Mathematics and Statistics Vol. 10(5), pp. 1116 - 1120
DOI: 10.13189/ms.2022.100522
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On a Weak Solution of a Fractional-order Temporal Equation


Iqbal M. Batiha 1,2,*, Zainouba Chebana 3, Taki-Eddine Oussaeif 3, Adel Ouannas 3, Iqbal H. Jebril 1
1 Mathematics Department, Al Zaytoonah University of Jordan, Queen Alia Airport St 594, Amman 11733, Jordan
2 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
3 Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi, Algeria

ABSTRACT

Several real-world phenomena emerging in engineering and science fields can be described successfully by developing certain models using fractional-order partial differential equations. The exact, analytical, semi-analytical or even numerical solutions for these models should be examined and investigated by distinguishing between their solvablities and non-solvabilities. In this paper, we aim to establish some sufficient conditions for exploring the existence and uniqueness of solution for a class of initial-boundary value problems with Dirichlet condition. The gained results from this research paper are established for the class of fractional-order partial differential equations by a method based on Lax Milgram theorem, which relies in its construction on properties of the symmetric part of the bilinear form. Lax Milgram theorem is deemed as a mathematical scheme that can be used to examine the existence and uniqueness of weak solutions for fractional-order partial differential equations. These equations are formulated here in view of the Caputo fractional-order derivative operator, which its inverse operator is the Riemann-Louville fractional-order integral one. The results of this paper will be supportive for mathematical analyzers and researchers when a fractional-order partial differential equation is handled in terms of finding its exact, analytical, semi-analytical or numerical solution.

KEYWORDS
Fractional Partial Differential Equation, Lax Milgram Theorem, Existence, Uniqueness

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Iqbal M. Batiha , Zainouba Chebana , Taki-Eddine Oussaeif , Adel Ouannas , Iqbal H. Jebril , "On a Weak Solution of a Fractional-order Temporal Equation," Mathematics and Statistics, Vol. 10, No. 5, pp. 1116 - 1120, 2022. DOI: 10.13189/ms.2022.100522.

(b). APA Format:
Iqbal M. Batiha , Zainouba Chebana , Taki-Eddine Oussaeif , Adel Ouannas , Iqbal H. Jebril (2022). On a Weak Solution of a Fractional-order Temporal Equation. Mathematics and Statistics, 10(5), 1116 - 1120. DOI: 10.13189/ms.2022.100522.