations, and the symmetrically coupled KdV equations. These solutions are expressed in terms of hyperbolic, trigonometric, and rational functions. The method provides a simple way to obtain exact solutions to important nonlinear PDEs.
The document discusses square matrices and determinants. It begins by noting that square matrices are the only matrices that can have inverses. It then presents an algorithm for calculating the inverse of a square matrix A by forming the partitioned matrix (A|I) and applying Gauss-Jordan reduction. The document also discusses determinants, defining them recursively as the sum of products of diagonal entries with signs depending on row/column position, for matrices larger than 1x1. Complexity increases exponentially with matrix size.
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.
The document discusses linear transformations and their applications in mathematics for artificial intelligence. It begins by introducing linear transformations and how matrices can be used to define functions. It describes how a matrix A can define a linear transformation fA that maps vectors in Rn to vectors in Rm. It also defines key concepts for linear transformations like the kernel, range, row space, and column space. The document will continue exploring topics like the derivative of transformations, linear regression, principal component analysis, and singular value decomposition.
A Generalized Sampling Theorem Over Galois Field Domains for Experimental Des...csandit
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEY
This document provides an outline and introduction to a course on mathematics for artificial intelligence, with a focus on vector spaces and linear algebra. It discusses:
1. A brief history of linear algebra, from ancient Babylonians solving systems of equations to modern definitions of matrices.
2. The definition of a vector space as a set that can be added and multiplied by elements of a field, with properties like closure under addition and scalar multiplication.
3. Examples of using matrices and vectors to model systems of linear equations and probabilities of transitions between web pages.
4. The importance of linear algebra concepts like bases, dimensions, and eigenvectors/eigenvalues for machine learning applications involving feature vectors and least squares error.
This document discusses stability estimates for identifying point sources using measurements from the Helmholtz equation. It begins by introducing the inverse problem of determining source terms from boundary measurements of the electromagnetic field. It then proves identifiability, showing that a single boundary measurement can uniquely determine the sources. Next, it establishes a local Lipschitz stability result, demonstrating continuous dependence of the sources on the measurements. Finally, it proposes a numerical inversion algorithm using a reciprocity gap concept to quickly and stably identify multiple sources.
International Journal of Engineering Research and DevelopmentIJERD Editor
This document presents a wavelet-Galerkin technique for solving Neumann boundary value problems using wavelet bases. It begins with an introduction to wavelets and their history. It then discusses the wavelet-Galerkin method, describing how to represent functions and their derivatives as expansions in a wavelet basis. The technique is applied to solve one-dimensional Neumann Helmholtz and two-dimensional Neumann Poisson boundary value problems. Results show the method matches exact solutions for some parameters in Helmholtz problems but not all parameters in Poisson problems.
Lagrange's method solves constrained optimization problems by forming an augmented function that combines the objective function and constraints, using Lagrange multipliers (λ) as weighting factors. The method finds extrema by taking partial derivatives of the augmented function with respect to the objective variables and λ, setting the results equal to zero. This produces a system of equations that can be solved simultaneously to identify values that satisfy the constraint and optimize the original objective function.
The document discusses the dimension of vector spaces. It defines the dimension of a vector space V as the number of vectors in a basis for V. It states that any set containing more vectors than the dimension must be linearly dependent, and that every basis of V must consist of exactly the dimension number of vectors. Examples of finding the dimension of various vector spaces and subspaces are provided.
This document discusses numerical methods for solving nonlinear equations. It begins by stating that nonlinear equations cannot generally be solved analytically and must instead be approached using iterative methods. It then covers the general principles of iterative methods, including convergence conditions, order of convergence, and stop criteria. The document focuses on several common iterative methods, including first-order methods like the chord, classic chord, and regula falsi methods, as well as the second-order Newton-Raphson method. It also briefly discusses solving systems of nonlinear equations.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIIRai University
This document discusses methods for finding the roots or zeros of equations, including the bisection method, Newton-Raphson method, and regula-falsi method. It provides definitions and steps for each method. The bisection method works by repeatedly bisecting the interval that contains the root. Newton-Raphson uses successive approximations to iteratively find better estimates for the root. Regula-falsi is based on finding the x-intercept of the chord between two points on the function graph. Examples are provided to demonstrate applying each method to find the roots of equations.
This document presents a stability theorem for large solutions of the three-dimensional incompressible magnetohydrodynamic equations. The theorem proves that if one solution decays to zero and another solution is initially close, then the perturbed solution will remain close. The proof uses Sobolev spaces and interpolation inequalities. As consequences of the theorem, the stability result can be extended to cases where the domain is all of R3 or satisfies certain conditions. Further work could look at extending results on regularity and stability of fluid flows to the MHD case using different function spaces like Besov spaces.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
Las TIC se refieren al conjunto de recursos necesarios para manipular, convertir, almacenar, administrar, transmitir e encontrar información, incluyendo ordenadores, programas, redes y servicios. Las NTI comprenden tecnologías físicas como ordenadores y telecomunicaciones, así como sistemas operativos y programas, mientras que las NTIC han llegado a influir en todas las actividades sociales y económicas, especialmente la educación.
Este documento presenta un juego de preguntas y respuestas múltiples para que los clientes potenciales ayuden a crear y planificar su boda ideal de una manera divertida. El juego implica elegir opciones para la ubicación de la proposición, el número de invitados, el menú, la ropa de los novios, la luna de miel y otros detalles. Al final, los usuarios pueden votar por sus creaciones favoritas y las más populares recibirán descuentos en los servicios de organización de bodas.
Los medios de comunicación han evolucionado desde formas primitivas hasta medios tecnológicos modernos, y continuarán evolucionando en el futuro. Han pasado de instrumentos iniciales a formas más avanzadas como la radio, la televisión e Internet, permitiendo una comunicación más rápida y a gran escala.
The Engineer of Industrial Universtiy of Santander, Elkin Santafe, give us a little summary about direct methods for the solution of systems of equations
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
This document provides an introduction and outline for a discussion of orthonormal bases and eigenvectors. It begins with an overview of orthonormal bases, including definitions of the dot product, norm, orthogonal vectors and subspaces, and orthogonal complements. It also discusses the relationship between the null space and row space of a matrix. The document then provides an introduction to eigenvectors and outlines topics that will be covered, including what eigenvectors are useful for and how to find and use them.
The document describes the (G'/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations (PDEs) arising in mathematical physics. The method involves expressing the solution as a polynomial in (G'/G), where G satisfies a second order linear ordinary differential equation. The method is demonstrated by using it to find traveling wave solutions of the variable coefficients KdV (vcKdV) equation, the modified dispersive water wave (MDWW) equations, and the symmetrically coupled KdV equations. These solutions are expressed in terms of hyperbolic, trigonometric, and rational functions. The method provides a simple way to obtain exact solutions to important nonlinear PDEs.
The document discusses square matrices and determinants. It begins by noting that square matrices are the only matrices that can have inverses. It then presents an algorithm for calculating the inverse of a square matrix A by forming the partitioned matrix (A|I) and applying Gauss-Jordan reduction. The document also discusses determinants, defining them recursively as the sum of products of diagonal entries with signs depending on row/column position, for matrices larger than 1x1. Complexity increases exponentially with matrix size.
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.
The document discusses linear transformations and their applications in mathematics for artificial intelligence. It begins by introducing linear transformations and how matrices can be used to define functions. It describes how a matrix A can define a linear transformation fA that maps vectors in Rn to vectors in Rm. It also defines key concepts for linear transformations like the kernel, range, row space, and column space. The document will continue exploring topics like the derivative of transformations, linear regression, principal component analysis, and singular value decomposition.
A Generalized Sampling Theorem Over Galois Field Domains for Experimental Des...csandit
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEY
This document provides an outline and introduction to a course on mathematics for artificial intelligence, with a focus on vector spaces and linear algebra. It discusses:
1. A brief history of linear algebra, from ancient Babylonians solving systems of equations to modern definitions of matrices.
2. The definition of a vector space as a set that can be added and multiplied by elements of a field, with properties like closure under addition and scalar multiplication.
3. Examples of using matrices and vectors to model systems of linear equations and probabilities of transitions between web pages.
4. The importance of linear algebra concepts like bases, dimensions, and eigenvectors/eigenvalues for machine learning applications involving feature vectors and least squares error.
This document discusses stability estimates for identifying point sources using measurements from the Helmholtz equation. It begins by introducing the inverse problem of determining source terms from boundary measurements of the electromagnetic field. It then proves identifiability, showing that a single boundary measurement can uniquely determine the sources. Next, it establishes a local Lipschitz stability result, demonstrating continuous dependence of the sources on the measurements. Finally, it proposes a numerical inversion algorithm using a reciprocity gap concept to quickly and stably identify multiple sources.
International Journal of Engineering Research and DevelopmentIJERD Editor
This document presents a wavelet-Galerkin technique for solving Neumann boundary value problems using wavelet bases. It begins with an introduction to wavelets and their history. It then discusses the wavelet-Galerkin method, describing how to represent functions and their derivatives as expansions in a wavelet basis. The technique is applied to solve one-dimensional Neumann Helmholtz and two-dimensional Neumann Poisson boundary value problems. Results show the method matches exact solutions for some parameters in Helmholtz problems but not all parameters in Poisson problems.
Lagrange's method solves constrained optimization problems by forming an augmented function that combines the objective function and constraints, using Lagrange multipliers (λ) as weighting factors. The method finds extrema by taking partial derivatives of the augmented function with respect to the objective variables and λ, setting the results equal to zero. This produces a system of equations that can be solved simultaneously to identify values that satisfy the constraint and optimize the original objective function.
The document discusses the dimension of vector spaces. It defines the dimension of a vector space V as the number of vectors in a basis for V. It states that any set containing more vectors than the dimension must be linearly dependent, and that every basis of V must consist of exactly the dimension number of vectors. Examples of finding the dimension of various vector spaces and subspaces are provided.
This document discusses numerical methods for solving nonlinear equations. It begins by stating that nonlinear equations cannot generally be solved analytically and must instead be approached using iterative methods. It then covers the general principles of iterative methods, including convergence conditions, order of convergence, and stop criteria. The document focuses on several common iterative methods, including first-order methods like the chord, classic chord, and regula falsi methods, as well as the second-order Newton-Raphson method. It also briefly discusses solving systems of nonlinear equations.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIIRai University
This document discusses methods for finding the roots or zeros of equations, including the bisection method, Newton-Raphson method, and regula-falsi method. It provides definitions and steps for each method. The bisection method works by repeatedly bisecting the interval that contains the root. Newton-Raphson uses successive approximations to iteratively find better estimates for the root. Regula-falsi is based on finding the x-intercept of the chord between two points on the function graph. Examples are provided to demonstrate applying each method to find the roots of equations.
This document presents a stability theorem for large solutions of the three-dimensional incompressible magnetohydrodynamic equations. The theorem proves that if one solution decays to zero and another solution is initially close, then the perturbed solution will remain close. The proof uses Sobolev spaces and interpolation inequalities. As consequences of the theorem, the stability result can be extended to cases where the domain is all of R3 or satisfies certain conditions. Further work could look at extending results on regularity and stability of fluid flows to the MHD case using different function spaces like Besov spaces.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
Las TIC se refieren al conjunto de recursos necesarios para manipular, convertir, almacenar, administrar, transmitir e encontrar información, incluyendo ordenadores, programas, redes y servicios. Las NTI comprenden tecnologías físicas como ordenadores y telecomunicaciones, así como sistemas operativos y programas, mientras que las NTIC han llegado a influir en todas las actividades sociales y económicas, especialmente la educación.
Este documento presenta un juego de preguntas y respuestas múltiples para que los clientes potenciales ayuden a crear y planificar su boda ideal de una manera divertida. El juego implica elegir opciones para la ubicación de la proposición, el número de invitados, el menú, la ropa de los novios, la luna de miel y otros detalles. Al final, los usuarios pueden votar por sus creaciones favoritas y las más populares recibirán descuentos en los servicios de organización de bodas.
Los medios de comunicación han evolucionado desde formas primitivas hasta medios tecnológicos modernos, y continuarán evolucionando en el futuro. Han pasado de instrumentos iniciales a formas más avanzadas como la radio, la televisión e Internet, permitiendo una comunicación más rápida y a gran escala.
Este documento habla sobre los estilos de aprendizaje y enseñanza. Explica que los estilos educativos se aprenden a través de la interacción con otros y que existen dos clases principales de estilos: cognitivos y afectivos. También menciona algunos tipos de estilos cognitivos comúnmente estudiados como el control flexible, nivelamiento-agudización, y dependencia-independencia. El documento concluye explicando que es posible la auto y heteroevaluación de los estilos de aprendizaje de acuerdo a investigaciones c
El Salvador es un pequeño país centroamericano ubicado entre Guatemala y Honduras. Tiene un clima tropical cálido y una economía que depende en gran medida de la agricultura. La cultura salvadoreña es una mezcla de influencias indígenas y españolas, y la gastronomía nacional se centra en platos como las pupusas y tamales. El Salvador también tiene una rica historia literaria y artística.
Los medios de comunicación han evolucionado desde formas primitivas hasta medios tecnológicos modernos, y continuarán evolucionando en el futuro a medida que la tecnología avance. Los primeros medios incluían formas simples de transmitir información, mientras que hoy existen muchas más opciones tecnológicas como resultado de la constante evolución de los medios a lo largo de la historia.
Este documento describe un sistema de gestión centralizado para plantas de agua que permite a las compañías recopilar y analizar información en tiempo real sobre el estado de sus plantas para reducir costos y mejorar la calidad del servicio. El sistema integra datos automáticos y manuales de las plantas en una plataforma central que monitorea parámetros como la calidad del agua y el consumo energético para generar alertas y reportes. Esto mejora la reacción ante fallas, optimiza el mantenimiento y la operación de las plantas, y ayuda a cu
An Optimization Model for A Proposed Trigeneration System IJERA Editor
This document proposes an optimization model for a trigeneration system at a university campus. The trigeneration system would provide electricity, heating, and cooling and use both renewable and non-renewable energy sources. The proposed system includes a power generation unit, waste heat recovery system, absorption chiller, electrical chiller, photovoltaic systems, and thermal collector. The optimization model is formulated as a linear program to minimize total system costs over a one-year period under constraints of meeting energy demands and not exceeding the capacities of system components. The objective is to determine the optimal operation of the trigeneration system components to satisfy the university's electricity, heating, and cooling loads at lowest cost.
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Analytical and Exact solutions of a certain class of coupled nonlinear PDEs using Adomian-Pad´e method
1. Khadijah M. Abualnaja. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 12, ( Part -1) December 2016, pp.01-10
www.ijera.com 1 | P a g e
Analytical and Exact solutions of a certain class of coupled nonlinear
PDEs using Adomian-Pad´e method
Khadijah M. Abualnaja
Department of Mathematics and Statistics, College of Science, Taif University, Taif, KSA
ABSTRACT
The purpose of this study is to introduce a modification of the Adomian decomposition method using
Pad´e approximation and Laplace transform to obtain a closed form of the solutions of nonlinear
partial differential equations. Several test examples are given; illustrative examples and the
coupled nonlinear system of Burger’s equations. The obtained results ensure that this modification
is capable for solving the nonlinear PDEs that have wide application in physics and engineering.
Keywords: Pad´e approximation; Laplace transform; Adomian decomposition method; nonlinear
Burger’s equations.
I. INTRODUCTION
In this paper, the Adomian
decomposition method (ADM) [3] is used to
solve nonlinear partial differential equations
(NPDEs). It is based on the search for a
solution in the form of a series and on
decomposing the nonlinear operator into a
series in which the terms are calculated
recursively using the Adomian’s polynomials
([4], [7]). In recent years, a growing interest
towards the applications of this method in
nonlinear problems has been devoted by
engineering practice, see ([1], [2], [6], [9],
[11], [12]). This method has many advantages
such as, fast convergence, doesn’t require
discretizations of space-time variables, which
gives rise to rounding off errors, and doesn’t
require solving the resultant nonlinear system
of discrete equations. Unlike, the traditional
methods, the computation in this method
doesn’t require a large computer memory.
Also, we are going to improve the accuracy of
ADM by using the Pad´e approximation and
apply the improved method to solve some
physical models of NPDEs, namely, illustrative
examples and the system of Burger’s
equations.
II. ANALYSIS OF THE ADOMIAN DECOMPOSITION METHOD
In this section, the basic idea of ADM to solve NPDEs is introduced as follows: Consider the
following nonlinear partial differential equation
(x, t),N (u) = fu(x, t) +xtA (1)
with suitable initial and boundary conditions. Where xtA is the partial differential operator and N(u)
is the non-linear term.
Let xt+ Rx+ Lt= LxtA where tL is the highest partial derivative with respect to t (assume
of order n), Lx is the highest partial derivative with respect to x and xtR is the remainder from the
operator, therefore, (1) will take the following operator form
Lt u(x,t) = f(x,t) −(Lx + Rxt)u(x,t) −N(u). (2)
By applying the inverse operator on both sides of (2), we get
u(x,t) = φ(x,t) + 1
t
L [f(x,t) −(Lx + Rxt)u(x,t) −N(u)] (3)
where the inverse operator is defined by:
t
dtdt
t
foldnt
L
0
,...(.)
0
......
(.)
1
and φ(x,t) is the solution of the homogeneous differential equation Lt u(x,t) = 0 with the same initial
conditions of (1).
RESEARCH ARTICLE OPEN ACCESS
2. Khadijah M. Abualnaja. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 12, ( Part -1) December 2016, pp.01-10
www.ijera.com 2 | P a g e
t
Now, ADM suggests that the solution can be expressed as an infinite series of the following form
,
n
(x,t)
n
uu(x,t)
0
(4)
and the nonlinear term expanded in terms of Adomian’s polynomials
n
A
0
,
n
AN(u)
n
(5)
where these polynomials are defined by the relation
., n
λ
]
n
i
i
u
i
λN(
n
d λ
n
d
[
n!
n
A 0
0
0
1
(6)
Substituting from (4) and (5) in (3), we obtain
],
0
-
0
)
xt
R+x(L-t)(x,f[
1
+t)(x,=
0
n
n
A
n
(x,t)
n
u
t
L
n
(x,t)
n
u
equating the similar terms in both sides of the above equation, we get the following recurrence
relation
t)](x,f[
1
t
L
+t)(x,=t)(x,u 0
.n],
n
A(x,t)
n
)uxt+ Rx(L(x,t) =nu 1
11
(7)
Now, we present some basic definitions of the Taylor series method, Pad´e approximation ([5], [13]),
needed in the next sections of the paper.
III. THE PADÉ APPROXIMANTS ON THE SERIES SOLUTION
The general setup in approximation theory is that a function f is given and that one wants to
approximate it with a simpler function g but in such a way that the difference between f and g is small. The
advantage is that the simpler function g can be handled without too many difficulties, but the disadvantage is
that one loses some information since f and g are different.
Definition 1.
When we obtain the truncated series solution of order at least L+M in t by ADM, we will use it to obtain
(x,t)
M
L
PA
Padé approximation for the solutionu(x,t) . The Padé approximation ([4], [25]) are a particular
type of rational fraction approximation to the value of the function. The idea is to match the Taylor series
expansion as far as possible.
We denote the
M
L
PA to
0i
i
x
i
aR(x) by:
,PA
(x)
M
Q
(x)
L
P
(x)
M
L
(8)
where (x)
L
P is a polynomial of degree at most L and (x)
M
Q is a polynomial of degree at most M.
.
M
x
M
q...
2
x
2
qx
1
q1(x)
M
Q,
L
x
L
p...
2
x
2
px
1
p
0
p(x)
L
P (9)
To determine the coefficients of (x)
L
P and (x)
M
Q , we may multiply (8) by (x)
M
Q , which linearizes the
coefficient equations. We can write out (8) in more detail as:
3. Khadijah M. Abualnaja. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 12, ( Part -1) December 2016, pp.01-10
www.ijera.com 3 | P a g e
0,
M
q
L
a...
1
q
1-ML
a
ML
a
...
0,
M
q
2M-L
a...
1
q
1L
a
2L
a
0,
M
q
1M-L
a...
1
q
L
a
1L
a
(10)
.
L
p
L
q
0
a...
1
q
1-L
a
L
a
....
,
2
p
2
q
0
a
1
q
1
a
2
a
,
1
p
1
q
0
a
1
a
,
0
p
0
a
(11)
To solve these equations, we start with
Eq.(10), which is a set of linear equations for all the
unknown q's. Once the q's are known, then Eq.(11)
gives an explicit formula for the unknown p's,
which complete the solution. Each choice of L,
degree of the numerator and M, degree of the
denominator, leads to an approximant. The major
difficulty in applying the technique is how to direct
the choice in order to obtain the best approximant.
This needs the use of a criterion for the choice
depending on the shape of the solution. A criterion
which has worked well here is the choice of
M
L
approximation such that L=M. We construct the
approximation by built-in utilities of Mathematica
in the following sections. If ADM truncated Taylor
series of the exact solution with enough terms and
the solution can be written as the ratio of two
polynomials with no common factors, then the
Padé approximation for the truncated series provide
the exact solution. Even when the exact solution
cannot be expressed as the ratio of two
polynomials, the Padé approximation for the ADM
truncated series usually greatly improve the
accuracy and enlarge the convergence domain of
the solutions.
IV. THE MODIFIED ALGORITHM
OF ADM
In spite of the advantages of ADM, it
has some drawbacks. By using ADM, we get
a series, in practice a truncated series
solution. The series often coincides with the
Taylor expansion of the true solution at point
x = 0, in the initial value case. Although the
series can be rapidly convergent in a very
small region, it has very slow convergent rate
in the wider region we examine and
the truncated series solution is an inaccurate
solution in that region, which will greatly
restrict the application area of the method.
All the truncated series solutions have the
same problem. Many examples given can be
used to support this assertion [12].
Pad´e approximation [5] approximates
any function by a ratio of two polynomials.
The coefficients of the powers occurring in
the polynomials are determined by the
coefficients in the Taylor expansion.
Generally, the Pade approximation can enlarge
the convergence domain of the truncated
Taylor series and can improve greatly the
convergent rate of the truncated Taylor series
[10].
The suggested modification of ADM can be
done by using the following algorithm.
Algorithm
Step 1. Solve the differential equation using
the standard ADM;
Step 2. Truncate the obtained series solution
by ADM;
Step 3. Take the Laplace transform of the
truncated series;
Step 4. Find the Pad´e approximation of the
previous step;
Step 5. Take the inverse Laplace transform.
This modification often gets the closed form of
the exact solution of the differential equation.
Now, we implement this algorithm to some
examples of linear and nonlinear differential
equations to illustrate our modification.
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V. ILLUSTRATIVE EXAMPLES
Example 1:
Consider the following differential equation
,u
dt
ud
0
2
2
(12)
subject to the initial conditions 1000 ) =(u,) =u( . We can rewrite (12) in an operator form as
follows
,) =u(t) + u(ttL 0 (13)
where the differential operator tL is
2
2
dt
d
tL .
By applying the inverse operator on (13) and using the initial conditions, we can derive
u(t),
t
L)(t) + tuu(t) = u(
1
00
(14)
where the operator
1
t
L is an integral operator and given by
t t
dtdt
t
L
0 0
.(.)(.)
1
The ADM [3] assumes that the unknown solution can be expressed by an infinite series of the
following form
.
0n
(t)
n
uu(t)
(15)
Substituting from (15), into (14) and equating the similar terms in both sides, we get the following
recurrence relation
.n(t),nutL(t) =
n
u),(t)+tu(t)=u(u 0
1
1
00
0
(16)
Now, from the recurrence relationship (16) we obtain the following components of the solution u(t)
t,00)(
0
)(t)+tuu(tu
,
!
t
(t)u
t
L(t)u
3
3
0
1
1
...,
!5
5
)(
1
1
)(
2
t
tu
t
Ltu
.
)!12(
12
)1(
)(
1
)(
1
n
n
t
n
tnu
t
Lt
n
u
By the same procedure we can obtain the components of the solution. Therefore, the approximate
solution can be readily obtained by
.
0
)!12(
12
)1(
)(
n
k
k
k
t
k
(t)ntu (17)
Which is the partial sum of the Taylor series of the solution u(t) at t=0. Figure 1 shows the error
between the exact solution u(t) and (t),
5
from this figure we can find that the error at t ∈ [0, 5] is
nearly to 0, but at t∈ [5,10] the error takes large values.
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Figure 1: The error between the exact solution u(t) and the solution by ADM, .
5
(t)
Now, because (12) is an oscillatory system, here we can apply Laplace transform [8] to (t)
n
which
yields
.
22
1
)1(
...
6
1
4
1
2
1
]£[
n
s
n
sss
(t)
n
For the sake simplicity, let
t
s
1
then:
.
221
)1(...
642
]£[
n
t
n
ttt(t)
n
Its ]
n
n
[
1
1
Pad´e approximations with n>0 yields .
2
1
2
1
1
t
t
]
n
n
[
Replace
s
t
1
, we obtain ]
n
n
[
1
1
in terms of s as follows:
,
2
1
1
1
1
s
]
n
n
[
by using the inverse Laplace transform to ]
n
n
[
1
1
we obtain the true solution of (12):
u(t) = sin(t).
Example 2:
Consider the following nonlinear partial differential equation
tu u+xxuttu 2 = g(x,t) =−2sin2(x)sin(t)cos(t), (18)
with the initial conditions u(x, 0) = sin(x), ut(x,0) = 0.
The exact solution of this equation is u(x t) =sin(x) cos(t).
First, to overcome the complicated excitation from the term g(x,t), which can cause difficult
integrations and proliferation of terms, we use the Taylor expansion of the function g(x,t) at t = 0,
in the following form
...),
7
315
45
15
23
3
2
)((
2
sin2
0
)(),(
ttttx
k
k
tx
k
gtxg
in this case, (18) reduces to the form
.
k
k
(x)t
k
g=tu u+xxuttu
0
2 (19)
We can rewrite (19) in an operator form as follows
,
0
)(
k
k
(x)t
k
g=uu + NxLutL (20)
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where the differential operator
2
2
t
tL
the nonlinear term is tuuN (u) = 2 . By applying the
inverse operator on (20) and using the initial conditions, we can derive
.
0
])([
1
)0,()0,(),(
k
k
(x)t
k
guN
xx
u
t
Lx
t
tuxutxu (21)
The ADM [3] assumes that the unknown solution can be expressed by an infinite series of the form
(4) and the nonlinear operator term can be decomposed by an infinite series of polynomials, given by
(5), where the components 0nt),(x,nu will be determined recurrently and An are the so-called
Adomian’s polynomials of ,...
2
u,
1
u,
0
u defined by (6), these polynomials can be constructed for all
nonlinearity according to algorithm set by Adomian.
Substituting from (4), (5) into (21) and equating the similar terms in both sides of the equation, we
getthefollowingrecurrence relation
u0(x, t) = u(x,0), 0,n],
n
t(x)ng+nA[
1
),(
1
nxx
u
t
Ltx
n
u (22)
where the first An Adomian’s polynomials that represent the nonlinear term N (u) = 2uut
are given by
t
uu=A
00
2
0
,
),
t
u+ u
t
u(u=A
0110
2
1
),
t
u+ u
t
u+ u
t
u(u=A
201102
2
2
), ...
t
u+ u
t
u+ u
t
u+ u
t
u( u=A
21301203
2
3
other polynomials can be generated in a like manner.
Now from the recurrence relationship (22) we obtain the following resulting components
(x),
t
(x,t)u(x),(x,t)u sin
!2
2
1
sin
0
(x),
t
(x,t)u(x),
t
(x,t)u sin
!6
6
3
sin
!4
4
2
.sin
!10
10
5
sin
!8
8
4
(x)
t
(x,t)u(x),
t
(x,t)u
Having n,...,,(x,t), i
i
u ,210 , the solution is as follows
.
n
i
(x,t)
i
u(x,t)
n
u(x,t)
1
0
(23)
In our calculations we use the truncated series (x,t)
4
ˆ to order t8, i.e.,
,sin
8
8
6
6
4
4
2
2
1
4
ˆ (x))
!
t
!
t
!
t
!
t
((x,t) (24)
which coincides with the first five terms of the Taylor series of the solution u(x,t) at t = 0. Here, we
apply Laplace transformation to ),(
4
ˆ tx , which yields
.sin
9
1
7
1
5
1
3
11
]
4
ˆ£[ (x))
sssss
((x,t)
For the sake simplicity, let
t
s
1
then:
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]
4
ˆ£[ (x,t) = (t −t3 + t5 −t7 + t9) sin(x).
All of the ]
M
L
[ Pad´e approximants of the above equation with L > 0, M > 1 and L+M <10 yields
.sin
2
1
(x)
t
t
]
M
L
[
Replace
s
t
1
, we obtain ]
M
L
[ in terms of s as follows:
.sin
2
1
(x)
s
s
]
M
L
[
By using the inverse Laplace transform to ]
M
L
[ we obtain the true solution of (18)
u(x,t) = sin(x)cos(t).
VI. IMPLEMENTATION THE MODIFICATION OF ADM TO SYSTEM OF
BURGERS’ EQUATIONS
Consider the following coupled nonlinear system of Burgers’ equations
ut −uxx −2uux + (uv)x = 0, (25)
vt −vxx −2vvx + (uv)x = 0, (26)
subject to the following initial conditions
u(x,0) = v(x,0) = sin(x). (27)
Now, we will solve the system (25)-(27) by using the above algorithm as follows:
Step 1. Solve the system (25)-(27) by using ADM
In this step, we apply ADM to the system (25)-(27), so, we rewrite it in the following operator form
0,=v)(u,
1
N+xxuu
t
L (28)
0.=v)(u,
2
N+xxvv
t
L (29)
By using the inverse operator, we can write the above system (28)-(29) in the following form
u(x,t) = u(x,0)+ 1
t
L [uxx] − ]
1
[
1
(u, v)N
t
L
, (30)
v(x,t) = v(x,0)+ 1
t
L [vxx] − ]
2
[
1
(u, v)N
t
L
, (31)
where the inverse operator is defined by
t
dt
t
L
0
,(.)(.)
1
and the nonlinear terms (u, v)N
1
and (u, v)N
2
are defined by
x(uv)+x2uu
1
(u, v)N , x(uv)+x2vv
2
(u, v)N . (32)
The ADM suggests that the solutions u(x,t) and v(x,t) be decomposed by an infinite series of
components
,
0
),(),(,
0
),(),(
n
txnvtxv
n
txnutxu (33)
and the nonlinear terms defined in (32) are decomposed by the infinite series:
,
m
m
A(u, v)N,
m
m
A(u, v)N
0
22
0
11
(33)
where uk(x,t) and vk(x,t), k ≥ 0 are the components of u(x,t) and v(x,t) that will be elegantly
determined, Akn are called Adomian’s polynomials and defined by
, . . .,,m, k
λ
)]
i
i
v
i
λ),
i
i
u
i
λ(
k
N
m
d λ
m
d
[
m!
km
A 210,2,1
0
00
1
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From the above considerations, the decomposition method defines the components uk(x,t) and
vk(x,t) for k ≥0, by the following recursive relations
u0(x,t) = u(x,0), un+1(x,t) = 1
t
L [unxx] −
1
t
L [
n
A
1
], (34)
v0(x,t) = v(x,0), vn+1(x,t) = 1
t
L [vnxx]−
1
t
L [
n
A
2
]. (35)
This will enable us to determine the components un(x,t) and vn(x,t) recurrently. However, in many
cases, the exact solution in a closed form may be obtained.
For numerical comparison purpose, we construct the solutions u(x,t) and v(x,t)
,, (x,t)
n
Lim
n
v(x,t)(x,t)
n
Lim
n
u(x,t)
where
1.n,
1
0
,
1
0
n
i
(x,t)
i
v(x,t)
n
n
i
(x,t)
i
u(x,t)
n
(36)
Now, by using the above procedure to the system (25)-(26), we can derive the solution of this system
as follows
u0(x,t) = sin(x), v0(x,t)=sin(x),
u1(x,t) = −tsin(x), v1(x,t) = −tsin(x),
(x).
!
t
(x,t)v(x),
!
t
(x,t)u sin
2
2
2
sin
2
2
2
The rest of components of the iterative formulas (34)-(35) are obtained in the same manner using the
Mathematica package version 5. Where the first Adomian’s polynomials of Ain, i = 1,2 are given by
A10 = 2u0u0x −(u0v0x +v0u0x),
A20 = 2v0v0x −(u0v0x +v0u0x),
A11 = 2(u0u1x +u1u0x)−(u0v1x +u1v0x +v0u1x +v1u0x),
A21 =2(v0v1x +v1v0x)−(u0v1x +u1v0x +v0u1x +v1u0x).
In the same manner, we can compute other components of Ain, i =1,2.
Step 2. Truncate the series solution obtained by ADM
We have applied ADM by using the fourth order approximation only, i.e., the approximate
solutions is
(x),)
!
t
!
t
!
t
t(
n
i
(x,t)
i
uU(x,t)u(x,t) sin
4
4
3
3
2
2
1
1
0
(37)
.sin
4
4
3
3
2
2
1
1
0
(x))
!
t
!
t
!
t
t(
n
i
(x,t)
i
vV(x,t)v(x,t)
(38)
The behavior of the error between the exact solution and the solution obtained by ADM in the
regions 0 ≤ x ≤ 1 and 0 ≤ t ≤ 5 is shown in the figures (2 and 3). The numerical results are
obtained by using fourth order approximation only from the formulas (34)-(35). From these figures,
we achieved a very good approximation for the solution of the system at the small values of time t, but
at the large values of the time t, the error takes large values.
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Figure 2: The error−u = |u(x,t) −U(x,t)|.
Figure 3: The error−v = |v(x,t) −V (x,t)|.
Step 3. Take the Laplace transform of the equations (37)-(38)
sin(x),)
5
s
1
4
s
1
3
s
1
2
s
1
s
1
(=t)](x,£[U (39)
sin(x),)
5
s
1
4
s
1
3
s
1
2
s
1
s
1
(=t)](x,£[V (40)
For the sake simplicity, let
t
s
1
then
sin(x),)
5
t
4
t
3
t
2
t(t=t)](x,£[U (41)
sin(x).)
5
t
4
t
3
t
2
t(t=t)](x,£[V (42)
Step 4. Find the Pad´e approximation of the equations (41)-(42)
All of the ]
M
L
[ Pad´e approximants of the above equation with L > 0, M >0 yields
.sin
1
(x)
t
t
]
M
L
[
Replace
s
t
1
, we obtain ]
M
L
[ in terms of s as follows:
.sin
1
1
(x)
s
]
M
L
[
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Step 5. Take the inverse Laplace transform
By using the inverse Laplace transform to ]
M
L
[ , we obtain the true solution
u(x,t) = e−t sin(x), v(x,t) = e−t sin(x),
which arethe same exact solution of the system (25)-(26).
VII. SUMMARYANDCONCLUSIONS
In this paper, we presented a
modification of ADM, this modification
considerably capable for solving a wide-range
class of linear and nonlinear differential
equations; especially the ones of high
nonlinearity order in engineering and physics
problems. This purpose was satisfied by
solving physical model of nonlinear coupled
system of PDEs. From the obtained results, we
can conclude that the application of Pad´e
approximants to the truncated series
solution (from ADM) greatly improve the
convergence domain and accuracy of the
solution. Also, we can conclude that the
approximate solutions of the presented
problems are excellent agreement with the
exact solution. Finally, we point out that the
corresponding analytical and numerical
solutions are obtained according to the
iteration equations using Mathematica 5.
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