A numerical method based on finite difference scheme for solving fractional Cable equation
Subject Areas : StatisticsLeyla Azami 1 , Amir Hosein Refahi Sheikhani 2 , Hashem Saberi Najafi 3
1 - Department of Applied Mathematics, Faculty of Mathematics Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran
2 - Department of Applied Mathematics,Faculty of Mathematical Sciences,Lahidjan Branch,Islamic Azad University,Lahijan,Iran
3 - Department of Applied Mathematics, Faculty of Mathematics Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Keywords: پتانسیل غشا, همگرایی, پایداری, آنالیز فوریه, مشتق کسری کاپوتو-فابریزیو,
Abstract :
In this paper, we proposed a numerical approximation to solve a type of fractional Cable equations based on the finite difference scheme that can be easily generalized to other fractional cable equations. The fractional derivatives are considered in the Caputo-Fabrizio type. By using the forward difference formula, the fractional derivative is approximated and then we can discretize the fractional equation with the mentioned approximation and difference scheme In order to demonstrate the stability and convergence, several Theorems and Lemmas are provided. Finally, two numerical examples of the cable equation with different boundary and initial conditions are used to demonstrate the effective and accuracy of the scheme. The obtained numerical results are compared with the analytical solution. Figures presented in both examples show consistency of numerical and analytical solutions that can be clearly seen. Tables also show a small error between the numerical and analytical solution which show that our proposed method is highly accurate.
[1] R. Ozarslan, E. Bas, D. Baleanu, B. Acay, Fractional physical problems including wind-influenced projectile motion with Mittag-Leffler kernel, J. AIMS Math, 5(1), 2019, 467-481.
[2] M. A. Firoozjaee, H. Jafari, A. Lia, D. Baleanu, Numerical approach of Fokker- Planck equation 1with Caputo-Fabrizio derivative using Ritz, J. Com. App. Math 339, 2018, 367-373.
[3] H. Aminikhah, A. H. Refahi Sheikhani, H. Rezazadeh, Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives, J. Scien. Iran, 23(3), 2016, 1048-1054.
[4] H. Aminikhah, A. H. Refahi Sheikhani, H. Rezazadeh, Exact solution of some nonlinear systems of partial differential equations by using the functional variable method, J. Math, 56(79), 2014, 103-116.
[5] H. Aminikhah, A. H. Refahi Sheikhani, H. Rezazadeh, Stability analysis of distributed order fractional chen system, J. Scien.World, 2013, 1-13.
[6] P. Gholamin, A. H. Refahi Sheikhani, A new three-dimensional chaotic system: Dynamical properties and simulation, J. Phy, 55(4), 2017, 1300-1309.
[7] I. Karatay, N. Kale, A new difference scheme for fractional cable equation and stability analysis, J. App. Math. Res. 4 (1), 2015, 52-57.
[8] S. Kumar, J. F. Gomez-Aguilar, Numerical solution of Caputo-Fabrizio time fractional distributed Order reaction-diffusion equation via Quasi Wavelet based numerical method, J. App. Comp. Mecha. 6(4), 2020, 848-861.
[9] K. Moaddy, I. Hashim, S. Momani, Non-standard difference scheme for solving fractional-order Rossler chaotic and hyperchaotic systems, J. Com. Math. Appl, 62, 2011, 1068-1074.
[10] K. Owolabi, Riemann-Liouville fractional derivative and application to model chaotic differential equations, J. Frac. Diff. Appl, 4(2), 2018, 99-1
[11] R. Almeida, A. B. Malinowska, T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, J. Math. Meth. Appl. Scien, 2018, 1-20.
[12] H. Aminikhah, A. Refahi Sheikhani, H. Rezazadeh, Exact solutions for the fractional differential equations by using the first integral method, J. Non. Eng, 4(1), 2015, 15-22.
[13] S. I. Muslih, O. P. Agrawal, Riesz Fractional Derivatives and Fractional Dimensional Space, J. Theo. Phys, 49, 2010, 270-275.
[14] S. Zeng, D. Baleanu, Y. Bai, G. Wu, Fractional differential equations of Caputo-Katugampola type and numerical solutions, J. App. Math. Comp, 315, 2017, 549-554.
[15] A. Atangana, K. M. Owolabi, New numerical approach for fractional differential equations, J. Math. Na, 2018, 1-21.
[16] M. Mashoof, A. H. Refahi Sheikhani, H. Saberi Najafi, Stability analysis of distributed- order Hilfer-Prabhakar systems based on inertia theory, J. Math. Notes, 104(1-2), 2018, 74-85.
[17] H. Khan, R. Shah, P. Kumam, D. Baleanu, M. Arif: An efficient analytical technique, for the solution of fractional-order telegram equations, J. Mathematics, 2019, 1-19.
[18] R. Shah, H. Khan, M. Arif, P. Kumam, Application of Laplace-Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations, J. Entropy, 2019, 1-17 .
[19] M. Modanli, A. Akgul, Numerical solution of fractional telegram differential equations by theta-method, J. Eur. Phy. Spec. Top, 226, 2017, 3693-3703.
[20] S. Yadav, R. K. Pandey, A. K. Shukla, Numerical approximations of Atangana-Baleanu Caputo derivative and its application, J. Cha. Soli. Fract, 118, 2019, 58-64.
[21] M. Modanli, Two numerical methods for fractional partial differential equation with nonlocal boundary value problem, J. Adv. Diff. equ, 2018, 1-19.
[22] A. Atangana, R. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, J. Adv. Diff. equ, 2016, 1-13.
[23] A. Atangana, J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, J. Adv. Mech. Eng. 7(10), 2015, 1-7.
[24] Y. Zhou, Z. Luo, A Crank–Nicolson finite difference scheme for the Riesz space fractional-order parabolic-type sine-Gordon equation, J. Ad. Diff. Equ, 2018, 1-7.
[25] U. Ali, F. A. Abdullah, A. I. Smail, Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation, J. Inter. Appr. Scien. Comp, 2, 2017, 18-29.
[26] S. M. Oishi, J. Y. Yuan, J. A. Cuminato, D. E. Stewart, Stability analysis of Crank–Nicolson and Euler schemes for time-dependent diffusion equations, J. Num. Math, 55(2), 2015, 487-513.
[27] I. Karatay, N. Kale, Finite Difference Method of Fractional Parabolic Partial Differential Equations with Variable Coefficients. J. Cont. Math. Scien, 9(16), 2014, 767-776.
[28] I. Karatay, N. Kale, S. Bayramoglu, A new difference scheme for time fractional heat equations based on the Crank-Nicholson method. J. Theo. App, 16, 2013, 892-910.
[29] Z. Liu, A. Cheng, X. Li, A fast-high order compact difference method for the fractional Cable equation, J. Num. Metho. Par. Diff. equ, 2018, 1-3
[30] W. Rall, Core conductor theory and Cable properties of neurons, in: R. Poeter(Ed), Handbook of Physiology: The Nervous System, American Physiological Society, Bethesda, MD, 1977, 39-97, (Chapter 3).
[31] S.Vitali, F. Mainardi, G. C. Castellani, Time fractional Cable equation and applications in neurophysiology, J. Chaos Soli. Fract, 102, 2017, 467-472.
[32] C. M. Chen, F. Liu, K. Brrage, Numerical analysis for a variable-order nonlinear Cable equation, J. Com. Appl. Math, 236, 2011, 209-224.
[33] R. Khanbabaee, M. Tabesh, Introduction to neurophysics. J. Physics Research, 15(4), 2016, 347-371.
