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E-raamat: Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations [Taylor & Francis e-raamat]

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  • Formaat: 273 pages, 104 Tables, black and white; 8 Illustrations, color; 34 Illustrations, black and white
  • Ilmumisaeg: 31-Jan-2018
  • Kirjastus: CRC Press
  • ISBN-13: 9781315167183
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  • Taylor & Francis e-raamat
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Wavelet Methods for Solving Partial Differential Equations and Fractional Differential EquationsSuurem pilt
  • Formaat: 273 pages, 104 Tables, black and white; 8 Illustrations, color; 34 Illustrations, black and white
  • Ilmumisaeg: 31-Jan-2018
  • Kirjastus: CRC Press
  • ISBN-13: 9781315167183
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9781315167183
The main focus of the book is to implement wavelet based transform methods for solving the problem of fractional order partial differential equations arising in modelling real physical phenomena. This book explores the analytical and numerical approximate solution obtained by wavelet methods for both classical and fractional order partial differential equations.
List of Figures
xi
List of Tables
xv
Preface xxiii
Acknowledgements xxvii
Mathematical Preliminary xxix
1 Numerous Analytical and Numerical Methods
1(22)
1.1 Introduction
1(1)
1.2 Part I: Fundamental Idea of Various Analytical Methods
1(9)
1.2.1 Variational Iteration Method (VIM)
1(1)
1.2.2 First Integral Method
2(1)
1.2.2.1 Algorithm of FIM
3(1)
1.2.3 Homotopy Perturbation Method (HPM)
4(1)
1.2.4 Optimal Homotopy Asymptotic Method (OHAM)
5(3)
1.2.5 Homotopy Analysis Method (HAM)
8(2)
1.3 Part II: Fundamental Idea of Various Numerical Methods
10(13)
1.3.1 Haar Wavelets and the Operational Matrices
11(2)
1.3.1.1 Function Approximation
13(1)
1.3.1.2 Operational Matrix of the General Order Integration
14(1)
1.3.2 Legendre Wavelets
15(1)
1.3.2.1 Function Approximation
16(1)
1.3.2.2 Operational Matrix of the General Order Integration
17(1)
1.3.3 Chebyshev Wavelets
18(1)
1.3.3.1 Function Approximation
19(1)
1.3.3.2 Operational Matrix of the General Order Integration
19(1)
1.3.4 Hermite Wavelets
20(1)
1.3.4.1 Function Approximation
21(1)
1.3.4.2 Operational Matrix of the General-Order Integration
21(2)
2 Numerical Solution of Partial Differential Equations by Haar Wavelet Method
23(40)
2.1 Introduction
23(1)
2.2 Outline of Present Study
24(1)
2.2.1 Burgers' Equation
24(1)
2.2.2 Modified Burgers' Equation
24(1)
2.2.3 Burgers--Huxley and Huxley Equations
25(1)
2.2.4 Modified Korteweg-de Vries (mKdV) Equation
25(1)
2.3 Application of the Haar Wavelet Method to Obtain the Numerical Solution of Burgers' Equation
25(6)
2.3.1 Numerical Results and Discussion for Burgers' Equation
30(1)
2.4 Haar Wavelet-Based Scheme for Modified Burgers' Equation
31(6)
2.4.1 Numerical Results for Modified Burgers' Equation
35(2)
2.5 Analytical and Numerical Methods for Solving the Burgers--Huxley Equation
37(7)
2.5.1 Application of Variational Iteration Method for Solving the Burgers--Huxley Equation
38(1)
2.5.2 Application of Haar Wavelet Method for Solving the Burgers--Huxley Equation
39(2)
2.5.3 Numerical Results for the Burgers-Huxley Equation
41(3)
2.6 Application of Analytical and Numerical Methods for Solving the Huxley Equation
44(5)
2.6.1 Application of Variational Iteration Method for Solving the Huxley Equation
44(1)
2.6.2 Application of the Haar Wavelet Method for Solving the Huxley Equation
45(2)
2.6.3 Numerical Results for the Huxley Equation
47(2)
2.7 Numerical Solution of the Generalized mKdV Equation
49(10)
2.7.1 Numerical Results of the mKdV Equation
53(6)
2.8 Error of Collocation Method
59(2)
2.9 Error Analysis
61(1)
2.10 Conclusion
62(1)
3 Numerical Solution of a System of Partial Differential Equations
63(26)
3.1 Introduction
63(1)
3.2 Overview of the Problem
64(1)
3.3 Analytical Solution of a System of Nonlinear Partial Differential Equations
65(5)
3.3.1 Application of HPM to Boussinesq--Burgers' Equations
65(2)
3.3.2 Application of OHAM to Boussinesq--Burgers' Equations
67(3)
3.4 Convergence of HPM
70(2)
3.5 Convergence of OHAM
72(1)
3.6 Numerical Results and Discussions
73(1)
3.7 Numerical Approach to Boussinesq--Burgers' Equations
73(8)
3.8 Convergence of Haar Wavelet Approximation
81(2)
3.9 Numerical Results
83(5)
3.10 Conclusion
88(1)
4 Numerical Solution of Fractional Differential Equations by the Haar Wavelet Method
89(32)
4.1 Introduction to Fractional Calculus
89(1)
4.2 Fractional Derivative and Integration
90(4)
4.2.1 Riemann--Liouville Integral and Derivative Operator
90(2)
4.2.2 Caputo Fractional Derivative
92(1)
4.2.3 Grunwald--Letnikov Fractional Derivative
92(1)
4.2.4 Riesz Fractional Derivative
93(1)
4.3 Outline of the Present Study
94(1)
4.4 Application of Analytical and Numerical Techniques to the Fractional Burgers--Fisher Equation
95(5)
4.4.1 Haar Wavelet-Based Scheme for the Fractional Burgers--Fisher Equation
95(3)
4.4.2 Application of Optimal Homotopy Asymptotic Method to the Time-Fractional Burgers--Fisher Equation
98(2)
4.5 Numerical Results for a Fractional Burgers-Fisher Equation
100(1)
4.6 Application of Analytical and Numerical Methods to a Fractional Fisher's Type Equation
101(5)
4.6.1 Haar Wavelet-Based Scheme for the Generalized Fisher's Equation
101(4)
4.6.2 Application of OHAM to the Generalized Fisher's Equation
105(1)
4.7 Numerical Results for a Fractional Fisher's Equation
106(1)
4.8 Solution of a Fractional FPE
107(6)
4.8.1 Application of Haar Wavelets to Time-Fractional FPE
107(5)
4.8.2 Application of a Two-Dimensional Haar Wavelet for Solving Time- and Space-Fractional FPE
112(1)
4.9 Numerical Results for a Fractional FPE
113(1)
4.10 Convergence Analysis of the Two-Dimensional Haar Wavelet Method
114(4)
4.11 Conclusion
118(3)
5 Application of Legendre Wavelet Methods for the Numerical Solution of Fractional Differential Equations
121(46)
5.1 Introduction
121(1)
5.2 Outline of the Present Study
121(2)
5.3 Solution of a Time-Fractional Parabolic Partial Differential Equation
123(5)
5.3.1 Application of HPM to Find the Exact Solution of Fractional Order Parabolic PDE
123(2)
5.3.2 Application of a Two-Dimensional Haar Wavelet for the Numerical Solution of a Fractional PDE
125(3)
5.3.3 Application of Two-Dimensional Legendre Wavelet for Solving Fractional PDE
128(1)
5.4 Numerical Results of Fractional Order PDE
128(6)
5.5 Implementation of Legendre Wavelets for Solving a Fractional KBK Equation
134(3)
5.6 Numerical Results and Discussion of a Time-Fractional KBK Equation
137(1)
5.7 Application of Analytical and Numerical Methods for Solving the Time-Fractional sKdV Equation
138(4)
5.7.1 Implementation of the Legendre Wavelet Method for a Numerical Solution of the Fractional sKdV Equation
138(2)
5.7.2 Comparison with HAM for a Solution of the Time-Fractional sKdV Equation
140(2)
5.8 Numerical Results and Discussion of the Time-Fractional sKdV Equation
142(5)
5.9 Convergence of Legendre Wavelet
147(4)
5.10 Solution of Fractional KK Equation Using Legendre Multiwavelets
151(2)
5.10.1 Introduction of Legendre Multiwavelets
151(1)
5.10.2 Function Approximation
151(1)
5.10.3 Operational Matrix of the General Order Integration
152(1)
5.11 Application of Analytical and Numerical Methods for Solving the Time-Fractional KK Equation
153(3)
5.11.1 Solution of the Fractional KK Equation Using Legendre Multiwavelets
153(2)
5.11.2 Comparison with OHAM for a Solution of the Time-Fractional KK Equation
155(1)
5.12 Numerical Results of the Fractional KK Equation
156(5)
5.13 Conclusion
161(6)
6 Application of Chebyshev Wavelet Methods for Numerical Simulation of Fractional Differential Equations
167(28)
6.1 Introduction
167(1)
6.2 Outline of the Present Study
168(1)
6.3 Formulation of a Time-Fractional SK Equation
169(2)
6.4 Application of Analytical and Numerical Methods for Solving a Fractional SK Equation
171(4)
6.4.1 Implementation of Chebyshev Wavelet on a Time-Fractional SK Equation
171(2)
6.4.2 Comparison with HAM for the Solution of a Time-Fractional SK Equation
173(2)
6.5 Numerical Results of a Fractional SK Equation
175(1)
6.6 Application of the Two-Dimensional Chebyshev Wavelet Method on a Time-Fractional CH Equation
175(4)
6.7 Numerical Results and Discussion
179(3)
6.8 Implementation of the Two-Dimensional Chebyshev Wavelet Method for an Approximate Solution of a Riesz Space-Fractional SGE
182(2)
6.9 Numerical Results and Discussion
184(7)
6.10 Convergence Analysis of a Chebyshev Wavelet
191(2)
6.11 Conclusion
193(2)
7 Application of the Hermite Wavelet Method for Numerical Simulation of Fractional Differential Equations
195(44)
7.1 Introduction
195(1)
7.2 Algorithm of Hermite Wavelet Method
196(2)
7.3 Application of Analytical and Numerical Methods for Solving a Time-Fractional Modified Fornberg--Whitham Equation
198(3)
7.3.1 Two-Dimensional Hermite Wavelet Method for Solving a Nonlinear Time-Fractional Modified Fornberg--Whitham Equation
198(2)
7.3.2 Comparison with OHAM for the Solution of Time-Fractional Modified Fornberg--Whitham Equation
200(1)
7.4 Numerical Results and Discussion
201(3)
7.5 Application of Analytical Methods to Determine the Exact Solutions of a Time-Fractional Modified Fornberg--Whitham Equation
204(10)
7.5.1 Implementation of the FIM for Solving a Fractional Modified Fornberg--Whitham Equation
204(9)
7.5.2 Implementation of OHAM for Approximate Solution of Fractional Modified Fornberg--Whitham Equation
213(1)
7.6 Numerical Results and Discussion
214(6)
7.7 Application of Analytical and Numerical Methods for Solving a Time-Fractional Coupled Jaulent--Miodek Equation
220(6)
7.7.1 Two-Dimensional Hermite Wavelet Method for Solving Nonlinear Time-Fractional Coupled Jaulent--Miodek Equations
220(4)
7.7.2 Comparison with OHAM for the Solution of a Nonlinear Time-Fractional Coupled Jaulent--Miodek Equation
224(2)
7.8 Numerical Results and Discussion
226(8)
7.9 Convergence of a Hermite Wavelet
234(2)
7.10 Conclusion
236(3)
8 Implementation of the Petrov--Galerkin Method for Solving Fractional Partial Differential Equations
239(18)
8.1 Introduction
239(2)
8.2 Implementation of the Petrov--Galerkin Method for the Numerical Solution of the Time-Fractional KdVB Equation
241(4)
8.3 Numerical Results and Discussion
245(1)
8.4 Implementation of the Petrov--Galerkin Method for the Numerical Solution of the Time-Fractional STO Equation
246(6)
8.5 Numerical Results and Discussion
252(3)
8.6 Conclusion
255(2)
References 257(10)
Index 267
Dr. Santanu Saha Ray is an associate professor in the Department of Mathematics at the National Institute of Technology in Rourkela, India. He is a member of the Society for Industrial and Applied Mathematics and the American Mathematical Society. He is also the editor-in-chief of the International Journal of Applied and Computational Mathematics and the author of numerous journal articles and two books: Graph Theory with Algorithms and Its Applications: In Applied Science and Technology and Fractional Calculus with Applications for Nuclear Reactor Dynamics. His research interests include fractional calculus, mathematical modeling, mathematical physics, stochastic modeling, integral equations, and wavelet transforms. Dr. Saha Ray earned his PhD from Jadavpur University.
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