Nonlinear Differential Equations in Physics (eBook)

This book discusses various novel analytical and numerical methods for solving partial and fractional differential equations. Moreover, it presents selected numerical methods for solving stochastic point kinetic equations in nuclear reactor dynamics by using Euler-Maruyama and strong-order Taylor numerical methods. The book also shows how to arrive at new, exact solutions to various fractional differential equations, such as the time-fractional Burgers-Hopf equation, the (3+1)-dimensional time-fractional Khokhlov-Zabolotskaya-Kuznetsov equation, (3+1)-dimensional time-fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov equation, fractional (2+1)-dimensional Davey-Stewartson equation, and integrable Davey-Stewartson-type equation. 

Many of the methods discussed are analytical-numerical, namely the modified decomposition method, a new two-step Adomian decomposition method, new approach to the Adomian decomposition method, modified homotopy analysis method with Fourier transform, modified fractional reduced differential transform method (MFRDTM), coupled fractional reduced differential transform method (CFRDTM), optimal homotopy asymptotic method, first integral method, and a solution procedure based on Haar wavelets and the operational matrices with function approximation. The book proposes for the first time a generalized order operational matrix of Haar wavelets, as well as new techniques (MFRDTM and CFRDTM) for solving fractional differential equations. Numerical methods used to solve stochastic point kinetic equations, like the Wiener process, Euler-Maruyama, and order 1.5 strong Taylor methods, are also discussed. 

Santanu Saha Ray is Professor and Head of the Department of Mathematics, National Institute of Technology Rourkela, Odisha, India. An elected Fellow of the Institute of Mathematics and its Applications, UK, since 2018, Prof. Saha Ray is also a member of the Society for Industrial and Applied Mathematics (SIAM), American Mathematical Society (AMS) and the International Association of Engineers (IAENG). With over 18 years of experience in teaching undergraduate and graduate students and 17 years of research in mathematics, his focus areas are fractional calculus, differential equations, wavelet transforms, stochastic differential equations, integral equations, nuclear reactor kinetics with simulation, numerical analysis, operations research, mathematical modeling, mathematical physics, and computer applications. 

The editor-in-chief of the International Journal of Applied and Computational Mathematics and associate editor of Mathematical Sciences (both published by Springer), Prof. Saha Ray has published research papers in various international journals of repute. In addition, he has authored six books: one with Springer and five with other publishers-Graph Theory with Algorithms and Its Applications: In Applied Science and Technology (Springer); Fractional Calculus with Applications for Nuclear Reactor Dynamics; Numerical Analysis with Algorithms and Programming; Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations; Generalized Fractional Order Differential Equations Arising in Physical Models; and Novel Methods for Solving Linear and Nonlinear Integral Equations (with other publishers). He has served as principal investigator for various sponsored research projects funded by government agencies. He received an IOP Publishing Top Cited Author Award in 2018, which recognizes outstanding authors using citations recorded in Web of Science.

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This book discusses various novel analytical and numerical methods for solving partial and fractional differential equations. Moreover, it presents selected numerical methods for solving stochastic point kinetic equations in nuclear reactor dynamics by using Euler-Maruyama and strong-order Taylor numerical methods. The book also shows how to arrive at new, exact solutions to various fractional differential equations, such as the time-fractional Burgers-Hopf equation, the (3+1)-dimensional time-fractional Khokhlov-Zabolotskaya-Kuznetsov equation, (3+1)-dimensional time-fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov equation, fractional (2+1)-dimensional Davey-Stewartson equation, and integrable Davey-Stewartson-type equation. Many of the methods discussed are analytical-numerical, namely the modified decomposition method, a new two-step Adomian decomposition method, new approach to the Adomian decomposition method, modified homotopy analysis method with Fourier transform, modified fractional reduced differential transform method (MFRDTM), coupled fractional reduced differential transform method (CFRDTM), optimal homotopy asymptotic method, first integral method, and a solution procedure based on Haar wavelets and the operational matrices with function approximation. The book proposes for the first time a generalized order operational matrix of Haar wavelets, as well as new techniques (MFRDTM and CFRDTM) for solving fractional differential equations. Numerical methods used to solve stochastic point kinetic equations, like the Wiener process, Euler-Maruyama, and order 1.5 strong Taylor methods, are also discussed.