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New York: Springer, 2019. — 188 p.The book focuses on how to implement discrete wavelet transform methods in order to solve problems of reaction–diffusion equations and fractional-order differential equations that arise when modelling real physical phenomena. It explores the analytical and numerical approximate solutions obtained by wavelet methods for both classical and fractional-order differential equations; provides comprehensive information on the conceptual basis of wavelet theory and its applications; and strikes a sensible balance between mathematical rigour and the practical applications of wavelet theory.

The book is divided into 11 chapters, the first three of which are devoted to the mathematical foundations and basics of wavelet theory. The remaining chapters provide wavelet-based numerical methods for linear, nonlinear, and fractional reaction–diffusion problems. Given its scope and format, the book is ideally suited as a text for undergraduate and graduate students of mathematics and engineering.

G. Hariharan has been a Senior Assistant Professor at the Department of Mathematics, SASTRA University, Thanjavur, Tamil Nadu, India, since 2003. He previously served as a lecturer at Adhiparasakthi Engineering College, Melmaruvathur, Tamil Nadu. Dr Hariharan received his MSc and PhD degrees in Mathematics from Bharathidasan University, Trichy, and SASTRA University, in 1999 and 2010, respectively. He has over 20 years of teaching experience at the undergraduate and graduate levels at several educational institutions and engineering institutes, as well as 16 years of research experience in applied mathematics. He is a life member of the Indian Society for Technical Education (ISTE), Ramanujan Mathematical Society (RMS), International Association of Engineers (IAENG), and the Indian Society of Structural Engineers (ISSE). Dr Hariharan has served as the Principal Investigator of projects for e.g. the DRDO-NRB (Naval Research Board) and Government of India, and has contributed research papers on several interdisciplinary topics such as wavelet methods, mathematical modelling, fractional calculus, enzyme kinetics, ship dynamics, and population dynamics. He has published over 85 peer-reviewed research papers on differential equations and applications in various leading international journals, including: Applied Mathematics and Computation, Electrochimica Acta, Ocean Engineering, Journal of Computational and Nonlinear Dynamics, MATCH-Communications in Mathematical and Computer Chemistry, Aerospace and Space Sciences, and the Arabian Journal for Science and Engineering. In addition, Dr Hariharan serves on the editorial boards of several prominent journals, including: Communications in Numerical Analysis, International Journal of Modern Mathematical Sciences, International Journal of Computer Applications, and International Journal of BioinformaticsContents :Preface

Acknowledgements

Contents

About the Author

Nomenclature

Reaction–Diffusion Equations (RDEs)

Importance of Reaction–Diffusion (RD) Problems

Fractional Differential Equation (FDE)

Mathematical Tools to Solve Fractional and Nonlinear Reaction–Diffusion Equations

Zero-Order Deformation Equation

Higher-Order Deformation Equation

A Few Numerical Examples (Chebyshev Wavelet Method for Solving Reaction–Diffusion Equations (RDEs))

References

Wavelet Analysis

Comparison Between Fourier Transform (FT) and Wavelet Transform (WT)

Evolution of Wavelets

Continuous-Time Wavelets

Desirable Properties of Wavelets

Multi-resolution Analysis (MRA)

Haar Wavelets

Function Approximation

Wavelet Method for Solving a Few Reaction–Diffusion Problems—Status and Achievements

Importance of Operational Matrix Methods for Solving Reaction–Diffusion Equations

References

Some Properties of Second Kind Chebyshev Polynomials and Their Shifted Forms

Function Approximation

Operational Matrices of Derivatives for M = , k =

Convergence Theorem for Chebyshev Wavelets

Accuracy Estimation

Legendre Wavelet Method (LWM)

Operational Matrices of Derivatives for M = , k =

Convergence Theorem for Legendre Wavelets

-D Legendre Wavelets

Block-Pulse Functions (BPFs)

Approximating the Nonlinear Term

References

Introduction

Haar Wavelet and Its Properties

Function Approximation

Method of Solution

Conclusion

References

Introduction

Nonlinear Stability Analysis of Lane–Emden Equation of First Kind (Emden–Fowler Equation)

The Emden–Fowler Equation as an Autonomous System

Application of Stability Analysis to Emden–Fowler Equation

Order of the SKCW Method []

Solving Linear Second-Order Two-Point Boundary Value Problems by SKCWM

Test Problems

Conclusion

References

Introduction

Chebyshev Wavelets

Method of Solution by the CWM and LWM

Numerical Experiments

References

Introduction

The General Nonlinear Parabolic PDEs

Parabolic Equation with Exponential Nonlinearity

The FitzHugh–Nagumo Equation

The Burgers Equation

The Burgers–Fisher Equation

References

Introduction

Solving Fisher’s and Fractional Fisher’s Equations by the LLWM

Convergence Analysis and Error Estimation [, ]

Illustrative Examples

Conclusion

Appendix: Basic Idea of Homotopy Analysis Method (HAM)

References

Introduction

Solving Fitzhugh–Nagumo (FN) Equation by the Haar Wavelet Method (HWM)

Solving Fitzhugh–Nagumo (FN) Equation by the LLWM

Convergence Analysis

Numerical Examples

Conclusion

References

Introduction

Wavelets

Legendre Wavelets

Two-dimensional Legendre Wavelets

Approximating the Nonlinear Term

Mathematical Model and the Method of Solution

Illustrative Example

References

Introduction

Mathematical Formulation of the Problem

Solution of the Boundary Problem by Shifted Second Kind Chebyshev Wavelets

Shifted Second Kind Chebyshev Operational Matrix of Derivatives

Method of Solution

Concluding Remarks

References

The book is divided into 11 chapters, the first three of which are devoted to the mathematical foundations and basics of wavelet theory. The remaining chapters provide wavelet-based numerical methods for linear, nonlinear, and fractional reaction–diffusion problems. Given its scope and format, the book is ideally suited as a text for undergraduate and graduate students of mathematics and engineering.

G. Hariharan has been a Senior Assistant Professor at the Department of Mathematics, SASTRA University, Thanjavur, Tamil Nadu, India, since 2003. He previously served as a lecturer at Adhiparasakthi Engineering College, Melmaruvathur, Tamil Nadu. Dr Hariharan received his MSc and PhD degrees in Mathematics from Bharathidasan University, Trichy, and SASTRA University, in 1999 and 2010, respectively. He has over 20 years of teaching experience at the undergraduate and graduate levels at several educational institutions and engineering institutes, as well as 16 years of research experience in applied mathematics. He is a life member of the Indian Society for Technical Education (ISTE), Ramanujan Mathematical Society (RMS), International Association of Engineers (IAENG), and the Indian Society of Structural Engineers (ISSE). Dr Hariharan has served as the Principal Investigator of projects for e.g. the DRDO-NRB (Naval Research Board) and Government of India, and has contributed research papers on several interdisciplinary topics such as wavelet methods, mathematical modelling, fractional calculus, enzyme kinetics, ship dynamics, and population dynamics. He has published over 85 peer-reviewed research papers on differential equations and applications in various leading international journals, including: Applied Mathematics and Computation, Electrochimica Acta, Ocean Engineering, Journal of Computational and Nonlinear Dynamics, MATCH-Communications in Mathematical and Computer Chemistry, Aerospace and Space Sciences, and the Arabian Journal for Science and Engineering. In addition, Dr Hariharan serves on the editorial boards of several prominent journals, including: Communications in Numerical Analysis, International Journal of Modern Mathematical Sciences, International Journal of Computer Applications, and International Journal of BioinformaticsContents :Preface

Acknowledgements

Contents

About the Author

Nomenclature

Reaction–Diffusion Equations (RDEs)

Importance of Reaction–Diffusion (RD) Problems

Fractional Differential Equation (FDE)

Mathematical Tools to Solve Fractional and Nonlinear Reaction–Diffusion Equations

Zero-Order Deformation Equation

Higher-Order Deformation Equation

A Few Numerical Examples (Chebyshev Wavelet Method for Solving Reaction–Diffusion Equations (RDEs))

References

Wavelet Analysis

Comparison Between Fourier Transform (FT) and Wavelet Transform (WT)

Evolution of Wavelets

Continuous-Time Wavelets

Desirable Properties of Wavelets

Multi-resolution Analysis (MRA)

Haar Wavelets

Function Approximation

Wavelet Method for Solving a Few Reaction–Diffusion Problems—Status and Achievements

Importance of Operational Matrix Methods for Solving Reaction–Diffusion Equations

References

Some Properties of Second Kind Chebyshev Polynomials and Their Shifted Forms

Function Approximation

Operational Matrices of Derivatives for M = , k =

Convergence Theorem for Chebyshev Wavelets

Accuracy Estimation

Legendre Wavelet Method (LWM)

Operational Matrices of Derivatives for M = , k =

Convergence Theorem for Legendre Wavelets

-D Legendre Wavelets

Block-Pulse Functions (BPFs)

Approximating the Nonlinear Term

References

Introduction

Haar Wavelet and Its Properties

Function Approximation

Method of Solution

Conclusion

References

Introduction

Nonlinear Stability Analysis of Lane–Emden Equation of First Kind (Emden–Fowler Equation)

The Emden–Fowler Equation as an Autonomous System

Application of Stability Analysis to Emden–Fowler Equation

Order of the SKCW Method []

Solving Linear Second-Order Two-Point Boundary Value Problems by SKCWM

Test Problems

Conclusion

References

Introduction

Chebyshev Wavelets

Method of Solution by the CWM and LWM

Numerical Experiments

References

Introduction

The General Nonlinear Parabolic PDEs

Parabolic Equation with Exponential Nonlinearity

The FitzHugh–Nagumo Equation

The Burgers Equation

The Burgers–Fisher Equation

References

Introduction

Solving Fisher’s and Fractional Fisher’s Equations by the LLWM

Convergence Analysis and Error Estimation [, ]

Illustrative Examples

Conclusion

Appendix: Basic Idea of Homotopy Analysis Method (HAM)

References

Introduction

Solving Fitzhugh–Nagumo (FN) Equation by the Haar Wavelet Method (HWM)

Solving Fitzhugh–Nagumo (FN) Equation by the LLWM

Convergence Analysis

Numerical Examples

Conclusion

References

Introduction

Wavelets

Legendre Wavelets

Two-dimensional Legendre Wavelets

Approximating the Nonlinear Term

Mathematical Model and the Method of Solution

Illustrative Example

References

Introduction

Mathematical Formulation of the Problem

Solution of the Boundary Problem by Shifted Second Kind Chebyshev Wavelets

Shifted Second Kind Chebyshev Operational Matrix of Derivatives

Method of Solution

Concluding Remarks

References

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