Nonlinear Differential Equations in Physics

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 EulerMaruyama 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 BurgersHopf equation, the (3+1)-dimensional time-fractional KhokhlovZabolotskayaKuznetsov equation, (3+1)-dimensional time-fractional KdVKhokhlovZabolotskayaKuznetsov equation, fractional (2+1)-dimensional DaveyStewartson equation, and integrable DaveyStewartson-type equation.

Many of the methods discussed are analyticalnumerical, 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, EulerMaruyama, and order 1.5 strong Taylor methods, are also discussed.

Discusses key concepts of various novel methods in fractional calculus Helps in solving fractional differential equations used for the accurate modeling of physical phenomena Highlights research into the applications of wavelet methods and fractional differential equations Offers a valuable resource for graduate and research students, as well as scientists and engineers in the fields of applied mathematics, applied physics and engineering

Autorentext

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 outstandingauthors using citations recorded in Web of Science.

Inhalt

1. Mathematical Preliminaries.- 2. New Approaches for Decomposition Method for the Solution of Differential Equations.- 3. Numerical Solution of Fractional Differential Equations by using New Wavelet Operational Matrix of General Order.- 4. Numerical Solutions of Riesz Fractional Partial Differential Equations.- 5. New Exact Solutions of Fractional Order Partial Differential Equations.- 6. An Investigation for New Double Periodic Solutions of Coupled Schrödinger-Boussinesq Equations.- 7. New Techniques for Fractional Reduced Differential Transform Method.- 7. A Novel Approach with Time-Splitting Fourier Spectral Method for Riesz Fractional Differential Equations.- 9. Numerical Simulation of Stochastic Point Kinetic Equation in the Dynamical System of Nuclear Reactor.