Applied Numerical Mathematics ( IF 2.8 ) Pub Date : 2021-06-30 , DOI: 10.1016/j.apnum.2021.06.010 Ahmed S. Hendy , Mahmoud A. Zaky , Rob H. De Staelen
This paper is devoted to introducing a novel methodology to prove the convergence and stability of a Crank–Nicolson difference approximation for a class of multi-term time-fractional diffusion equations with nonlinear delay and space fractional derivatives in case of sufficient smooth solutions. The temporal fractional derivatives are approximated by a specific form of L1 scheme at . A fourth-order difference approximation for the spatial fractional derivatives is employed by using the weighted average of the shifted Grünwald formulae. This methodology is based on a class of discrete fractional Grönwall inequalities convenient with the quadrature formula used to approximate the Caputo derivative at . In the present work, the method of energy inequalities is utilized to show that the used difference scheme is stable and converges to the exact solution with order , in the case that satisfies , which means that , such that is the maximum α-th order in the multi-order fractional operators.
中文翻译:
具有时滞的非线性多项时空分数阶偏微分方程的高阶有限差分求解器数值分析的一般框架
本文致力于介绍一种新的方法来证明 Crank-Nicolson 差分近似对于一类具有非线性延迟和空间分数阶导数的多项式时间分数扩散方程在足够光滑解的情况下的收敛性和稳定性。时间分数阶导数近似于特定形式的 L1 方案在. 通过使用移位 Grünwald 公式的加权平均值,采用了空间分数阶导数的四阶差分近似。该方法基于一类离散分数 Grönwall 不等式,使用正交公式可以方便地在以下位置逼近 Caputo 导数. 在目前的工作中,能量不等式的方法被用来证明所使用的差分格式是稳定的,并且收敛到有阶的精确解,在这种情况下 满足 , 意思就是 ,这样 是多阶小数运算符中的最大α阶。