The Numerical Method Of Fractional Order Differential Equation

We introduce the concept of fractional order briefly at first, and use it into standard heat conduction equation. Then we built the time-space fractional heat conduction equation, prove the stability and convergence of the scheme. Numerical solutions are obtained from the model, and the plots of temperature field curves are given. The results show that the temperature field is very sensitive to the order of fractional derivative.The fractional order is also intorduced into the seepage flow mechanics. Two flow models of fluids in fractal reservoirs are established with time fractional derivative and time-space fractional derivatives respectively. An implicit scheme and an explicit one are proposed for the flow models with time fractional derivative by using finite difference methods. And the numerical solution is obtained. For the flow models with time-space fractional derivatives, an explicit scheme is considered and the numerical solution is also obtained. An infinitely large and a finite closed fractal reservoir with a constant rate production are discussed for the two flow models. The plots of typical pressure curves and pressure dynamic characteristic analyses are given. This paper addresses the change rules of pressure with the change of time fractional derivativeα, space fractional derivativeγand spectral dimension respectively. The results can be provided as theoretical basis for fractal reservoirs development. ds...