Advanced Numerical and Semi-Analytical Methods for Differential Equations

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-analytically.

Autorentext

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.

Inhalt
Acknowledgments xi

Preface xiii

1 Basic Numerical Methods 1

1.1 Introduction 1

1.2 Ordinary Differential Equation 2

1.3 Euler Method 2

1.4 Improved Euler Method 5

1.5 RungeKutta Methods 7

1.5.1 Midpoint Method 7

1.5.2 RungeKutta Fourth Order 8

1.6 Multistep Methods 10

1.6.1 AdamsBashforth Method 10

1.6.2 AdamsMoulton Method 10

1.7 Higher-Order ODE 13

References 16

2 Integral Transforms 19

2.1 Introduction 19

2.2 Laplace Transform 19

2.2.1 Solution of Differential Equations Using Laplace Transforms 20

2.3 Fourier Transform 25

2.3.1 Solution of Partial Differential Equations Using Fourier Transforms 26

References 28

3 Weighted Residual Methods 31

3.1 Introduction 31

3.2 Collocation Method 33

3.3 Subdomain Method 35

3.4 Least-square Method 37

3.5 Galerkin Method 39

3.6 Comparison of WRMs 40

References 42

4 Boundary Characteristics Orthogonal Polynomials 45

4.1 Introduction 45

4.2 GramSchmidt Orthogonalization Process 45

4.3 Generation of BCOPs 46

4.4 Galerkin's Method with BCOPs 46

4.5 RayleighRitz Method with BCOPs 48

References 51

5 Finite Difference Method 53

5.1 Introduction 53

5.2 Finite Difference Schemes 53

5.2.1 Finite Difference Schemes for Ordinary Differential Equations 54

5.2.1.1 Forward Difference Scheme 54

5.2.1.2 Backward Difference Scheme 55

5.2.1.3 Central Difference Scheme 55

5.2.2 Finite Difference Schemes for Partial Differential Equations 55

5.3 Explicit and Implicit Finite Difference Schemes 55

5.3.1 Explicit Finite Difference Method 56

5.3.2 Implicit Finite Difference Method 57

References 61

6 Finite Element Method 63

6.1 Introduction 63

6.2 Finite Element Procedure 63

6.3 Galerkin Finite Element Method 65

6.3.1 Ordinary Differential Equation 65

6.3.2 Partial Differential Equation 71

6.4 Structural Analysis Using FEM 76

6.4.1 Static Analysis 76

6.4.2 Dynamic Analysis 78

References 79

7 Finite Volume Method 81

7.1 Introduction 81

7.2 Discretization Techniques of FVM 82

7.3 General Form of Finite Volume Method 82

7.3.1 Solution Process Algorithm 83

7.4 One-Dimensional ConvectionDiffusion Problem 84

7.4.1 Grid Generation 84

7.4.2 Solution Procedure of ConvectionDiffusion Problem 84

References 89

8 Boundary Element Method 91

8.1 Introduction 91

8.2 Boundary Representation and Background Theory of BEM 91

8.2.1 Linear Differential Operator 92

8.2.2 The Fundamental Solution 93

8.2.2.1 Heaviside Function 93

8.2.2.2 Dirac Delta Function 93

8.2.2.3 Finding the Fundamental Solution 94

8.2.3 Green's Function 95

8.2.3.1 Green's Integral Formula 95

8.3 Derivation of the Boundary Element Method 96

8.3.1 BEM Algorithm 96

References 100

9 AkbariGanji's Method 103

9.1 Introduction 103

9.2 Nonlinear Ordinary Differential Equations 104

9.2.1 Preliminaries 104

9.2.2 AGM Approach 104

9.3 Numerical Examples 105

9.3.1 Unforced Nonlinear Differential Equations 105

9.3.2 Forced Nonlinear Differential Equation 107

References 109 10 Exp-Function Method **11...