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Existence and continuity results for Kirchhoff parabolic equation with Caputo–Fabrizio operator
Existence and continuity results for Kirchhoff parabolic equation with Caputo–Fabrizio operator
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Date
2023
Authors
Nguyen Huy Tuan
Anh Tuan Nguyen
Nguyen Huu Can
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Research Projects
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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."
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Keywords
"Caputo–Fabrizio operator,
Kirchhoff equation,
Banach fixed point theory,
Sobolev embeddings"