一类时间分数阶电报方程解的初始奇异性

The Initial Singularity of Solutions to a Class of Time-Fractional Telegraph Equations

摘要: 时间分数阶电报方程在描述具有遗传与记忆特性的反常扩散过程中具有重要作用。然而，此类方程的解在初始时刻附近往往存在奇异性，这源于分数阶微分算子的非局部性质，并且会显著影响传统数值方法的精度。本文主要研究一类时间分数阶电报方程的解在初始时刻的奇异性，通过运用Laplace变换法，将方程的解进行分解，构造出关键的积分算子，通过分析这些积分算子及其各阶时间导数在$t\to 0^{+}$ 时的渐近行为，最终严格证明了解本身及其直至三阶时间导数在初始时刻的奇异性质，并给出了其奇异强度的精确估计。该正则性分析为后续设计能够有效处理此类奇异性、基于非均匀时间网格的高效数值格式提供了理论依据。

Abstract: The time-fractional telegraph equations play a crucial role in describing anomalous diffusion processes with hereditary and memory properties. However, the solutions of such equations often exhibit singularities near the initial time, which stems from the non-local nature of fractional differential operators and can significantly affect the accuracy of traditional numerical methods. This paper focuses on studying the singularity of solutions to a class of time-fractional telegraph equations at the initial moment. By employing the Laplace transform method, the solution of the equation is decomposed, and key integral operators are constructed. Through analyzing the asymptotic behavior of these integral operators and their time derivatives of various orders as $t\to 0^{+}$ , it is rigorously proven that the solution itself and its time derivatives up to the third order possess singular properties at the initial time, with precise estimates of the singularity strength provided. This regularity analysis offers a theoretical foundation for the subsequent design of efficient numerical schemes based on non-uniform temporal grids that can effectively handle such singularities.