High Order Compact Difference Method For The Time Fractional Dispersive Equations

The fractional order differential equations are a generalization of the integralorder differential equations. Duing to the global correlation, it can be well used todescribe the specific process of various physics models. The theory and numericalmethods of the fractional order differential equations become a hot field at present.The high order compact difference methods and the local correlative compactdifference methods for the time fractional linear dispersive equations and the highorder compact difference methods for the time fractional order nonlinear KdVequations are studied in this thesis.In the first part of the paper, the high order compact difference methods of thetime-fractional order linear dispersive equations are discussed. By using a four orderand a six order difference approximation to the partial derivative of the spatialvariables，locally quadratic interpolation approximation to time fractional orderCaputo derivative, a fourth-order four-point stencil and a sixth-order five-point stencilcompact implicit difference schemes are derived for the time fractional lineardispersive equations. It is shown that the convergence order of the two compactimplicit difference schemes areO(τ~2+h~4)andO(τ~2+h~6), respectively. Thenumerical experiments show that the present compact implicit difference schemes arestable and with high accuracy.In the second part of the paper, duing to the global correlation of the fractionalorder differential equations, it needs to have a large storage. A discrete numericalmethods of short-term memory for the fractional differential equations is presented tosolve the time fractional order Caputo derivative. It only need store part history dataand its discretization error is well controled. The discrete methods of fractionaldifferential equations are calculated for a long time. To a certain extent, it can solvethe problem of storage. The methods of this paper can applied this local correlation compact difference method to the time fractional order linear dispersive equations.The numerical experiments show that the present methods can save memory space,which is accurate and effective.In the third part of the paper, the high order compact difference methods for thetime-fractional order nonlinear KdV equations are researched. A fourth-orderfour-point stencil compact difference schemes are devised for solving the timefractional nonlinear KdV equations. It is shown that the convergence order of thecompact implicit difference schemes isO(τ~2+h~4). The numerical experiments showthat the present compact difference schemes are effective and with good stability.