A New Iterative Algorithm for the Time-Fractional Fisher Equation Including Small Delay

Year 2022, Volume: 10 Issue: 1, 1 - 10, 15.04.2022

Ali Demir , Mine Aylin Bayrak

Abstract

This study aims to establish a numerical solution of time fractional Fisher equation with small delay by utilizing residual power series method (RPSM). First of all, replacing the term including small delay by in Taylor series expansion of it, we reduce the problem into a fractional Fisher equation without delay. Secondly, applying RPSM, the coefficients of the series are determined which converges to the solution of the equation rapidly. Effectiveness and accuracy of this algorithm are illustrated by presented examples.

Keywords

Fisher equation, Fractional delay differential equation, Residual power series method

References

• [1] R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol. 27 (1983), 201–210.
• [2] R. L. Bagley and P. J. Torvik, Fractional calculus a different approach to the analysis of viscoelastically damped structures, AIAA J. 21(5) (1983), 741–748.
• [3] R. L. Bagley and P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J. 23 (1985), 918–925.
• [4] R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng. 32 (2004), 1–104.
• [5] D. A. Robinson,The use of control systems analysis in neurophysiology of eye movements, Ann. Rev. Neurosci. 4 (1981), 462–503. [6] I. Podlubny, Fractional Differential equations, Academic Press, San Diego, 1999.
• [7] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam, 2006.
• [8] V. Daftardar-Gejji and A. Babakhani, Analysis of a system of fractional differential equations, J. Math. Anal. Appl. 293(2)(2004), 511–522.
• [9] Z. Odibat, Approximations of fractional integrals and Caputo fractional derivatives, Appl. Math. Comput. 178(2)(2006), 527–533.
• [10] J. Sabatier, O. P. Agrawal and JAT Machado (eds), Advances in fractional calculus: theoretical developments and applications in physics and engineering. Dordrecht: Springer, 2007.
• [11] S. G. Samko , A. A Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Amsterdam: Gordon and Breach, 1993.
• [12] D. Baleanu ,K. Diethelm and E. Scalas, et al., Fractional: calculus models and numerical methods (complexity, nonlinearity, and chaos). Boston, MA: World Scientific, 2012.
• [13] M. Yavuz, T. A. Sulaiman, F. Usta and H. Bulut, Analysis and Numerical Computations of the Fractional Regularized Long Wave Equation with Damping Term, Math. Meth. Appl. Sci. In Press, https://doi.org/10.1002/mma.6343.
• [14] T. A. Sulaiman, M. Yavuz, H. Bulut and H. M. Baskonus, Investigation of the fractional coupled viscous Burgers’ equation involving Mittag-Leffler kernel, Physica A. 527 (2019),121-126.
• [15] F. Usta, Numerical Analysis of Fractional Volterra Integral Equations via Bernstein Approximation Method, J. Comput.d Appl. Math. 384 (2021),113198, https://doi.org/10.1016/j.cam.2020.113198.
• [16] F. Usta and M. Z. Sarıkaya, The Analytical Solution of Van der Pol and Lienard Differential Equations within Conformable Fractional Operator by Retarded Integral Inequalities, Demonstratio Mathematica, 52(1)(2019), 204–212.
• [17] F. Usta, Numerical solution of fractional elliptic PDE’s by the collocation method, Applications Appl. Math.: An Intern. J., 12(1) (2017), 470-478.
• [18] O. Abu Arqub , Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math. 5 (2013), 31–52.
• [19] O. Abu Arqub , A. El-Ajou and A. Bataineh, et al., A representation of the exact solution of generalized Lane Emden equations using a new analytical method, Abstr. Appl. Anal. 2013(2013), 378593.
• [20] A. El-Ajou , O. Abu Arqub and Z. Al Zhour et al., New results on fractional power series: theories and applications, Entropy 16 (2013), 5305–5323.
• [21] O. Abu Arqub , A. El-Ajou , Z. Al Zhour , et al., Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique, Entropy 16 (2014), 471–493.
• [22] O. Abu Arqub , A. El-Ajou and S. Momani , Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, J. Comput. Phys. 293 (2015), 385–399.
• [23] A. El-Ajou , O. Abu Arqub and S. Momani , Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: a new iterative algorithm, J. Comput. Phys. 293 (2015), 81–95.
• [24] M. Alquran , K. Al-Khaled and J. Chattopadhyay , Analytical solutions of fractional population diffusion model: residual power series, Nonlin Stud 22 (2015), 31–39
• [25] A. El-Ajou , O. Abu Arqub , S. Momani , et al., A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Comput. 257 (2015), 119–133.
• [26] R. A.Fisher, The wave of advance of advantageous genes, Annu. Eugen. 7 (1937), 355-369.
• [27] I. C. Sungu and H. Demir , New approach and solution technique to solve time fractional nonlinear reaction-diffusion equations, Math. Probl. Eng. 2015 (2015), 457013.
• [28] S. Z. Rida , A.M.A El-Sayed and A.A.M. Arafa, On the solutions of time-fractional reaction–diffusion equations, Commun. Nonlinear Sci. 15 (2010), 3847–3854.
• [29] N.A. Khan , F. Ayaz , L. Jin , et al., On approximate solutions for the time-fractional reaction-diffusion equation of Fisher type, Int. J. Phys. Sci. 6 (2011), 2483–2496.
• [30] V. K. Baranwal , R. K. Pandey , M. P. Tripathi , et al., An analytic algorithm for time fractional nonlinear reaction–diffusion equation based on a new iterative method, Commun. Nonlinear Sci. 17 (2012), 3906–3921.
• [31] M. Merdan , Solutions of time-fractional reaction– diffusion equation with modified Riemann-Liouville derivative, Int. J. Phys. Sci. 7 (2012), 2317–2326.
• [32] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary order, Academic Press, California, 1974.
• [33] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
• [34] S. S. Ray, Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 1130–1295.
• [35] R. Magin , X. Feng and D. Baleanu , Solving fractional order Bloch equation, Concept magnetic Res. 34A (2009), 16-23.
• [36] H. Poincar´e, Sur les int´egrales irr´eguli`eres, Acta mathematica 8(1)(1886), 295-344.
• [37] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables (Vol. 55, p. 319). Washington, DC: National bureau of standards, 1886.
• [38] W. Eckhaus, Matched asymptotic expansions and singular perturbations, Elsevier, 2011.
• [39] F. Say, Late-order terms of second order ODEs in terms of pre-factors, Hacettepe J. Math. Statistics, 50(2) (2021), 342-350.
• [40] M. Van Dyke, Higher approximations in boundary-layer theory part 3. parabola in uniform stream, J. Fluid Mech. 19(1) (1964), 145-159.
• [41] E. J. Hinch, Perturbation Methods,Cambridge University Press, 1991.
• [42] M. M. Al Qurashi , Z. Korpnar , D. Baleanu and M. Inc, A new iterative algorithm on the time-fractional Fisher equation: Residual power series method, Adv. Mech. Eng. 9(9) (2017), 1-8.