Nonlinear Differential Equations in Physics: Novel Methods for Finding Solutions

Description

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. Front Matter ....Pages i-xxxi Mathematical Preliminaries (Santanu Saha Ray)....Pages 1-53 New Approaches for Decomposition Method for the Solution of Differential Equations (Santanu Saha Ray)....Pages 55-85 Numerical Solution of Fractional Differential Equations by Using New Wavelet Operational Matrix of General Order (Santanu Saha Ray)....Pages 87-118 Numerical Solutions of Riesz Fractional Partial Differential Equations (Santanu Saha Ray)....Pages 119-154 New Exact Solutions of Fractional-Order Partial Differential Equations (Santanu Saha Ray)....Pages 155-197 New Exact Traveling Wave Solutions of the Coupled Schrödinger–Boussinesq Equations and Tzitzéica-Type Evolution Equations (Santanu Saha Ray)....Pages 199-229 New Techniques on Fractional Reduced Differential Transform Method (Santanu Saha Ray)....Pages 231-333 A Novel Approach with Time-Splitting Fourier Spectral Method for Riesz Fractional Differential Equations (Santanu Saha Ray)....Pages 335-373 Numerical Simulation of Stochastic Point Kinetics Equation in the Dynamical System of Nuclear Reactor (Santanu Saha Ray)....Pages 375-388