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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 EulerâMaruyama 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 BurgersâHopf equation, the (3+1)-dimensional time-fractional KhokhlovâZabolotskayaâKuznetsov equation, (3+1)-dimensional time-fractional KdVâKhokhlovâZabolotskayaâKuznetsov equation, fractional (2+1)-dimensional DaveyâStewartson equation, and integrable DaveyâStewartson-type equation. Many of the methods discussed are analyticalânumerical, 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, EulerâMaruyama, and order 1.5 strong Taylor methods, are also discussed. |
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