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.

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 Runge–Kutta Methods 7

1.5.1 Midpoint Method 7

1.5.2 Runge–Kutta Fourth Order 8

1.6 Multistep Methods 10

1.6.1 Adams–Bashforth Method 10

1.6.2 Adams–Moulton 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 Gram–Schmidt Orthogonalization Process 45

4.3 Generation of BCOPs 46

4.4 Galerkin’s Method with BCOPs 46

4.5 Rayleigh–Ritz 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 Convection–Diffusion Problem 84

7.4.1 Grid Generation 84

7.4.2 Solution Procedure of Convection–Diffusion 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 Akbari–Ganji’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 111

10.1 Introduction 111

10.2 Basics of Exp-Function Method 111

10.3 Numerical Examples 112

References 117

11 Adomian Decomposition Method 119

11.1 Introduction 119

11.2 ADM for ODEs 119

11.3 Solving System of ODEs by ADM 123

11.4 ADM for Solving Partial Differential Equations 125

11.5 ADM for System of PDEs 127

References 130

12 Homotopy Perturbation Method 131

12.1 Introduction 131

12.2 Basic Idea of HPM 131

12.3 Numerical Examples 133

References 138

13 Variational Iteration Method 141

13.1 Introduction 141

13.2 VIM Procedure 141

13.3 Numerical Examples 142

References 146

14 Homotopy Analysis Method 149

14.1 Introduction 149

14.2 HAM Procedure 149

14.3 Numerical Examples 151

References 156

15 Differential Quadrature Method 157

15.1 Introduction 157

15.2 DQM Procedure 157

15.3 Numerical Examples 159

References 165

16 Wavelet Method 167

16.1 Introduction 167

16.2 HaarWavelet 168

16.3 Wavelet–Collocation Method 170

References 175

17 Hybrid Methods 177

17.1 Introduction 177

17.2 Homotopy Perturbation Transform Method 177

17.3 Laplace Adomian Decomposition Method 182

References 186

18 Preliminaries of Fractal Differential Equations 189

18.1 Introduction to Fractal 189

18.1.1 Triadic Koch Curve 190

18.1.2 Sierpinski Gasket 190

18.2 Fractal Differential Equations 191

18.2.1 Heat Equation 192

18.2.2 Wave Equation 194

References 194

19 Differential Equations with Interval Uncertainty 197

19.1 Introduction 197

19.2 Interval Differential Equations 197

19.2.1 Interval Arithmetic 198

19.3 Generalized Hukuhara Differentiability of IDEs 198

19.3.1 Modeling IDEs by Hukuhara Differentiability 199

19.3.1.1 Solving by Integral Form 199

19.3.1.2 Solving by Differential Form 199

19.4 Analytical Methods for IDEs 201

19.4.1 General form of nth-order IDEs 202

19.4.2 Method Based on Addition and Subtraction of Intervals 202

References 206

20 Differential Equations with Fuzzy Uncertainty 209

20.1 Introduction 209

20.2 Solving Fuzzy Linear System of Differential Equations 209

20.2.1 𝛼-Cut of TFN 209

20.2.2 Fuzzy Linear System of Differential Equations (FLSDEs) 210

20.2.3 Solution Procedure for FLSDE 211

References 215

21 Interval Finite Element Method 217

21.1 Introduction 217

21.1.1 Preliminaries 218

21.1.1.1 Proper and Improper Interval 218

21.1.1.2 Interval System of Linear Equations 218

21.1.1.3 Generalized Interval Eigenvalue Problem 219

21.2 Interval Galerkin FEM 219

21.3 Structural Analysis Using IFEM 223

21.3.1 Static Analysis 223

21.3.2 Dynamic Analysis 225

References 227

Index 231

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.