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Full 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.
Contents
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.
