A 2D Convolution Kernel Determination Problem for the Time-Fractional Diffusion Equation

IF 0.8 Q2 MATHEMATICS
D. K. Durdiev, M. Akylbayev, Zh. Maxumova, A. Iskakova
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引用次数: 0

Abstract

In this article, two dimensional inverse problem of determining convolution kernel in the fractional diffusion equation with the time-fractional Caputo derivative is studied. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation with Riemann–Liouville derivative is constructed. Using the formulas of asymptotic expansions for the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function, which was used for studying the inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. The local existence and global uniqueness results are proven by the aid of fixed point argument in suitable functional classes. Also the stability estimate is obtained.

时间-分数扩散方程的二维卷积核确定问题
摘要 本文研究了具有时间分数卡普托导数的分数扩散方程中确定卷积核的二维反问题。为了表示直接问题的解,构建了具有黎曼-刘维尔导数的时间分数扩散方程的基本解。利用基本解及其导数的渐近展开公式,可以根据未知核函数的规范得到直接问题解的估计值,该估计值用于研究逆问题。逆问题被简化为 Volterra 类型的等价积分方程。借助适当函数类的定点论证,证明了局部存在性和全局唯一性结果。同时还获得了稳定性估计。
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来源期刊
CiteScore
1.50
自引率
42.90%
发文量
127
期刊介绍: Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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