Advanced Numerical and Semi-Analytical Methods for Differential Equations

Snehashish Chakraverty, Nisha Mahato, Perumandla Karunakar, Tharasi Dilleswar Rao

• Publisher: Wiley
• Year: 2019
• ISBN:
  9781119423430
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

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.

1
Basic Numerical Methods

1.1 Introduction

Differentialequations form the backbone of various science and engineering problems viz. structural mechanics, image processing, control theory, stationary analysis of circuits, etc. Generally, engineering problems are modeled in terms of mathematical functions or using relationships between the function and its derivatives. For instance, in structural mechanics the governing equation of motion

(1.1)

associated with Figure 1.1 is expressed in the form of differential equation with respect to the rate of change in time.

Figure 1.1 Mechanical system.

Here m, c, and k are mass, damping, stiffness parameters, respectively, and f(t) is the external force applied on the mechanical system.

There exist various techniques for solving simple differential equations analytically. Modeling of differential equations to compute exact solutions may be found in Refs. [1–3]. But, due to complexity of problems in n...

Table of contents

• Cover
• Table of Contents
• Acknowledgments
• Preface
• 1 Basic Numerical Methods
• 2 Integral Transforms
• 3 Weighted Residual Methods
• 4 Boundary Characteristics Orthogonal Polynomials
• 5 Finite Difference Method
• 6 Finite Element Method
• 7 Finite Volume Method
• 8 Boundary Element Method
• 9 Akbari–Ganji's Method
• 10 Exp‐Function Method
• 11 Adomian Decomposition Method
• 12 Homotopy Perturbation Method
• 13 Variational Iteration Method
• 14 Homotopy Analysis Method
• 15 Differential Quadrature Method
• 16 Wavelet Method
• 17 Hybrid Methods
• 18 Preliminaries of Fractal Differential Equations
• 19 Differential Equations with Interval Uncertainty
• 20 Differential Equations with Fuzzy Uncertainty
• 21 Interval Finite Element Method
• Index
• End User License Agreement