Geodesic
Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe method
Nanosistemy: fizika, himiâ, matematika, Tome 15 (2024) no. 1, pp. 5-15
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This paper aims to study the existence and uniqueness of a weak solution for the boundary value problem of a time fractional equation involving the Caputo fractional derivative with an integral operator. By utilizing the discretization method, we first derive some a priori estimates for the approximate solutions at the points $(x, t_j)$. We then evaluate the accuracy of the proposed method to demonstrate that the implemented sequence of $\alpha$-Rothe functions converges in a certain sense, and its limit is the solution (in a weak sense) of our problem. It must be pointed out that the constructed L1 scheme is designed to approximate the Caputo fractional derivative mentioned in the problem.
Keywords: weak solution, a priori estimates, Rothe's method.
Mots-clés : Fractional diffusion equation 
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     title = {Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the {Rothe} method},
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Y. Bekakra; A. Bouziani. Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe method. Nanosistemy: fizika, himiâ, matematika, Tome 15 (2024) no. 1, pp. 5-15. http://geodesic.mathdoc.fr/item/NANO_2024_15_1_a0/