Well-posed results for nonlocal fractional parabolic equation involving Caputo-Fabrizio operator

Volume 26, Issue 4, pp 357--367 http://dx.doi.org/10.22436/jmcs.026.04.04

Publication Date: January 06, 2022 Submission Date: September 18, 2021 Revision Date: October 19, 2021 Accteptance Date: November 05, 2021

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Authors

T. T. Phong - Division of Applied Mathematic, Thu Dau Mot University, Binh Duong province, Viet Nam. L. D. Long - Division of Applied Mathematic, Thu Dau Mot University, Binh Duong province, Viet Nam.

Abstract

In this paper, we study the parabolic problem associated with non-local conditions, with the Caputo-Fabrizio derivative. Equations on the sphere have many important applications in physics, phenomena, and oceanography. The main motivation for us to study non-local boundary value problems comes from two main reasons: the first reason is that current major interest in several application areas. The second reason is to study approximation for the terminal value problem. With some given data, we prove that the problem has only the solution for two cases. In case \(\epsilon = 0,\) we prove the problem has a local solution. In case \(\epsilon > 0,\) then the problem has a global solution. The main tools and techniques in our demonstration are of using Banach's fixed point theorem in conjunction with some Fourier series analysis involved some evaluation of spherical harmonic function. Several upper and lower upper limit techniques for the Mittag-Lefler functions are also applied.

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Keywords

• Nonlocal parabolic equation
• Banach fixed point theory
• sphere
• regularity

MSC

•  35R11
•  35B65
•  26A33

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