Advanced Numerical and Semi-Analytical Methods for Differential Equations

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.

• Acknowledgments xiPreface xiii1 Basic Numerical Methods 11.1 Introduction 11.2 Ordinary Differential Equation 21.3 Euler Method 21.4 Improved Euler Method 51.5 Runge–Kutta Methods 71.5.1 Midpoint Method 71.5.2 Runge–Kutta Fourth Order 81.6 Multistep Methods 101.6.1 Adams–Bashforth Method 101.6.2 Adams–Moulton Method 101.7 Higher-Order ODE 13References 162 Integral Transforms 192.1 Introduction 192.2 Laplace Transform 192.2.1 Solution of Differential Equations Using Laplace Transforms 202.3 Fourier Transform 252.3.1 Solution of Partial Differential Equations Using Fourier Transforms 26References 283 Weighted Residual Methods 313.1 Introduction 313.2 Collocation Method 333.3 Subdomain Method 353.4 Least-square Method 373.5 Galerkin Method 393.6 Comparison of WRMs 40References 424 Boundary Characteristics Orthogonal Polynomials 454.1 Introduction 454.2 Gram–Schmidt Orthogonalization Process 454.3 Generation of BCOPs 464.4 Galerkin’s Method with BCOPs 464.5 Rayleigh–Ritz Method with BCOPs 48References 515 Finite Difference Method 535.1 Introduction 535.2 Finite Difference Schemes 535.2.1 Finite Difference Schemes for Ordinary Differential Equations 545.2.1.1 Forward Difference Scheme 545.2.1.2 Backward Difference Scheme 555.2.1.3 Central Difference Scheme 555.2.2 Finite Difference Schemes for Partial Differential Equations 555.3 Explicit and Implicit Finite Difference Schemes 555.3.1 Explicit Finite Difference Method 565.3.2 Implicit Finite Difference Method 57References 616 Finite Element Method 636.1 Introduction 636.2 Finite Element Procedure 636.3 Galerkin Finite Element Method 656.3.1 Ordinary Differential Equation 656.3.2 Partial Differential Equation 716.4 Structural Analysis Using FEM 766.4.1 Static Analysis 766.4.2 Dynamic Analysis 78References 797 Finite Volume Method 817.1 Introduction 817.2 Discretization Techniques of FVM 827.3 General Form of Finite Volume Method 827.3.1 Solution Process Algorithm 837.4 One-Dimensional Convection–Diffusion Problem 847.4.1 Grid Generation 847.4.2 Solution Procedure of Convection–Diffusion Problem 84References 898 Boundary Element Method 918.1 Introduction 918.2 Boundary Representation and Background Theory of BEM 918.2.1 Linear Differential Operator 928.2.2 The Fundamental Solution 938.2.2.1 Heaviside Function 938.2.2.2 Dirac Delta Function 938.2.2.3 Finding the Fundamental Solution 948.2.3 Green’s Function 958.2.3.1 Green’s Integral Formula 958.3 Derivation of the Boundary Element Method 968.3.1 BEM Algorithm 96References 1009 Akbari–Ganji’s Method 1039.1 Introduction 1039.2 Nonlinear Ordinary Differential Equations 1049.2.1 Preliminaries 1049.2.2 AGM Approach 1049.3 Numerical Examples 1059.3.1 Unforced Nonlinear Differential Equations 1059.3.2 Forced Nonlinear Differential Equation 107References 10910 Exp-Function Method 11110.1 Introduction 11110.2 Basics of Exp-Function Method 11110.3 Numerical Examples 112References 11711 Adomian Decomposition Method 11911.1 Introduction 11911.2 ADM for ODEs 11911.3 Solving System of ODEs by ADM 12311.4 ADM for Solving Partial Differential Equations 12511.5 ADM for System of PDEs 127References 13012 Homotopy Perturbation Method 13112.1 Introduction 13112.2 Basic Idea of HPM 13112.3 Numerical Examples 133References 13813 Variational Iteration Method 14113.1 Introduction 14113.2 VIM Procedure 14113.3 Numerical Examples 142References 14614 Homotopy Analysis Method 14914.1 Introduction 14914.2 HAM Procedure 14914.3 Numerical Examples 151References 15615 Differential Quadrature Method 15715.1 Introduction 15715.2 DQM Procedure 15715.3 Numerical Examples 159References 16516 Wavelet Method 16716.1 Introduction 16716.2 HaarWavelet 16816.3 Wavelet–Collocation Method 170References 17517 Hybrid Methods 17717.1 Introduction 17717.2 Homotopy Perturbation Transform Method 17717.3 Laplace Adomian Decomposition Method 182References 18618 Preliminaries of Fractal Differential Equations 18918.1 Introduction to Fractal 18918.1.1 Triadic Koch Curve 19018.1.2 Sierpinski Gasket 19018.2 Fractal Differential Equations 19118.2.1 Heat Equation 19218.2.2 Wave Equation 194References 19419 Differential Equations with Interval Uncertainty 19719.1 Introduction 19719.2 Interval Differential Equations 19719.2.1 Interval Arithmetic 19819.3 Generalized Hukuhara Differentiability of IDEs 19819.3.1 Modeling IDEs by Hukuhara Differentiability 19919.3.1.1 Solving by Integral Form 19919.3.1.2 Solving by Differential Form 19919.4 Analytical Methods for IDEs 20119.4.1 General form of nth-order IDEs 20219.4.2 Method Based on Addition and Subtraction of Intervals 202References 20620 Differential Equations with Fuzzy Uncertainty 20920.1 Introduction 20920.2 Solving Fuzzy Linear System of Differential Equations 20920.2.1 𝛼-Cut of TFN 20920.2.2 Fuzzy Linear System of Differential Equations (FLSDEs) 21020.2.3 Solution Procedure for FLSDE 211References 21521 Interval Finite Element Method 21721.1 Introduction 21721.1.1 Preliminaries 21821.1.1.1 Proper and Improper Interval 21821.1.1.2 Interval System of Linear Equations 21821.1.1.3 Generalized Interval Eigenvalue Problem 21921.2 Interval Galerkin FEM 21921.3 Structural Analysis Using IFEM 22321.3.1 Static Analysis 22321.3.2 Dynamic Analysis 225References 227Index 231