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On study the existing result for the time fractional equation using the topological degree method | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 2، دوره 16، شماره 1، فروردین 2025، صفحه 11-22 اصل مقاله (506.3 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2023.28891.4017 | ||
| نویسندگان | ||
| Chaima Saadi* ؛ Hakim Lakhal؛ Kamel Slimani | ||
| Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS), University 20 August 1955 Skikda, Algeria | ||
| تاریخ دریافت: 13 آبان 1401، تاریخ پذیرش: 25 آذر 1402 | ||
| چکیده | ||
| In this article, we suggest introducing the Riemann-Liouville time fractional derivative to a fractional equation (FPDE) involving a fractional Laplacian. Our work is divided into two parts. In the first part, the existence and uniqueness of time fractional linear equations are demonstrated, and the Galerkin approach is proposed to deal with them. In the second part, we investigate the existence results of the time fractional semilinear equation. To solve This problem, the Leray-Schauder degree method has been used with some conditions on the semilinear term. | ||
| کلیدواژهها | ||
| Time fractional derivative؛ Galerkin method؛ Leray-Schauder degree؛ Riemann-Liouville derivative؛ fractional semilinear parabolic equation | ||
| مراجع | ||
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[1] E. Abada, H. Lakhal, and M. Maouni, Topological, degree method for fractional Laplacian system, Bull. Math. Anal. Appl. 13 (2021), 10–19. [2] E. Abada, H. Lakhal, and M. Maouni, Existence and uniqueness of solution for a nonlinear fractional problem involving the distributional Riesz derivative, Math. Methods Appl. Sci. 45 (2022), 6181–6193. [3] L. Abatangelo, V. Felli, and B. Noris, On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets, Commun. Contemp. Math. 22 (2020), no. 8, 1950071. [4] B. Barrios, E. Colorado, A. de Pablo, and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differ. Equ. 252 (2012), 6133–6162. [5] K. Diethelm and N.J. Ford, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. [6] L. Djilali and A. Rougirel, Galerkin method for time fractional diffusion equations, J. Elliptic Parabol. Equ. 4 (2018), 349–368. [7] S. Dob, H. Lakhal, and M. Maouni, Existence and Uniqueness of solutions for a nonlinear fractional Elliptic system, Malays. J. Math. Sci., 15 (2021), 347–356. [8] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [9] A.-A. Kilbas, H.-M. Srivastava, and J.-J. Trujillo, Theory and Applications of Fractional Differential Equation, Elsevier, Amsterdam, 2006. [10] H. Lakhal, B. Khodja, and W. Gharbi, Existence results of nontrivial solutions for a semi linear elliptic system at resonance, J. Adv. Res. Dyn. Control Syst. 5 (2013), 1–12. [11] A.-M. Nakhushev, Fractional Calculus and its Application, FIZMATLIT, Moscow, 2003. [In Russian] [12] K.-B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [13] Y. Ouedjedi, A. Rougirel, and K. Benmeriem, Galerkin method for time fractional semilinear equations, Fract. Calc. Appl. Anal. 24 (2021),755–774. [14] C. Saadi, H. Lakhal, K. Slimani, and S. Dob, Existence and uniqueness of distributional solution for semilinear fractional elliptic equation involving new operator and some numerical results, Math. Methods Appl. Sci. 45 (2022), 3843–3854. [15] C. Saadi, H. Lakhal, K. Slimani, and S. Dob, Distributional solution for semilinear system involving fractional gradient and a numerical example, Int. J. Nonlinear Anal. Appl. 14 (2023), no. 1, 1175–1185. [16] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102. [17] K. Slimani, C. Saadi, and H. Lakhal, Existence results for convection-reaction fractional problem involving the distributional Riesz derivative, Math. Methods Appl. Sci. 45 (2022), 10247–10255. | ||
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