Existence and continuity results for Kirchhoff parabolic equation with Caputo–Fabrizio operator

Abstract

This paper deals with the solution of the Kirchhoff parabolic equation involving the Caputo–Fabrizio fractional derivative with non-singular kernel. We represent the mild solution by equation operators which are defined via Fourier series. The main results of this work are about the existence, uniqueness and the continuity property with respect to the derivative order of the mild solution. The result of the existence of a solution consists of two parts: The existence of a local solution with a nonlinear source function. The existence of a global solution in the case of a linear source function. To achieve global results, we had to introduce a new space with norm contains weighted function. The key analysis is based on the Banach fixed point theorem and some Sobolev embeddings. The order-continuity problem has two main results: The first result proves the continuity related to order α∈(0,1). The remaining results examine the continuity when the fractional order is approaching 1−. Finally, we illustrate the achieved theoretical results by some numerical examples.