Construction of Numerical Approximation for Solving Fractional Partial Differential Equations

Abstract

In this thesis we have solve fractional-partial-differential(FPD) equations. In Chapter 2 for nonlinear time-space fractional Schr¨odinger(TSFS) equation an implicit finite difference (IFD) scheme is developed which is unconditionally-stable at order O(t2􀀀b + ¯h2) where t is time and ¯h is space step-size. Computational cost is lower for non-linear part by developing the explicit implicit scheme (EIS). And for the coupled TSFS system IFD scheme is developed which is also unconditionally-stable. The efficiency and accuracy of developed schemes are verified by numerical experiments. In Chapter 3 one step exponential time differencing (ETD) in temporal and two step ETD and in spatial fractional centered-difference scheme(CDS) are applied to solve the semilinear TSFS equation. Effectiveness of the numerical method is enhanced by two parametric mittag-leffler-function(MLF) which is evaluated by Pad ´ e-approximations. Effectiveness of the develop scheme is given by several examples. In Chapter 4 nonlinear TSFS equation is resolved by a Crank Nicolson-difference(CND) method. Stability and truncation-error of method are described. For improving the effectiveness of calculation a three level Linearized-difference-method (LDM) is also developed. In Chapter 5 a compact finite difference numerical scheme is developed for time fractional fourth order diffusion equation. In this paper spatial domain is discretized by Stepenhnson’s scheme and for time derivative fractional trapezoid formula is used. There is also presented complete stability and convergence analysis at the order of O(t2 +h4). The accuracy and effectiveness of proposed scheme is analyzed by numerical experiments showing that our numerical experiments are agreed with our analytical results.