MIXED-TYPE DIFFERENTIAL EQUATION WITH FRACTIONAL DERIVATIVE IN THE CAPUTO SENSE - Alpysbayeva , A.E.; Murzambetova , M.B.; Ryskan, A. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2025)
The purpose of this paper is to study boundary value problems for a mixed-type differential equation with fractional derivatives in the sense of Caputo and to demonstrate the existence and uniqueness of its solution. Such equations, including fractional derivatives, have significant potential to describe various physical processes in which the effects of memory and heredity are evident, such as abnormal diffusion, heat transfer, and relaxation phenomena. The paper presents an analytical approach to solving the problem based on the method of separating variables by representing the solution as a Fourier series. As a result, the conditions for the uniqueness of the solution were established and strictly proved, which, if certain conditions are met, ensure the ambiguity of the task. Additionally, the uniform convergence of the obtained series of solutions is demonstrated under the specified conditions. The results obtained can be used in the theory of differential equations and further applied research.