A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions
A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions
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DOI：
10.1007/s10255-022-1102-8
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发表时间：
2022-07
期刊：
影响因子：
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通讯作者：
Dakang Cen;Zhi-Bo Wang;Yan Mo
中科院分区：
文献类型：
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作者：
Dakang Cen;Zhi-Bo Wang;Yan Mo
本文给出了一种求解初始时刻具有弱奇异性的非线性分数次积分-微分方程的紧致有限差分格式，精度阶为O(N−(2−α)+M−4)，其中N，M表示时空方向上的网格数目，α∈(0，1)为分数阶。为了在满足解的正则性要求的基础上恢复全部精度，我们分别采用L1方法和梯形积积分(PI)规则对Caputo导数和Riemann-Liouville积分进行离散，并用牛顿线性化方法对非线性项进行了仔细的处理。基于离散分数Grönwall不等式和Riemann-Liouville分数次积分的保留离散系数，用能量法分析了该格式的稳定性和收敛性。数值算例也证实了理论结果的正确性。
In this paper, a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed, withO(N−(2−α)+M−4) accuracy order, whereN, Mdenote the numbers of grids in temporal and spatial direction,α∈ (0, 1) is the fractional order. To recover the full accuracy based on the regularity requirement of the solution, we adopt theL1 method and the trapezoidal product integration (PI) rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral, respectively, further handle the nonlinear term carefully by the Newton linearized method. Based on the discrete fractional Grönwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral, the stability and convergence of the proposed scheme are analyzed by the energy method. Theoretical results are also confirmed by a numerical example.