We construct the two unique piecewise $C^3$-splines of degree six and eight to approximate the solution of fractional integro-differential equations. Existence and uniqueness of the solution of second order fractional integro-differential equation has been proved. Convergence analysis of the presented methods have been discussed. Illustrative examples are included to demonstrate the validity and applicability of the presented approaches.

中文翻译：

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$C^3$样条求解二阶分数阶积分微分方程

我们分段构造两个唯一 $C^3$-六阶和八阶样条曲线近似分数阶积分微分方程的解。证明了二阶分数阶积分微分方程解的存在唯一性。已经讨论了所提出方法的收敛性分析。包括说明性示例以证明所提出方法的有效性和适用性。