Advanced Numerical and Semi-Analytical Methods for Differential Equations

Advanced Numerical and Semi-Analytical Methods for Differential Equations

by Snehashish Chakraverty, Nisha Mahato, Perumandla Karunakar, Tharasi Dilleswar Rao
Released April 2019
Publisher(s): Wiley
ISBN: 9781119423423

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Book description

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.

Table of contents

  1. Cover
  2. Acknowledgments
  3. Preface
  4. 1 Basic Numerical Methods
    1. 1.1 Introduction
    2. 1.2 Ordinary Differential Equation
    3. 1.3 Euler Method
    4. 1.4 Improved Euler Method
    5. 1.5 Runge–Kutta Methods
    6. 1.6 Multistep Methods
    7. 1.7 Higher‐Order ODE
    8. References
  5. 2 Integral Transforms
    1. 2.1 Introduction
    2. 2.2 Laplace Transform
    3. 2.3 Fourier Transform
    4. References
  6. 3 Weighted Residual Methods
    1. 3.1 Introduction
    2. 3.2 Collocation Method
    3. 3.3 Subdomain Method
    4. 3.4 Least‐square Method
    5. 3.5 Galerkin Method
    6. 3.6 Comparison of WRMs
    7. Exercise
    8. References
  7. 4 Boundary Characteristics Orthogonal Polynomials
    1. 4.1 Introduction
    2. 4.2 Gram–Schmidt Orthogonalization Process
    3. 4.3 Generation of BCOPs
    4. 4.4 Galerkin's Method with BCOPs
    5. 4.5 Rayleigh–Ritz Method with BCOPs
    6. Exercise
    7. References
  8. 5 Finite Difference Method
    1. 5.1 Introduction
    2. 5.2 Finite Difference Schemes
    3. 5.3 Explicit and Implicit Finite Difference Schemes
    4. References
  9. 6 Finite Element Method
    1. 6.1 Introduction
    2. 6.2 Finite Element Procedure
    3. 6.3 Galerkin Finite Element Method
    4. 6.4 Structural Analysis Using FEM
    5. References
  10. 7 Finite Volume Method
    1. 7.1 Introduction
    2. 7.2 Discretization Techniques of FVM
    3. 7.3 General Form of Finite Volume Method
    4. 7.4 One‐Dimensional Convection–Diffusion Problem
    5. References
  11. 8 Boundary Element Method
    1. 8.1 Introduction
    2. 8.2 Boundary Representation and Background Theory of BEM
    3. 8.3 Derivation of the Boundary Element Method
    4. References
  12. 9 Akbari–Ganji's Method
    1. 9.1 Introduction
    2. 9.2 Nonlinear Ordinary Differential Equations
    3. 9.3 Numerical Examples
    4. References
  13. 10 Exp‐Function Method
    1. 10.1 Introduction
    2. 10.2 Basics of Exp‐Function Method
    3. 10.3 Numerical Examples
    4. References
  14. 11 Adomian Decomposition Method
    1. 11.1 Introduction
    2. 11.2 ADM for ODEs
    3. 11.3 Solving System of ODEs by ADM
    4. 11.4 ADM for Solving Partial Differential Equations
    5. 11.5 ADM for System of PDEs
    6. Exercise
    7. References
  15. 12 Homotopy Perturbation Method
    1. 12.1 Introduction
    2. 12.2 Basic Idea of HPM
    3. 12.3 Numerical Examples
    4. References
  16. 13 Variational Iteration Method
    1. 13.1 Introduction
    2. 13.2 VIM Procedure
    3. 13.3 Numerical Examples
    4. References
  17. 14 Homotopy Analysis Method
    1. 14.1 Introduction
    2. 14.2 HAM Procedure
    3. 14.3 Numerical Examples
    4. Exercise
    5. References
  18. 15 Differential Quadrature Method
    1. 15.1 Introduction
    2. 15.2 DQM Procedure
    3. 15.3 Numerical Examples
    4. References
  19. 16 Wavelet Method
    1. 16.1 Introduction
    2. 16.2 Haar Wavelet
    3. 16.3 Wavelet–Collocation Method
    4. Exercise
    5. References
  20. 17 Hybrid Methods
    1. 17.1 Introduction
    2. 17.2 Homotopy Perturbation Transform Method
    3. 17.3 Laplace Adomian Decomposition Method
    4. References
  21. 18 Preliminaries of Fractal Differential Equations
    1. 18.1 Introduction to Fractal
    2. 18.2 Fractal Differential Equations
    3. References
  22. 19 Differential Equations with Interval Uncertainty
    1. 19.1 Introduction
    2. 19.2 Interval Differential Equations
    3. 19.3 Generalized Hukuhara Differentiability of IDEs
    4. 19.4 Analytical Methods for IDEs
    5. References
  23. 20 Differential Equations with Fuzzy Uncertainty
    1. 20.1 Introduction
    2. 20.2 Solving Fuzzy Linear System of Differential Equations
    3. References
  24. 21 Interval Finite Element Method
    1. 21.1 Introduction
    2. 21.2 Interval Galerkin FEM
    3. 21.3 Structural Analysis Using IFEM
    4. References
  25. Index
  26. End User License Agreement

Product information

  • Title: Advanced Numerical and Semi-Analytical Methods for Differential Equations
  • Author(s): Snehashish Chakraverty, Nisha Mahato, Perumandla Karunakar, Tharasi Dilleswar Rao
  • Release date: April 2019
  • Publisher(s): Wiley
  • ISBN: 9781119423423