时间分数阶Cable方程修正格式的误差分析

考虑时间分数阶Cable方程在修正的二阶向后差分格式下的误差分析.利用连续Laplace变换、反Laplace变换方法得到方程的准确解,类似得到空间有限元半离散解;运用Lubich的修正方法引入此分数阶微分方程的修正格式,离散的Laplace变换、反Laplace变换法得到Cable方程的时间离散解,进而讨论了时间离散下L₂范数的误差估计,得到二阶收敛阶,并用数值算例验证了定理的结论.这个结论比不修正的情形下一阶收敛阶要高.

Error analysis of modified schemes for time-fractional Cable equation

Error analysis of the modified second-order backward difference scheme for the time-fractional Cable equation is carried out. By using continuous Laplace transform and inverse Laplace transform, the exact solution of the equation is obtained, and the finite element semidiscrete solution is obtained similarly. Then Lubich's correction method is used to get the modified form of the fractional differential equation. The discrete solutions of the Cable equation are obtained by means of the discrete Laplace transform and the inverse Laplace transform. Finally, the error estimates under the norm are discussed and the second order of convergence is obtained. Numerical results verification is finally performed to validate the theoretical findings discussed here, which proved that it is better than the first order convergence without modification.