Abstract
In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero whenδtends to zero.
Highlights
The aim of this study is to investigate the final value for the space fractional diffusion equation
The applications of the conformable derivative are interested in various models such as the harmonic oscillator, the damped oscillator, and the forced oscillator, electrical circuits, chaotic systems in dynamics, and quantum mechanics
See, e.g., [5], we must confirm that the study of the ODE problem with the conformable derivative is very different from the study of the PDE problem with a conformable derivative
Summary
We mention the PDEs with conformable derivative where D is a Sobolev space, such as L2ðΩÞ, Wγ,pðΩÞ, and DðAγÞ. When we study the PDE model, we often do with a multivariable function v : ð0, TÞ ⟶ D, where D is a Sobolev space. (i) This paper is the first study on the final value problem for a diffusion equation with a Kirchhoff-type equation and conformable derivative. One of the most difficult points is finding the appropriate functional spaces for the solution (ii) The second result is to investigating the regularized solution for our problem. It can be said that our article is one of the first results, giving a general and comprehensive picture, considering both the frequency and the inaccuracy of Kirchhoff’s diffusion equation with fractional time and space derivative. Convergence error between the regularized solution and the correct solution has been established, with some suitable conditions of input value data
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