A Second Order Finite Difference Approximation for the Fractional Diffusion Equation
A Second Order Finite Difference Approximation for the Fractional Diffusion Equation
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DOI：
10.7763/ijapm.2013.v3.212
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
H. M. Nasir;B. Gunawardana;H. M. N. P. Abeyrathna
中科院分区：
文献类型：
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作者：
H. M. Nasir;B. Gunawardana;H. M. N. P. Abeyrathna
摘要考虑一维分数阶扩散方程的一个近似。我们声称，并表明，有限差分近似从Grünwald-Letnikov制定，往往声称是一阶精度，实际上是一个二阶近似的分数导数在一个点远离网格点。利用这一事实，我们设计了分数阶扩散方程的二阶精度有限差分逼近。该方法也被证明是无条件稳定的。通过这种方法，我们对待三种情况下的差分近似在一个统一的设置。数值算例证明了所得结果的正确性。
Abstract—We consider an approximation of one-dimensional fractional diffusion equation. We claim and show that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. The proposed method is also shown to be unconditionally stable. By this approach, we treat three cases of difference approximations in a unified setting. The results obtained are justified by numerical examples.