Note on a time fractional diffusion equation with time dependent variables coefficients

Yıl 2021, , 600 - 610, 30.12.2021

Le Dinh Long

https://doi.org/10.31197/atnaa.972116

Cited By: 3

Öz

In this short paper, we study time fractional diffusion equations with time-dependent coefficients. The derivative operator that appears in the main equation is Riemann-Liouville. The main purpose of the paper is to prove the existence of a global solution. Due to the nonlocality of the derivative operator, we cannot represent the solution directly when the coefficient depends on time. Using some new transformations and techniques, we investigate the global solution. This paper can be considered as one of the first results on the topic related to problems with time-dependent coefficients. Our main tool is to apply Fourier analysis method and combine with some estimates of Mittag-Lefler functions and some Sobolev embeddings.

Anahtar Kelimeler

Fractional diffusion equation; Riemman-Liouville, Fractional diffusion equatio, regularity

Destekleyen Kurum

Thu Dau Mot University

Kaynakça

• [1] N.H. Tuan, Y. Zhou, T.N. Thach, N.H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data Commun. Nonlinear Sci. Numer. Simul. 78 (2019), 104873, 18 pp.
• [2] N.H. Tuan, L.N. Huynh, T.B. Ngoc, Y. Zhou, On a backward problem for nonlinear fractional diffusion equations Appl. Math. Lett. 92 (2019), 76-84.
• [3] T.B. Ngoc, Y. Zhou, D. O'Regan, N.H. Tuan, On a terminal value problem for pseudoparabolic equations involving Riemann-Liouville fractional derivatives, Appl. Math. Lett. 106 (2020), 106373, 9 pp.
• [4] J. Manimaran, L. Shangerganesh, A. Debbouche, Finite element error analysis of a time-fractional nonlocal diffusion equation with the Dirichlet energy, J. Comput. Appl. Math. 382 (2021), 113066, 11 pp
• [5] J. Manimaran, L. Shangerganesh, A. Debbouche, A time-fractional competition ecological model with cross-diffusion Math. Methods Appl. Sci. 43 (2020), no. 8, 5197-5211
• [6] N.H. Tuan, A. Debbouche, T.B. Ngoc, Existence and regularity of final value problems for time fractional wave equations Comput. Math. Appl. 78 (2019), no. 5, 1396-1414.
• [7] N.H. Tuan, T. Caraballo, On initial and terminal value problems for fractional nonclassical diffusion equations Proc. Amer. Math. Soc. 149 (2021), no. 1, 143-161.
• [8] T. Caraballo, T.B. Ngoc, N.H. Tuan, R. Wang, On a nonlinear Volterra integrodifferential equation involving fractional derivative with Mittag-Leffer kernel Proc. Amer. Math. Soc. 149 (2021), no. 08, 3317-3334.
• [9] I. Podlubny, Fractional differential equations, Academic Press, London, 1999.
• [10] B. D. Coleman, W. Noll, Foundations of linear viscoelasticity, Rev. Mod. Phys., 33(2) 239 (1961).
• [11] P. Clément, J. A. Nohel, Asymptotic behavior of solutions of nonlinear volterra equations with completely positive kernels, SIAM J. Math. Anal., 12(4) (1981), pp. 514-535.
• [12] X.L. Ding, J.J. Nieto, Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms, Frac. Calc. Appl. Anal. 21 (2018), pp. 312-335.
• [13] L.C.F. Ferreira, E.J. Villamizar-Roa, Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differ. Integral Equ., 19(12) (2006), pp. 1349-1370.
• [14] T. Jankowski, Fractional equations of Volterra type involving a Riemann-Liouville derivative Appl. Math. Lett. 26 (2013), no. 3, 344-350.
• [15] X. Wanga, L. Wanga, Q. Zeng, Fractional differential equations with integral boundary conditions, J. Nonlinear Sci. Appl. 8 (2015), 309-314.
• [16] C. Zhai, R. Jiang, Unique solutions for a new coupled system of fractional differential equations Adv. Difference Equ. 2018, Paper No. 1, 12 pp.
• [17] D. del-Castillo-Negrete, B. A. Carreras, V. E. Lynch; Nondiffusive transport in plasma turbulene: A fractional diffusion approach, Phys. Rev. Lett., 94 (2005), 065003.
• [18] S. Kou, Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins, Ann. Appl. Stat., 2 (2008), 501-535.
• [19] R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Star. Sol. B, 133 (1986), 425-430.
• [20] K. Sakamoto, M. Yamamoto, Initial value/boudary value problems for fractional diffusion- wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.
• [21] F.S. Bachir, S. Abbas, M. Benbachir, M. Benchohra, Hilfer-Hadamard Fractional Differential Equations, Existence and Attractivity, Advances in the Theory of Nonlinear Analysis and its Application, 2021, Vol 5 , Issue 1, Pages 49-57.
• [22] A. Salim, M. Benchohra, J. Lazreg, J. Henderson, Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Non-Instantaneous Impulses in Banach Spaces , Advances in the Theory of Nonlinear Analysis and its Application, Vol 4 , Issue 4, Pages 332-348, 2020.
• [23] Z. Baitichea, C. Derbazia, M. Benchohrab, ψ-Caputo Fractional Differential Equations with Multi-point Boundary Con- ditions by Topological Degree Theory, Results in Nonlinear Analysis 3 (2020) No. 4, 167-178
• [24] Y. Chen, H. Gao, M. Garrido-Atienza, B. Schmalfuÿ, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), pp. 79-98.
• [25] J.E. Lazreg, S. Abbas, M. Benchohra, and E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces , Open Mathematics 2021; 19: 363-372, https://doi.org/10.1515/math-2021-0040
• [26] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On The Solutions Of Fractional Differential Equations Via Geraghty Type Hybrid Contractions, Appl. Comput. Math., V.20, N.2, 2021,313-333
• [27] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences, https://doi.org/10.1002/mma.665
• [28] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions , RACSAM (2021) 115:155; https://doi.org/10.1007/ s13398-021-01095-3
• [29] Z. Baitiche, C. Derbazi, M. Benchohra, (2020). ψ-Caputo fractional di?erential equations with multi-point boundary conditions by Topological Degree Theory . Results in Nonlinear Analysis ,Volume 3, Issue 4, , (2020): 167-178.
• [30] A. Ardjouni , A. Djoudi, Existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional di?erential equations . Results in Nonlinear Analysis , 2 (3) (2019): 136-142.
• [31] S. Redhwan, S. Shaikh, M. Abdo, Some properties of Sadik transform and its applications of fractional-order dynamical systems in control theory, Advances in the Theory of Nonlinear Analysis and its Application , 4 (1) , (2020): 51-66.
• [32] T.B. Ngoc, V.V. Tri, Z. Hammouch, N.H. Can, Stability of a class of problems for timespace fractional pseudo-parabolic equation with datum measured at terminal time, Applied Numerical Mathematics, 167, (2021): 308-329.
• [33] E. Karapinar, H.D. Binh, N.L. Luc, N.H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Adv. Di?erence Equ., 70, 26 pp.
• [34] J. Patil, A. Chaudhari, A. Mohammed, B. Hardan, Upper and lower solution method for positive solution of generalized Caputo fractional di?erential equations. Advances in the Theory of Nonlinear Analysis and its Application, 4(4), 2020; 279-291.
• [35] S. Muthaiah, M. Murugesan, and N.G. Thangaraj, Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations. Advances in the Theory of Nonlinear Analysis and its Application, 3(3), 2019; pp.162-173.
• [36] E. Karapinar, H.D. Binh, N.H. Luc, and N.H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Advances in Di?erence Equations 2021, no. 1, (2021): 1-24.
• [37] H. Afshari, E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces, Advances in Di?erence Equations, 2020(1); 1-11.
• [38] H. Afshari, S. Kalantari, E. Karapinar, Solution of fractional differential equations via coupled fixed point, Electron. J.Differ. Equ, 286, No. 286, 2015; pp. 1-12.