伴随Volterra积分微分方程的最优阶分数后向配点法

中文摘要 AI 辅助

本文针对具有弱奇异核的伴随Volterra积分微分方程，开发并分析了一种最优阶分数后向配点法。后向Volterra结构连同弱奇异核在终端端点处引起分数幂奇异性，从而降低了精确解的经典正则性，并导致标准多项式配点法的阶数恶化。我们首先建立了一个刻画解在终端奇异行为的正则性结果，表明解是连续可微的，而其二阶导数在终端端点处可能表现出弱奇异性。受此正则性结构启发，我们引入终端分级网格，并构造了一种分数后向配点格式，其局部逼近空间适应端点奇异性。针对解及其导数，我们推导了严格的收敛性和超收敛性估计。通过适当选择分数参数和网格分级指数，所提方法达到了由局部逼近阶决定的最优收敛阶。数值实验证实了理论预测，并展示了该方法对伴随弱奇异Volterra积分微分方程的准确性和有效性。

英文摘要

This paper develops and analyzes an optimal-order fractional backward collocation method for adjoint Volterra integro-differential equations with weakly singular kernels. The backward Volterra structure, together with the weakly singular kernel, induces fractional-power singularities at the terminal endpoint, thereby reducing the classical regularity of the exact solution and causing order deterioration in standard polynomial collocation methods. We first establish a regularity result that characterizes the terminal singular behavior of the solution, showing that the solution is continuously differentiable, whereas its second derivative may exhibit a weak singularity at the terminal endpoint. Motivated by this regularity structure, we introduce a terminally graded mesh and construct a fractional backward collocation scheme whose local approximation space is adapted to the endpoint singularity. Rigorous convergence and superconvergence estimates are derived for both the solution and its derivative. With suitable choices of the fractional parameter and the mesh-grading exponent, the proposed method attains the optimal convergence orders dictated by the local approximation degree. Numerical experiments confirm the theoretical predictions and demonstrate the accuracy and effectiveness of the method for adjoint weakly singular Volterra integro-differential equations.

发表机构

• Imam Mohammad Ibn Saud Islamic University（伊玛目穆罕默德·伊本·沙特伊斯兰大学）

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