Advanced numerical and semi-analytical methods for differential equations / Snehashish Chakraverty [and three others].

Format:

Book

Author/Creator:

Chakraverty, Snehashish, author.

Language:

English

Subjects (All):

Differential equations.

Physical Description:

1 online resource (253 pages)

Edition:

1st edition

Place of Publication:

Hoboken, New Jersey : Wiley, 2019.

System Details:

text file

Summary:

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 methods Examines differential equations in an uncertain environment Includes a new scenario in which uncertainty (in term of intervals and fuzzy numbers) has been included in differential equations Contains solved example problems, as well as some unsolved problems for self-validation of the topics covered Advanced Numerical and Semi Analytical Methods for Differential Equations is an excellent text for graduate as well as post graduate students and researchers studying various methods for solving differential equations, numerically and semi-analyti...

Contents:

Cover

Title Page

Copyright

Contents

Acknowledgments

Preface

Chapter 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

1.5.1 Midpoint Method

1.5.2 Runge-Kutta Fourth Order

1.6 Multistep Methods

1.6.1 Adams-Bashforth Method

1.6.2 Adams-Moulton Method

1.7 Higher‐Order ODE

References

Chapter 2 Integral Transforms

2.1 Introduction

2.2 Laplace Transform

2.2.1 Solution of Differential Equations Using Laplace Transforms

2.3 Fourier Transform

2.3.1 Solution of Partial Differential Equations Using Fourier Transforms

Chapter 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

Chapter 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 BCOPs

Chapter 5 Finite Difference Method

5.1 Introduction

5.2 Finite Difference Schemes

5.2.1 Finite Difference Schemes for Ordinary Differential Equations

5.2.1.1 Forward Difference Scheme

5.2.1.2 Backward Difference Scheme

5.2.1.3 Central Difference Scheme

5.2.2 Finite Difference Schemes for Partial Differential Equations

5.3 Explicit and Implicit Finite Difference Schemes

5.3.1 Explicit Finite Difference Method

5.3.2 Implicit Finite Difference Method

Chapter 6 Finite Element Method

6.1 Introduction

6.2 Finite Element Procedure

6.3 Galerkin Finite Element Method

6.3.1 Ordinary Differential Equation

6.3.2 Partial Differential Equation

6.4 Structural Analysis Using FEM.

6.4.1 Static Analysis

6.4.2 Dynamic Analysis

Chapter 7 Finite Volume Method

7.1 Introduction

7.2 Discretization Techniques of FVM

7.3 General Form of Finite Volume Method

7.3.1 Solution Process Algorithm

7.4 One‐Dimensional Convection-Diffusion Problem

7.4.1 Grid Generation

7.4.2 Solution Procedure of Convection-Diffusion Problem

Chapter 8 Boundary Element Method

8.1 Introduction

8.2 Boundary Representation and Background Theory of BEM

8.2.1 Linear Differential Operator

8.2.2 The Fundamental Solution

8.2.2.1 Heaviside Function

8.2.2.2 Dirac Delta Function

8.2.2.3 Finding the Fundamental Solution

8.2.3 Green's Function

8.2.3.1 Green's Integral Formula

8.3 Derivation of the Boundary Element Method

8.3.1 BEM Algorithm

Chapter 9 Akbari-Ganji's Method

9.1 Introduction

9.2 Nonlinear Ordinary Differential Equations

9.2.1 Preliminaries

9.2.2 AGM Approach

9.3 Numerical Examples

9.3.1 Unforced Nonlinear Differential Equations

9.3.2 Forced Nonlinear Differential Equation

Chapter 10 Exp‐Function Method

10.1 Introduction

10.2 Basics of Exp‐Function Method

10.3 Numerical Examples

Chapter 11 Adomian Decomposition Method

11.1 Introduction

11.2 ADM for ODEs

11.3 Solving System of ODEs by ADM

11.4 ADM for Solving Partial Differential Equations

11.5 ADM for System of PDEs

Chapter 12 Homotopy Perturbation Method

12.1 Introduction

12.2 Basic Idea of HPM

12.3 Numerical Examples

Chapter 13 Variational Iteration Method

13.1 Introduction

13.2 VIM Procedure

13.3 Numerical Examples

Chapter 14 Homotopy Analysis Method

14.1 Introduction

14.2 HAM Procedure

14.3 Numerical Examples

References.

Chapter 15 Differential Quadrature Method

15.1 Introduction

15.2 DQM Procedure

15.3 Numerical Examples

Chapter 16 Wavelet Method

16.1 Introduction

16.2 Haar Wavelet

16.3 Wavelet-Collocation Method

Chapter 17 Hybrid Methods

17.1 Introduction

17.2 Homotopy Perturbation Transform Method

17.3 Laplace Adomian Decomposition Method

Chapter 18 Preliminaries of Fractal Differential Equations

18.1 Introduction to Fractal

18.1.1 Triadic Koch Curve

18.1.2 Sierpinski Gasket

18.2 Fractal Differential Equations

18.2.1 Heat Equation

18.2.2 Wave Equation

Chapter 19 Differential Equations with Interval Uncertainty

19.1 Introduction

19.2 Interval Differential Equations

19.2.1 Interval Arithmetic

19.3 Generalized Hukuhara Differentiability of IDEs

19.3.1 Modeling IDEs by Hukuhara Differentiability

19.3.1.1 Solving by Integral Form

19.3.1.2 Solving by Differential Form

19.4 Analytical Methods for IDEs

19.4.1 General form of nth‐order IDEs

19.4.2 Method Based on Addition and Subtraction of Intervals

Chapter 20 Differential Equations with Fuzzy Uncertainty

20.1 Introduction

20.2 Solving Fuzzy Linear System of Differential Equations

20.2.1 α‐Cut of TFN

20.2.2 Fuzzy Linear System of Differential Equations (FLSDEs)

20.2.3 Solution Procedure for FLSDE

Chapter 21 Interval Finite Element Method

21.1 Introduction

21.1.1 Preliminaries

21.1.1.1 Proper and Improper Interval

21.1.1.2 Interval System of Linear Equations

21.1.1.3 Generalized Interval Eigenvalue Problem

21.2 Interval Galerkin FEM

21.3 Structural Analysis Using IFEM

21.3.1 Static Analysis

21.3.2 Dynamic Analysis

Index

EULA.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781119423461

9781119423430

1119423430

9781119423447

1119423449

OCLC:

1090728073