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BL
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| Record Number
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890671
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| Main Entry
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Hariharan, G.
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| Title & Author
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Wavelet solutions for reaction-diffusion problems in science and engineering /\ G. Hariharan.
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| Publication Statement
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Singapore :: Springer,, 2019.
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| Series Statement
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Forum for Interdisciplinary Mathematics
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| Page. NO
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1 online resource (188 pages)
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| ISBN
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9789813299603
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: 9813299606
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9789813299597
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| Notes
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5.4.1 Solving Nonlinear Second-Order Two-Point Boundary Value Problems by the S2KCWM
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| Bibliographies/Indexes
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Includes bibliographical references.
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| Contents
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Intro; Preface; Acknowledgements; Contents; About the Author; Nomenclature; 1 Reaction-Diffusion (RD) Problems; 1.1 Reaction-Diffusion Equations (RDEs); 1.2 Importance of Reaction-Diffusion (RD) Problems; 1.3 A Few Familiar Reaction-Diffusion Equations (RDEs); 1.3.1 Nonlinear Singular Boundary Value Problem (Lane-Emden Type) and Wavelets; 1.4 Fractional Differential Equation (FDE); 1.5 Definitions of Fractional Derivatives and Integrals; 1.6 Mathematical Tools to Solve Fractional and Nonlinear Reaction-Diffusion Equations; 1.6.1 Basic Idea of Homotopy Analysis Method (HAM)
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1.6.2 Zero-Order Deformation Equation1.6.3 Higher-Order Deformation Equation; 1.6.4 A Few Numerical Examples (Chebyshev Wavelet Method for Solving Reaction-Diffusion Equations (RDEs)); References; 2 Wavelet Analysis-An Overview; 2.1 Wavelet Analysis; 2.2 Comparison Between Fourier Transform (FT) and Wavelet Transform (WT); 2.3 Wavelets and Multi-resolution Analysis (MRA); 2.4 Evolution of Wavelets; 2.5 Genesis of Wavelets; 2.6 Continuous-Time Wavelets; 2.7 Discrete Wavelet Transform (DWT); 2.8 Desirable Properties of Wavelets; 2.9 Multi-resolution Analysis (MRA)
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2.10 Discrete Wavelet Transforms Methods2.10.1 Haar Wavelets; 2.10.2 Function Approximation; 2.11 Wavelet Method for Solving a Few Reaction-Diffusion Problems-Status and Achievements; 2.11.1 Importance of Operational Matrix Methods for Solving Reaction-Diffusion Equations; References; 3 Shifted Chebyshev Wavelets and Shifted Legendre Wavelets-Preliminaries; 3.1 Introduction to Shifted Second Kind Chebyshev Wavelet Method (S2KCWM); 3.1.1 Some Properties of Second Kind Chebyshev Polynomials and Their Shifted Forms; 3.1.2 Shifted Second Kind Chebyshev Wavelets; 3.2 Function Approximation
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3.2.1 Operational Matrices of Derivatives for M = 2, k = 03.2.2 Operational Matrices of Derivatives for k = 0, M = 3; 3.3 Convergence Theorem for Chebyshev Wavelets; 3.3.1 Accuracy Estimation; 3.4 Legendre Wavelet Method (LWM); 3.4.1 Operational Matrices of Derivatives for M = 2, k = 0; 3.5 Convergence Theorem for Legendre Wavelets; 3.6 Error Analysis; 3.7 2-D Legendre Wavelets; 3.8 Block-Pulse Functions (BPFs); 3.9 Approximating the Nonlinear Term; 3.10 Approximation of Function; References; 4 Wavelet Method to Film-Pore Diffusion Model for Methylene Blue Adsorption onto Plant Leaf Powders
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4.1 Introduction4.2 Materials and Methods; 4.3 Haar Wavelet and Its Properties; 4.3.1 Function Approximation; 4.4 Method of Solution; 4.5 Conclusion; References; 5 An Efficient Wavelet-Based Spectral Method to Singular Boundary Value Problems; 5.1 Introduction; 5.2 Nonlinear Stability Analysis of Lane-Emden Equation of First Kind (Emden-Fowler Equation); 5.2.1 The Emden-Fowler Equation as an Autonomous System; 5.2.2 Application of Stability Analysis to Emden-Fowler Equation; 5.3 Order of the S2KCW Method [25]; 5.4 Solving Linear Second-Order Two-Point Boundary Value Problems by S2KCWM
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| Abstract
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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.
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| Subject
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Wavelets (Mathematics)
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| Subject
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Wavelets (Mathematics)
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| Dewey Classification
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515/.2433
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| LC Classification
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QA403.3
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