Advanced Numerical and Semi-Analytical Methods for Differential Equations

Snehashish Chakraverty , Nisha Mahato , Perumandla Karunakar , Tharasi Dilleswar Rao 지음

Wiley 2019년 03월 20일 출간

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Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs This student-friendly book deals with various approaches for solving differential equations numerically or semi-analytically depending on the type of equations and offers simple example problems to help readers along. Featuring both traditional and recent methods, Advanced Numerical and Semi Analytical Methods for Differential Equations begins with a review of basic numerical methods. It then looks at Laplace, Fourier, and weighted residual methods for solving differential equations. A new challenging method of Boundary Characteristics Orthogonal Polynomials (BCOPs) is introduced next. The book then discusses Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), and Boundary Element Method (BEM). Following that, analytical/semi analytic methods like Akbari Ganji's Method (AGM) and Exp-function are used to solve nonlinear differential equations. Nonlinear differential equations using semi-analytical methods are also addressed, namely Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), and Homotopy Analysis Method (HAM). Other topics covered include: emerging areas of research related to the solution of differential equations based on differential quadrature and wavelet approach; combined and hybrid methods for solving differential equations; as well as an overview of fractal differential equations. Further, uncertainty in term of intervals and fuzzy numbers have also been included, along with the interval finite element method. This book: Discusses various methods for solving linear and nonlinear ODEs and PDEs Covers basic numerical techniques for solving differential equations along with various discretization methods Investigates nonlinear differential equations using semi-analytical method

목차

1 Basic Numerical Methods 1.1 Introduction 1.2 Ordinary differential equation 1.3 Euler method 1.4 Improved Euler method 1.5 Runge-Kutta Methods 1.6 Multistep methods 1.7 Higher order ODE 2 Integral Transforms 2.1 Introduction 2.2 Laplace Transform 2.3 Fourier Tranform 3 Weighted Residual Methods 3.1 Introduction 3.2 Collocation method 3.3 Subdomain method 3.4 Least-square method 3.5 Galerkin method 3.6 Comparison of WRMs 4 Boundary Characteristics Orthogonal Polynomials 4.1 Introduction 4.2 Gram Schmidt Orthogonalization Process 4.3 Generation of BCOPs 4.4 Galerkin’s Method with BCOPs 4.5 Rayleigh-Ritz Method with BCOP’s 5 Finite Difference Method 5.1 Introduction 5.2 Finite Difference Scheme 5.3 Explicit and Implicit Finite Difference Schemes 6 Finite Element Method 6.1 Introduction 6.2 Finite element procedure 6.3 Galerkin finite element method 6.4 Structural analysis using FEM 7 Finite Volume Method 7.1 Introduction 7.2 Discretization Techniques of FVM 7.3 General Form of Finite Volume Method 7.4 One Dimensional Convection-diffusion problem 8 Boundary Element Method 8.1 Introduction 8.2 Boundary Representation and Background Theory of BEM 8.3 Derivation of the Boundary Element Method 9 Akbari Ganji’s Method 9.1 Introduction 9.2 Nonlinear ordinary differential equations 9.3 Numerical examples 10 Exp-function Method 10.1 Introduction 10.2 Basics of Exp-fucntion method 10.3 Numerical examples 11 Adomian Decomposition Method 11.1 Introduction 11.2 ADM for Ordinary Differential Equations (ODEs) 11.3 Solving system of

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저자(글) Snehashish Chakraverty

SNEHASHISH CHAKRAVERTY, PHD, is Professor in the Department of Mathematics at National Institute of Technology, Rourkela, Odisha, India. He is also the author of Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications and 12 other books. NISHA RANI MAHATO is a Senior Research Fellow in the Department of Mathematics at the National Institute of Technology, Rourkela, Odisha, India where she is pursuing her PhD. PERUMANDLA KARUNAKAR is a Senior Research Fellow in the Department of Mathematics at the National Institute of Technology, Rourkela, Odisha, India where he is pursuing his PhD. THARASI DILLESWAR RAO, is a Senior Research Fellow in the Department of Mathematics at the National Institute of Technology, Rourkela, Odisha, India where he is pursuing his PhD.

저자(글) Nisha Mahato

저자(글) Perumandla Karunakar

저자(글) Tharasi Dilleswar Rao