In this article, we propose a hybrid method based on a priori estimate and Fourier method to prove the existence and uniqueness of the solution of the linear one dimensional partial differential equation in the sense of Caputo-Fabrizio fractional derivative with the non-boundary condition. The functional analysis and Fourier methods are used. It is important to know that a priori estimate established in a nonclassical function space is a necessary tool to prove the uniqueness and dependence to continue with a strong solution of the studied problem.

fractional equation, non-boundary conditions, Fourier’s method, a priori estimate.

Received: September 21, 2021; Accepted: October 22, 2021; Published: November 5, 2021

How to cite this article: Moussa Zakari Djibibe, Bangan Soampa and Kokou Tcharie, On solvability of an evolution mixed problem for a certain parabolic fractional equation with weighted integral boundary conditions in Sobolev function spaces, Universal Journal of Mathematics and Mathematical Sciences 14(2) (2021), 107-119. DOI: 10.17654/UM014020107

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License 

References:

[1] B. Ahmad and J. Nieto, Existence results for nonlinear boundary value problems of fractional integro-differential equations with integral boundary conditions, Boundary Value Problems Vol. 2009, Article ID 708576, 11 pages.
[2] A. Samarkii, Some problems in differential equations theory, Differents. Uranv. 16(11) (1980), 1925-1935.
[3] A. Anguraj and P. Karthikeyan, Existence of solutions for fractional semilinear evolution boundary value problem, Commun. Appl. Anal. 14 (2010), 505-514.
[4] M. Belmekki and M. Benchohra, Existence results for fractional order semilinear functional differential equations, Proc. A. Razmadze Math. Inst. 146 (2008), 920.
[5] M. Benchohra, J. R. Graef and S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal. 87 (2008), 851-863.
[6] A. Bouziani and N. E. Benouar, Mixed problem with integral conditions for a third order parabolic equation, Kobe J. Math. 15(1) (1998), 47-58.
[7] A. Bouziani and N. E. Benouar, Probléme mixte avec conditions intgrales pour une classe déquations paraboliques, comptes rendus de l’Academie des Sciences, Paris t. 321, Srie I (1995), 1177-1182.
[8] A. Bouziani, Mixed problem for certain nonclassical equations with a small parameter, Bulletin de la Classe des Sciences, Acadmie Royale de Belgique 5 (1994), 389-400.
[9] A. Bouziani, Mixed problem with integral conditions for a certain parabolic equation, J. App. Math. and Stoch. Anal. 9 (1996), 323-330.
[10] Moussa Zakari Djibibe and Ahcene Merad, On solvability of the third pseudo-parabolic fractional equation with purely nonlocal conditions, Advances in Differential Equations and Control Processes 23(1) (2020), 87-104.
[11] M. Z. Djibibe, K. Tcharie and N. I. Yurchuk, Existence, Uniqueness and continuous dependance of solution of nonlocal boundary conditions of mixed problem for singular parabolic equation in nonclassical function spaces, Pioneer J. Adv. in Appl. Math. 7(1) (2013), 7-16.
[12] M. Z. Djibibe and K. Tcharie, On the solvability of an evolution problem with weighted integral boundary conditions in Sobolev function spaces with a priori estimate and Fouriers method, British J. Math. Comput. Sci. 3(4) (2013), 801-810.
[13] M. Z. Djibibe, K. Tcharie and N. I. Yurchuk, Continuous dependence of solutions to mixed boundary value problems for a parabolic equation, Electronic Journal of Differential Equations 2008(17) (2008), 1-10.
[14] N. J. Ford, J. Xiao and Y. Yan, A finite element method for time fractional partial differential equations, Fractional Calculus and Appl. Anal. 14(3) (2011), 454-474. doi : 10.2478/s13540-011-0028-2.
[15] J. H. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering 98, Dalian, China, 1998, pp. 288-291.
[16] J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15(1999), 86-90.
[17] X. J. Li and C. J. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Comm. Comput. Phys. 8(5) (2010), 1016-1051.
[18] D. Kaya and I. E. Inan, A numerical application of the decomposition method for the combined KdV-MKdV equation, Appl. Math. Comput. 168 (2005), 915-926.
[19] S. Mesloub, A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation, J. Math. Anal. Appl. 316 (2006), 189-209.
[20] A. Kiliman and H. E. Gadain, On the applications of Laplace and Sumudu transforms, J. Frankl. Inst. 347(5) (2010), 848-862.
[21] F. Dubois, A. C. Galucio and N. Point, Introduction à la Dérivée Fractionnaire: Théorie et Applications, 2010.
[22] V. E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theor. Math. Phys. 158(3) (2009), 355-359.
[23] A. Atangana and S. C. Oukouomi Noutchie, On multi-Laplace transform for solving nonlinear partial differential equations with mixed derivatives, Math. Probl. Eng. Volume 2014, Article ID 267843, 9 pages.
[24] H. Eltayeb and A. Kiliman, A note on solutions of wave, Laplaces and heat equations with convolution terms by using double Laplace transform, Appl. Math. Lett. 21(12) (2008), 1324-1329.
[25] S. Mesloub, A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation, J. Math. Anal. Appl. 316 (2006), 189-209.
[26] Bangan Soampa and Moussa Zakari Djibibe, Mixed problem with an pure integral two-space-variables condition for a third order fractional parabolic equation, MJM 8(1) (2020), 258-271.
[27] Bangan Soampa, Moussa Zakari Djibibe and Kokou Tcharie, Analytical approximation solution of pseudo-parabolic Fractional equation using a modified double Laplace decomposition method, Theo. Math. Appl. 10(1) (2020), 17-31.