Advanced numerical and semi analytical methods for differential equations / 
Snehashish Chakraverty (National Institute of Technology Rourkela, Odisha, India) [and three others]. 
 - 1 online resource. 
<br/><br/> Includes index.  ABOUT THE AUTHORS<br/>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.<br/><br/>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.<br/><br/>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.<br/><br/>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. 
<br/><br/>TABLE OF CONTENTS<br/>Acknowledgments xi<br/><br/>Preface xiii<br/><br/>1 Basic Numerical Methods 1<br/><br/>1.1 Introduction 1<br/><br/>1.2 Ordinary Differential Equation 2<br/><br/>1.3 Euler Method 2<br/><br/>1.4 Improved Euler Method 5<br/><br/>1.5 Runge–Kutta Methods 7<br/><br/>1.5.1 Midpoint Method 7<br/><br/>1.5.2 Runge–Kutta Fourth Order 8<br/><br/>1.6 Multistep Methods 10<br/><br/>1.6.1 Adams–Bashforth Method 10<br/><br/>1.6.2 Adams–Moulton Method 10<br/><br/>1.7 Higher-Order ODE 13<br/><br/>References 16<br/><br/>2 Integral Transforms 19<br/><br/>2.1 Introduction 19<br/><br/>2.2 Laplace Transform 19<br/><br/>2.2.1 Solution of Differential Equations Using Laplace Transforms 20<br/><br/>2.3 Fourier Transform 25<br/><br/>2.3.1 Solution of Partial Differential Equations Using Fourier Transforms 26<br/><br/>References 28<br/><br/>3 Weighted Residual Methods 31<br/><br/>3.1 Introduction 31<br/><br/>3.2 Collocation Method 33<br/><br/>3.3 Subdomain Method 35<br/><br/>3.4 Least-square Method 37<br/><br/>3.5 Galerkin Method 39<br/><br/>3.6 Comparison of WRMs 40<br/><br/>References 42<br/><br/>4 Boundary Characteristics Orthogonal Polynomials 45<br/><br/>4.1 Introduction 45<br/><br/>4.2 Gram–Schmidt Orthogonalization Process 45<br/><br/>4.3 Generation of BCOPs 46<br/><br/>4.4 Galerkin’s Method with BCOPs 46<br/><br/>4.5 Rayleigh–Ritz Method with BCOPs 48<br/><br/>References 51<br/><br/>5 Finite Difference Method 53<br/><br/>5.1 Introduction 53<br/><br/>5.2 Finite Difference Schemes 53<br/><br/>5.2.1 Finite Difference Schemes for Ordinary Differential Equations 54<br/><br/>5.2.1.1 Forward Difference Scheme 54<br/><br/>5.2.1.2 Backward Difference Scheme 55<br/><br/>5.2.1.3 Central Difference Scheme 55<br/><br/>5.2.2 Finite Difference Schemes for Partial Differential Equations 55<br/><br/>5.3 Explicit and Implicit Finite Difference Schemes 55<br/><br/>5.3.1 Explicit Finite Difference Method 56<br/><br/>5.3.2 Implicit Finite Difference Method 57<br/><br/>References 61<br/><br/>6 Finite Element Method 63<br/><br/>6.1 Introduction 63<br/><br/>6.2 Finite Element Procedure 63<br/><br/>6.3 Galerkin Finite Element Method 65<br/><br/>6.3.1 Ordinary Differential Equation 65<br/><br/>6.3.2 Partial Differential Equation 71<br/><br/>6.4 Structural Analysis Using FEM 76<br/><br/>6.4.1 Static Analysis 76<br/><br/>6.4.2 Dynamic Analysis 78<br/><br/>References 79<br/><br/>7 Finite Volume Method 81<br/><br/>7.1 Introduction 81<br/><br/>7.2 Discretization Techniques of FVM 82<br/><br/>7.3 General Form of Finite Volume Method 82<br/><br/>7.3.1 Solution Process Algorithm 83<br/><br/>7.4 One-Dimensional Convection–Diffusion Problem 84<br/><br/>7.4.1 Grid Generation 84<br/><br/>7.4.2 Solution Procedure of Convection–Diffusion Problem 84<br/><br/>References 89<br/><br/>8 Boundary Element Method 91<br/><br/>8.1 Introduction 91<br/><br/>8.2 Boundary Representation and Background Theory of BEM 91<br/><br/>8.2.1 Linear Differential Operator 92<br/><br/>8.2.2 The Fundamental Solution 93<br/><br/>8.2.2.1 Heaviside Function 93<br/><br/>8.2.2.2 Dirac Delta Function 93<br/><br/>8.2.2.3 Finding the Fundamental Solution 94<br/><br/>8.2.3 Green’s Function 95<br/><br/>8.2.3.1 Green’s Integral Formula 95<br/><br/>8.3 Derivation of the Boundary Element Method 96<br/><br/>8.3.1 BEM Algorithm 96<br/><br/>References 100<br/><br/>9 Akbari–Ganji’s Method 103<br/><br/>9.1 Introduction 103<br/><br/>9.2 Nonlinear Ordinary Differential Equations 104<br/><br/>9.2.1 Preliminaries 104<br/><br/>9.2.2 AGM Approach 104<br/><br/>9.3 Numerical Examples 105<br/><br/>9.3.1 Unforced Nonlinear Differential Equations 105<br/><br/>9.3.2 Forced Nonlinear Differential Equation 107<br/><br/>References 109<br/><br/>10 Exp-Function Method 111<br/><br/>10.1 Introduction 111<br/><br/>10.2 Basics of Exp-Function Method 111<br/><br/>10.3 Numerical Examples 112<br/><br/>References 117<br/><br/>11 Adomian Decomposition Method 119<br/><br/>11.1 Introduction 119<br/><br/>11.2 ADM for ODEs 119<br/><br/>11.3 Solving System of ODEs by ADM 123<br/><br/>11.4 ADM for Solving Partial Differential Equations 125<br/><br/>11.5 ADM for System of PDEs 127<br/><br/>References 130<br/><br/>12 Homotopy Perturbation Method 131<br/><br/>12.1 Introduction 131<br/><br/>12.2 Basic Idea of HPM 131<br/><br/>12.3 Numerical Examples 133<br/><br/>References 138<br/><br/>13 Variational Iteration Method 141<br/><br/>13.1 Introduction 141<br/><br/>13.2 VIM Procedure 141<br/><br/>13.3 Numerical Examples 142<br/><br/>References 146<br/><br/>14 Homotopy Analysis Method 149<br/><br/>14.1 Introduction 149<br/><br/>14.2 HAM Procedure 149<br/><br/>14.3 Numerical Examples 151<br/><br/>References 156<br/><br/>15 Differential Quadrature Method 157<br/><br/>15.1 Introduction 157<br/><br/>15.2 DQM Procedure 157<br/><br/>15.3 Numerical Examples 159<br/><br/>References 165<br/><br/>16 Wavelet Method 167<br/><br/>16.1 Introduction 167<br/><br/>16.2 HaarWavelet 168<br/><br/>16.3 Wavelet–Collocation Method 170<br/><br/>References 175<br/><br/>17 Hybrid Methods 177<br/><br/>17.1 Introduction 177<br/><br/>17.2 Homotopy Perturbation Transform Method 177<br/><br/>17.3 Laplace Adomian Decomposition Method 182<br/><br/>References 186<br/><br/>18 Preliminaries of Fractal Differential Equations 189<br/><br/>18.1 Introduction to Fractal 189<br/><br/>18.1.1 Triadic Koch Curve 190<br/><br/>18.1.2 Sierpinski Gasket 190<br/><br/>18.2 Fractal Differential Equations 191<br/><br/>18.2.1 Heat Equation 192<br/><br/>18.2.2 Wave Equation 194<br/><br/>References 194<br/><br/>19 Differential Equations with Interval Uncertainty 197<br/><br/>19.1 Introduction 197<br/><br/>19.2 Interval Differential Equations 197<br/><br/>19.2.1 Interval Arithmetic 198<br/><br/>19.3 Generalized Hukuhara Differentiability of IDEs 198<br/><br/>19.3.1 Modeling IDEs by Hukuhara Differentiability 199<br/><br/>19.3.1.1 Solving by Integral Form 199<br/><br/>19.3.1.2 Solving by Differential Form 199<br/><br/>19.4 Analytical Methods for IDEs 201<br/><br/>19.4.1 General form of nth-order IDEs 202<br/><br/>19.4.2 Method Based on Addition and Subtraction of Intervals 202<br/><br/>References 206<br/><br/>20 Differential Equations with Fuzzy Uncertainty 209<br/><br/>20.1 Introduction 209<br/><br/>20.2 Solving Fuzzy Linear System of Differential Equations 209<br/><br/>20.2.1 𝛼-Cut of TFN 209<br/><br/>20.2.2 Fuzzy Linear System of Differential Equations (FLSDEs) 210<br/><br/>20.2.3 Solution Procedure for FLSDE 211<br/><br/>References 215<br/><br/>21 Interval Finite Element Method 217<br/><br/>21.1 Introduction 217<br/><br/>21.1.1 Preliminaries 218<br/><br/>21.1.1.1 Proper and Improper Interval 218<br/><br/>21.1.1.2 Interval System of Linear Equations 218<br/><br/>21.1.1.3 Generalized Interval Eigenvalue Problem 219<br/><br/>21.2 Interval Galerkin FEM 219<br/><br/>21.3 Structural Analysis Using IFEM 223<br/><br/>21.3.1 Static Analysis 223<br/><br/>21.3.2 Dynamic Analysis 225<br/><br/>References 227<br/><br/>Index 231 
<br/><br/> Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs<br/><br/>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.<br/><br/>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:<br/><br/>Discusses various methods for solving linear and nonlinear ODEs and PDEs<br/>Covers basic numerical techniques for solving differential equations along with various discretization methods<br/>Investigates nonlinear differential equations using semi-analytical methods<br/>Examines differential equations in an uncertain environment<br/>Includes a new scenario in which uncertainty (in term of intervals and fuzzy numbers) has been included in differential equations<br/>Contains solved example problems, as well as some unsolved problems for self-validation of the topics covered <br/>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. 
<br/><br/><label>ISBN: </label>9781119423423  9781119423447 (Adobe PDF) 9781119423461 9781119423430 (ePub) 
<br/><br/><label>LCCN: </label>2019022130 
<br/><br/><label>Subjects--Topical Terms: </label><br/>Differential equations--Numerical solutions.
<br/><br/><label>Index Terms--Genre/Form: </label><br/>Electronic books.
<br/><br/><label>LC Class. No.: </label>QA372 
<br/><br/><label>Dewey Class. No.: </label>515/.35