Abstract
In this article, we prove the existence and uniqueness of a solution for 2-dimensional time-fractional differential equations with classical and integral boundary conditions. We start by writing this problem in the operator form and we choose suitable spaces and norms. Then we establish prior estimates from which we deduce the uniqueness of the strong solution. For the existence of solution for the fractional problem, we prove that the range of the operator generated by the considered problem is dense.
Highlights
Many physical phenomena bring us back to the study of fractional partial differential equations
Fractional diffusion equations appear widely in natural phenomena; these are suggested as mathematical models of physical problems in many fields, like the inhomogeneous fractional diffusion equations of the form
The study of existence and uniqueness of a solution for fractional differential equations starts by constructing variational formulation and choosing suitable spaces and norms
Summary
Many physical phenomena bring us back to the study of fractional partial differential equations. The study of existence and uniqueness of a solution for fractional differential equations starts by constructing variational formulation and choosing suitable spaces and norms. For our problem (2.1)–(2.4), we believe that the prior estimate method is the most powerful tool to prove the existence and uniqueness of the solution for fractional differential equation, and is more appropriate with classical and integral boundary conditions. 4, we establish the existence of the solution of problem (2.1)–(2.4), by proving that the closure of the range of the operator L generated by the considered problem is dense in the Hilbert space Y. Lemma 2.1 ([16]) For any absolutely continuous function v(t) on the interval [0, T], the following inequality holds: v(t)∂0βt v(t). Lemma 2.2 ([16]) Let a nonnegative absolutely continuous function y(t) satisfy the inequality.
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