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Hariharan G. Wavelet Solutions for Reaction-Diffusion Problems in Science and Engineering

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Hariharan G. Wavelet Solutions for Reaction-Diffusion Problems in Science and Engineering
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 Bioinformatics
Contents :
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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