Advanced Numerical and Semi-Analytical Methods for Differential Equations

Name: Advanced Numerical and Semi-Analytical Methods for Differential Equations
Brand: Wiley
Price: 95.99 EUR
Availability: OnlineOnly

Snehashish Chakraverty Nisha Mahato Perumandla Karunakar Tharasi Dilleswar Rao(Autor*in)

Wiley (Verlag)

1. Auflage

Erschienen am 20. März 2019

256 Seiten

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Beschreibung

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.

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Inhalt

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 a-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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