Advanced Numerical and Semi-Analytical Methods for Differential Equations.

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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- Chapter 10 Exp‐Function Method -- 10.1 Introduction -- 10.2 Basics of Exp‐Function Method -- 10.3 Numerical Examples -- References -- 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 -- References -- Chapter 12 Homotopy Perturbation Method -- 12.1 Introduction -- 12.2 Basic Idea of HPM -- 12.3 Numerical Examples -- References -- Chapter 13 Variational Iteration Method -- 13.1 Introduction -- 13.2 VIM Procedure -- 13.3 Numerical Examples -- References -- 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 -- References -- Chapter 16 Wavelet Method -- 16.1 Introduction -- 16.2 Haar Wavelet -- 16.3 Wavelet-Collocation Method -- References -- Chapter 17 Hybrid Methods -- 17.1 Introduction -- 17.2 Homotopy Perturbation Transform Method -- 17.3 Laplace Adomian Decomposition Method -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- Index -- EULA.

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