Table Of ContentWavelet Methods for Solving
Partial Differential Equations
and Fractional Differential
Equations
Wavelet Methods for Solving
Partial Differential Equations
and Fractional Differential
Equations
Dr. Santanu Saha Ray
Department of Mathematics
National Institute of Technology Rourkela
Rourkela, Odisha, India
Dr. Arun Kumar Gupta
School of Applied Sciences
KIIT University
Bhubaneswar, Odisha, India
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Saha Ray, Santanu, author. | Gupta, Arun Kumar (Mathematician), author.
Title: Wavelet methods for solving partial differential equations and
fractional differential equations / Santanu Saha Ray and Arun Kumar Gupta.
Description: Boca Raton : CRC Press, 2018. | Includes bibliographical
references and index.
Identifiers: LCCN 2017039698| ISBN 9781138053816 (hardback : alk. paper) |
ISBN 9781315167183 (ebook)
Subjects: LCSH: Differential equations, Partial. | Wavelets (Mathematics) |
Fractional differential equations. | Differential equations--Numerical
solutions.
Classification: LCC QA374 .S24 2018 | DDC 515/.353--dc23
LC record available at https://lccn.loc.gov/2017039698
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Contents
List of Figures .................................................................................................................................xi
List of Tables ..................................................................................................................................xv
Preface .........................................................................................................................................xxiii
Acknowledgements .................................................................................................................xxvii
Mathematical Preliminary ........................................................................................................xxix
1. Numerous Analytical and Numerical Methods ................................................................1
1.1 Introduction ...................................................................................................................1
1.2 Part I: Fundamental Idea of Various Analytical Methods .......................................1
1.2.1 Variational Iteration Method (VIM) ..............................................................1
1.2.2 First Integral Method .......................................................................................2
1.2.2.1 Algorithm of FIM .............................................................................3
1.2.3 Homotopy Perturbation Method (HPM) .....................................................4
1.2.4 Optimal Homotopy Asymptotic Method (OHAM) ....................................5
1.2.5 Homotopy Analysis Method (HAM) ............................................................8
1.3 Part II: Fundamental Idea of Various Numerical Methods ...................................10
1.3.1 Haar Wavelets and the Operational Matrices ............................................11
1.3.1.1 Function Approximation ...............................................................13
1.3.1.2 Operational Matrix of the General Order Integration ..............14
1.3.2 Legendre Wavelets .........................................................................................15
1.3.2.1 Function Approximation ...............................................................16
1.3.2.2 Operational Matrix of the General Order Integration ..............17
1.3.3 Chebyshev Wavelets ......................................................................................18
1.3.3.1 Function Approximation ...............................................................19
1.3.3.2 Operational Matrix of the General Order Integration ..............19
1.3.4 Hermite Wavelets ...........................................................................................20
1.3.4.1 Function Approximation ...............................................................21
1.3.4.2 Operational Matrix of the General-Order Integration ..............21
2. Numerical Solution of Partial Differential Equations by Haar Wavelet Method ....23
2.1 Introduction .................................................................................................................23
2.2 Outline of Present Study ............................................................................................24
2.2.1 Burgers’ Equation ..........................................................................................24
2.2.2 Modified Burgers’ Equation .........................................................................24
2.2.3 Burgers–Huxley and Huxley Equations .....................................................25
2.2.4 Modified Korteweg-de Vries (mKdV) Equation ........................................25
2.3 Application of the Haar Wavelet Method to Obtain the Numerical Solution
of Burgers’ Equation ...................................................................................................25
2.3.1 Numerical Results and Discussion for Burgers’ Equation .......................30
2.4 Haar Wavelet-Based Scheme for Modified Burgers’ Equation .............................31
2.4.1 Numerical Results for Modified Burgers’ Equation .................................35
2.5 Analytical and Numerical Methods for Solving the Burgers–Huxley Equation .......37
2.5.1 Application of Variational Iteration Method for Solving the
Burgers–Huxley Equation ............................................................................38
v
vi Contents
2.5.2 Application of Haar Wavelet Method for Solving the Burgers–Huxley
Equation ..........................................................................................................39
2.5.3 Numerical Results for the Burgers–Huxley Equation ..............................41
2.6 Application of Analytical and Numerical Methods for Solving the Huxley
Equation .......................................................................................................................44
2.6.1 Application of Variational Iteration Method for Solving the Huxley
Equation ..........................................................................................................44
2.6.2 Application of the Haar Wavelet Method for Solving the Huxley
Equation ..........................................................................................................45
2.6.3 Numerical Results for the Huxley Equation ..............................................47
2.7 Numerical Solution of the Generalized mKdV Equation .....................................49
2.7.1 Numerical Results of the mKdV Equation .................................................53
2.8 Error of Collocation Method .....................................................................................59
2.9 Error Analysis ..............................................................................................................61
2.10 Conclusion ...................................................................................................................62
3. Numerical Solution of a System of Partial Differential Equations ............................63
3.1 Introduction .................................................................................................................63
3.2 Overview of the Problem ...........................................................................................64
3.3 Analytical Solution of a System of Nonlinear Partial Differential Equations ....65
3.3.1 Application of HPM to Boussinesq–Burgers’ Equations .........................65
3.3.2 Application of OHAM to Boussinesq–Burgers’ Equations......................67
3.4 Convergence of HPM .................................................................................................70
3.5 Convergence of OHAM .............................................................................................72
3.6 Numerical Results and Discussions .........................................................................73
3.7 Numerical Approach to Boussinesq–Burgers’ Equations .....................................73
3.8 Convergence of Haar Wavelet Approximation.......................................................81
3.9 Numerical Results .......................................................................................................83
3.10 Conclusion ...................................................................................................................88
4. Numerical Solution of Fractional Differential Equations by the Haar
Wavelet Method ....................................................................................................................89
4.1 Introduction to Fractional Calculus ..........................................................................89
4.2 Fractional Derivative and Integration ......................................................................90
4.2.1 Riemann–Liouville Integral and Derivative Operator .............................90
4.2.2 Caputo Fractional Derivative .......................................................................92
4.2.3 Grünwald–Letnikov Fractional Derivative ................................................92
4.2.4 Riesz Fractional Derivative...........................................................................93
4.3 Outline of the Present Study .....................................................................................94
4.4 Application of Analytical and Numerical Techniques to the Fractional
Burgers–Fisher Equation..........................................................................................95
4.4.1 Haar Wavelet-Based Scheme for the Fractional Burgers–Fisher Equation ...95
4.4.2 Application of Optimal Homotopy Asymptotic Method to the
Time-Fractional Burgers–Fisher Equation ..................................................98
4.5 Numerical Results for a Fractional Burgers–Fisher Equation ............................100
4.6 Application of Analytical and Numerical Methods to a Fractional Fisher’s
Type Equation ............................................................................................................101
4.6.1 Haar Wavelet-Based Scheme for the Generalized Fisher’s Equation ...101
4.6.2 Application of OHAM to the Generalized Fisher’s Equation ...............105
Contents vii
4.7 Numerical Results for a Fractional Fisher’s Equation .........................................106
4.8 Solution of a Fractional FPE ....................................................................................107
4.8.1 Application of Haar Wavelets to Time-Fractional FPE ...........................107
4.8.2 Application of a Two-Dimensional Haar Wavelet for Solving
Time- and Space-Fractional FPE ................................................................112
4.9 Numerical Results for a Fractional FPE .................................................................113
4.10 Convergence Analysis of the Two-Dimensional Haar Wavelet Method ..........114
4.11 Conclusion .................................................................................................................118
5. Application of Legendre Wavelet Methods for the Numerical Solution
of Fractional Differential Equations ...............................................................................121
5.1 Introduction ...............................................................................................................121
5.2 Outline of the Present Study ...................................................................................121
5.3 Solution of a Time-Fractional Parabolic Partial Differential Equation ..............123
5.3.1 Application of HPM to Find the Exact Solution of Fractional
Order Parabolic PDE ...................................................................................123
5.3.2 Application of a Two-Dimensional Haar Wavelet for the Numerical
Solution of a Fractional PDE ......................................................................125
5.3.3 Application of Two-Dimensional Legendre Wavelet for Solving
Fractional PDE ..............................................................................................128
5.4 Numerical Results of Fractional Order PDE .........................................................128
5.5 Implementation of Legendre Wavelets for Solving a Fractional KBK Equation .....134
5.6 Numerical Results and Discussion of a Time-Fractional KBK Equation ..........137
5.7 Application of Analytical and Numerical Methods for Solving the
Time-Fractional sKdV Equation ..............................................................................138
5.7.1 Implementation of the Legendre Wavelet Method for a Numerical
Solution of the Fractional sKdV Equation ................................................138
5.7.2 Comparison with HAM for a Solution of the Time-Fractional
sKdV Equation .........................................................................................140
5.8 Numerical Results and Discussion of the Time-Fractional sKdV Equation .....142
5.9 Convergence of Legendre Wavelet .........................................................................147
5.10 Solution of Fractional KK Equation Using Legendre Multiwavelets ................151
5.10.1 Introduction of Legendre Multiwavelets .................................................151
5.10.2 Function Approximation ............................................................................151
5.10.3 Operational Matrix of the General Order Integration ............................152
5.11 Application of Analytical and Numerical Methods for Solving the
Time-Fractional KK Equation ..................................................................................153
5.11.1 Solution of the Fractional KK Equation Using Legendre
Multiwavelets ........................................................................................153
5.11.2 Comparison with OHAM for a Solution of the Time-Fractional
KK Equation ...........................................................................................155
5.12 Numerical Results of the Fractional KK Equation ...............................................156
5.13 Conclusion .................................................................................................................161
6. Application of Chebyshev Wavelet Methods for Numerical Simulation
of Fractional Differential Equations ...............................................................................167
6.1 Introduction ...............................................................................................................167
6.2 Outline of the Present Study ...................................................................................168
6.3 Formulation of a Time-Fractional SK Equation ....................................................169
viii Contents
6.4 Application of Analytical and Numerical Methods for Solving
a Fractional SK Equation ..........................................................................................171
6.4.1 Implementation of Chebyshev Wavelet on a Time-Fractional
SK Equation ................................................................................................171
6.4.2 Comparison with HAM for the Solution of a Time-Fractional SK
Equation .............................................................................................................173
6.5 Numerical Results of a Fractional SK Equation ...................................................175
6.6 Application of the Two-Dimensional Chebyshev Wavelet Method
on a Time-Fractional CH Equation .........................................................................175
6.7 Numerical Results and Discussion .........................................................................179
6.8 Implementation of the Two-Dimensional Chebyshev Wavelet Method
for an Approximate Solution of a Riesz Space-Fractional SGE ..........................182
6.9 Numerical Results and Discussion .........................................................................184
6.10 Convergence Analysis of a Chebyshev Wavelet ...................................................191
6.11 Conclusion .................................................................................................................193
7. Application of the Hermite Wavelet Method for Numerical Simulation
of Fractional Differential Equations ...............................................................................195
7.1 Introduction ...............................................................................................................195
7.2 Algorithm of Hermite Wavelet Method .................................................................196
7.3 Application of Analytical and Numerical Methods for Solving
a Time-Fractional Modified Fornberg–Whitham Equation ................................198
7.3.1 Two-Dimensional Hermite Wavelet Method for Solving a Nonlinear
Time-Fractional Modified Fornberg–Whitham Equation ......................198
7.3.2 Comparison with OHAM for the Solution of Time-Fractional
Modified Fornberg–Whitham Equation ...................................................200
7.4 Numerical Results and Discussion .........................................................................201
7.5 Application of Analytical Methods to Determine the Exact Solutions
of a Time-Fractional Modified Fornberg–Whitham Equation ............................204
7.5.1 Implementation of the FIM for Solving a Fractional Modified
Fornberg–Whitham Equation ....................................................................204
7.5.2 Implementation of OHAM for Approximate Solution of Fractional
Modified Fornberg–Whitham Equation ...................................................213
7.6 Numerical Results and Discussion .........................................................................214
7.7 Application of Analytical and Numerical Methods for Solving
a Time-Fractional Coupled Jaulent–Miodek Equation ........................................220
7.7.1 Two-Dimensional Hermite Wavelet Method for Solving Nonlinear
Time-Fractional Coupled Jaulent–Miodek Equations ............................220
7.7.2 Comparison with OHAM for the Solution of a Nonlinear
Time-Fractional Coupled Jaulent–Miodek Equation ..............................224
7.8 Numerical Results and Discussion .........................................................................226
7.9 Convergence of a Hermite Wavelet ........................................................................234
7.10 Conclusion .................................................................................................................236
8. Implementation of the Petrov–Galerkin Method for Solving Fractional
Partial Differential Equations ..........................................................................................239
8.1 Introduction ...............................................................................................................239
8.2 Implementation of the Petrov–Galerkin Method for the Numerical
Solution of the Time-Fractional KdVB Equation ..................................................241
Contents ix
8.3 Numerical Results and Discussion .........................................................................245
8.4 Implementation of the Petrov–Galerkin Method for the Numerical Solution
of the Time-Fractional STO Equation .....................................................................246
8.5 Numerical Results and Discussion .........................................................................252
8.6 Conclusion .................................................................................................................255
References ....................................................................................................................................257
Index .............................................................................................................................................267
