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
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