Finite Defference Methods For The Time Fractional Partial Differential Equations

The fractional derivatives are more suitable to describe the natural developing process withdependence on the historical data because of its nonlocal properties. Thus the fractional derivativesare nowadays applied to more and more technological and scientific problems. But numericalcomputation is difficult and complicated for the fractional partial differential equations correspondingto the dependence on the historical data and the global correlation of the fractional derivative. Finitedifference method has the advantage of simplicity and can be used to many kinds of problems. So it isimportant among the numerical methods for solving both fractional and integral partial differentialequations. This work is devoted to the finite difference methods for some temporal fractionaldiffudion equations.For the fractional diffusion-wave equation of order four with Caputo derivative,L1approximation is used to discretize the temporal fractional derivative and two discrete methods isadoped to the discretization of the spatial fourth order derivative: one is based on the direct utilizaitonof Taylor expansion; aother is based on the method of order reduction. Two Crank-Nicolson schemesare obtained. Then by using the new compact operator of the spatial fourth order derivative and thecompact operator of the spatial second order derivative, two compact difference scheme are reduced.Stability and convergence of these schemes are analyzed by the energy method. Numericalexperiments demonstrate the accuracy and effectiveness of these schemes. Similar work is done forthe fourth order fractional sudiffusion eauation and the corresponding results are got. During theresaerch, a simpler proof is given by using the new compact operator of the spatial fourth orderderivative intruduced in this paper.For the subdiffusion equaitons with Riemann-Liouville fractional derivative, some defferenceschemes are constructed in this paper by integerating the original equation and then using thenumerical integration idea. Two implicit schemes with orderO (2h2)andO (2h4)areconstructed in our paper. Then by some technical process, the stability and convergence of theschemes withO (2)are analyzed firstly by us in L norm which is a new result. For thefractional Cable equation, an implicit compact scheme is established by using spatial compactoperator. The stability and convergenc are given in L norm. Then a compact scheme with orderO (2h4)is considered and numerical results are listed to verify the effectiveness of theseschemes. For the fractional equation in polar coordinates, the singularity in the origin increases thedifficulty of the discretization. For the fractional equation with radial symmetry, the temporalfractional derivative is discretized byL1approximation and the method of order reduction isadopted in spatial direciton. Using an intermediate variable, a simple and accurate computing formulais obtained to the natrual derivative boudary conditions. The method used here to handle the thenatrual derivative boudary conditions at the origin is also suitable to the integeral equaitons in polarcoordinates. Then eliminating the intermediate variable, a self-contained simplicated scheme easy tocalculate can easily be obtained. For the fractional equation on a disk, complication of computation isincreased and the discrtization of the natrual derivative boundary conditions is difficult. Byconsidering the2periodic condition in the azimuthal direction, the finite Fourier transformation isadopt to transform the solving space to the Fourier mode. Then by means of FFT and IFFT package,the computation is fast completed. Numerical tests show that using the fast Fourier transformation isnecessary for a efficient solver. Especially for solving the compact scheme, the fast Fouriertransformation is more important.