Note on a Allen-Cahn equation with Caputo-Fabrizio derivative

Abstract

In this short note, we investigate the Allen-Cahn equation with the appearance of the Caputo-Fabizzio derivative. We obtain a local solution when the initial value is small enough. This is an equation that has many practical applications. The power term in the nonlinear component of the source function and the Caputo-Fabizzio operator combine to make finding the solution space more difficult than the classical problem. We discovered a new technique, connecting Hilbert scale and $L^p$ spaces, to overcome these difficulties. Evaluation of the smoothness of the solution was also performed. The research ideas in this paper can be used for many other models.

Keywords

• Allen-Cahn equation
• Fractional diffusion equation; Caputo-Fabizzio
• regularity

References

1. [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. [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. [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. [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. [5] J. Manimaran, L. Shangerganesh, A. Debbouche, A time-fractional competition ecological model with cross-di?usion Math. Methods Appl. Sci. 43 (2020), no. 8, 5197-5211.
6. [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. [7] I. Podlubny, Fractional differential equations, Academic Press, London, 1999.
8. [8] B. D. Coleman, W. Noll, Foundations of linear viscoelasticity, Rev. Mod. Phys., 33(2) 239 (1961).

9. [9] 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.
10. [10] X.L. Ding, J.J. Nieto, Analytical solutions for multi-term time-space fractional partial di?erential equations with nonlocal damping terms, Frac. Calc. Appl. Anal. 21 (2018), pp. 312-335.
11. [11] 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
12. [12] H. Afshari, ., Karapinar, 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)
13. [13] 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.
14. [14] B. Alqahtani, H. Aydi, E. Karapinar, V. Rakocevic, A Solution for Volterra Fractional Integral Equations by Hybrid Contractions Mathematics 2019, 7, 694.
15. [15] E. Karapinar, A. Fulga, M. Rashid, L. Shahid, H. Aydi, Large Contractions on Quasi-Metric Spaces with an Application to Nonlinear Fractional Differential-Equations Mathematics 2019, 7, 444.
16. [16] 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).
17. [17] E. Karapinar, T.Abdeljawad, F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Advances in Difference Equations, 2019, 2019:421
18. [18] A. Abdeljawad, R.P. Agarwal, E. Karapinar, P.S. Kumari, Solutions of he Nonlinear Integral Equation and Fractional Di?erential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space Symmetry 2019, 11, 686.
19. [19] N. Hung, H. Binh, N. Luc, A. Nguyen Thi Kieu, L. Long, Stochastic sub-diffusion equation with conformable derivative driven by standard Brownian motion. Advances in the Theory of Nonlinear Analysis and its Application, (2021) 5(3) , 287-299.
20. [20] V. Tri, Existence of an initial value problem for time-fractional Oldroyd-B fluid equation using Banach fixed point theorem. Advances in the Theory of Nonlinear Analysis and its Application, (2021) 5 (4) , 523-530.
21. [21] N.H. Tuan, N.H. Can, R. Wang, Y. Zhou, Initial value problem for fractional Volterra integro-differential equations with Caputo derivative, Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021030.
22. [22] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Di?er. Appl., 1(2) (2015), pp. 1-13.
23. [23] M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Di?er. Appl., 2(2) (2016), pp. 1-11.
24. [24] J. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Di?er. Appl., 1(2) (2015), pp. 87-92.
25. [25] M.S. Hashemi, E. Darvishi, M. Inc, A geometric numerical integration method for solving the Volterra integro-differential equations Int. J. Comput. Math. 95 (2018), no. 8, 1654-1665.
26. [26] T.M. Atanackovic, S. Pillipovi¢, D. Zorica, Properties of the Caputo-Fabrizio fractional derivative and its distributional settings, Fract. Calc. Appl. Anal., 21, (2018), pp. 29-44.
27. [27] F.B. Weissler, Semilinear evolution equations in Banach spaces J. Functional Analysis 32 (1979), no. 3, 277-296.
28. [28] Q. Du, J. Yang, Z. Zhou, Time-fractional Allen-Cahn equations: analysis and numerical methods J. Sci. Comput. 85 (2020), no. 2, Paper No. 42, 30 pp.
29. [29] S.M. Allen, J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (6) (1979) 1085-1095