Semilinear parabolic diffusion systems on the sphere with Caputo-Fabrizio operator

Yıl 2022, Cilt: 6 Sayı: 2, 148 - 156, 30.06.2022

Tran Binh

https://doi.org/10.31197/atnaa.1012869

Cited By: 1

Öz

PDEs on spheres have many important applications in physical phenomena, oceanography and meteorology, geophysics. In this paper, we study the parabolic systems with Caputo-Fabrizio derivative. In order to establish the existence of the mild solution, we use the Banach fixed point theorem and some analysis of Fourier series associated with several evaluations of the spherical harmonics function. Some of the techniques on upper and lower bounds of the Mittag-Lefler functions are also applied. This is one of the first research results on the systems of parabolic diffusion on the sphere.

Anahtar Kelimeler

parabolic systems; Banach fixed point theory, regularity., parabolic systems; Banach fixed point theory, regularity.

Kaynakça

• [1] 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
• [2] H. Afshari, E, Karapınar, A discussion on the existence of positive solutions of the boundary value problems via-Hilfer fractional derivative on b-metric spaces, Advances in Difference Equations volume 2020, Article number: 616 (2020)
• [3] H.Afshari, S. Kalantari, E. Karapinar; Solution of fractional differential equations via coupled fixed point, Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 286, pp. 1-12
• [4] B.Alqahtani, H. Aydi, E. Karapınar, V. Rakocevic, A Solution for Volterra Fractional Integral Equations by Hybrid Contractions Mathematics 2019, 7, 694.
• [5] E. Karapinar, A.Fulga,M. Rashid, L.Shahid, H. Aydi, Large Contractions on Quasi-Metric Spaces with an Application to Nonlinear Fractional DifferentialEquations Mathematics 2019, 7, 444.
• [6] A.Salim, B. Benchohra, E. Karapinar, J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations Adv Differ Equ 2020, 601 (2020)
• [7] E. Karapinar; T.Abdeljawad; F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Advances in Difference Equations, 2019, 2019:421
• [8] A.Abdeljawad, R.P. Agarwal, E. Karapinar, P.S.Kumari, Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space Symmetry 2019, 11, 686.
• [9] 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.
• [10] 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.
• [11] Q.T. Le Gia Approximation of parabolic PDEs on spheres using collocation method, Adv. Comput. Math., 22 (2005), 377–397.
• [12] Q.T. L. Gia, N.H. Tuan, T. Tran, Solving the backward heat equation on the unit sphere ANZIAM J. (E) 56 (2016), pp. C262–C278.
• [13] Q.T. Le Gia Approximation of parabolic PDEs on spheres using collocation method, Adv. Comput. Math., 22 (2005), 377–397.
• [14] Q.T. Le Gia Galerkin approximation of elliptic PDEs on spheres, J. Approx. Theory , 130 (2004), 125–149.
• [15] Q.T. Le Gia, I.H. Sloan, T. Tran, Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere Math. Comp. 78 (2009), no. 265, 79–101
• [16] Z. Brze´zniak, B. Goldys, Q.T. Le Gia, Random attractors for the stochastic Navier-Stokes equations on the 2D unit sphere J. Math. Fluid Mech. 20 (2018), no. 1, 227–253.
• [17] N.D. Phuong, N. H. Luc, Note on a Nonlocal Pseudo-Parabolic Equation on the Unit Sphere, Dynamic Systems and Applications 30 (2021) No.2, 295–304.
• [18] N.H. Luc, H. Jafari, P. Kumam, N.H. Tuan, On an initial value problem for time fractional pseudo-parabolic equation with Caputo derivarive, Mathematical Methods in the Applied Sciences, to appear.
• [19] O. Nikan, H. Jafari, A. Golbabai, Numerical analysis of the fractional evolution model for heat flow in materials with memory Alexandria Engineering Journal 59 (4), 2627–2637
• [20] R. M. Ganji, H. Jafari, S. Nemati, A new approach for solving integro-differential equations of variable order, Journal of Computational and Applied Mathematics 379, 1–13
• [21] H. Jafari, H. Tajadodi, R.M. Ganji, A numerical approach for solving variable order differential equations based on Bernstein polynomials Comput. Math. Methods 1 (2019), no. 5, e1055, 11 pp
• [22] Caputo M, Fabrizio M, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1(2) (2015), pp. 1–13.
• [23] M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2(2) (2016), pp. 1–11.
• [24] J. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1(2) (2015), pp. 87–92.
• [25] Caputo M, Fabrizio M, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1(2) (2015), pp. 1–13.
• [26] M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2(2) (2016), pp. 1–11.
• [27] J. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1(2) (2015), pp. 87–92.
• [28] T.M. Atanackovi´c, S. Pillipovi´c, D. Zorica, Properties of the Caputo-Fabrizio fractional derivative and its distributional settings, Fract. Calc. Appl. Anal., 21, (2018), pp. 29–44.
• [29] V. Gafiychuk and B. Datsko, Stability analysis and oscillatory structures in timefractional reaction-diffusion systems Phys. Rev. E 75 (2007), article 055201(R).
• [30] V. Gafiychuk, B. Datsko, and V. Meleshko, Mathematical modeling of timefractional reaction-diffusion systems J. Comput. Appl. Math. 220 (2008), 215–225.