On the numerical solution of nonlinear fractional-integro differential equations

Abstract

In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), has been employed to give approximate solutions of some nonlinear Fredholm and Volterra type fractional-integro differential equations (FIDEs). Comparing with the exact solution, the PIA produces reliable and accurate results for FIDEs. 

Keywords

• Fractional-integro differential equations,Caputo fractional derivative,Initial value problems

References

1. Aksoy Y. and Pakdemirli M., New perturbation-iteration solutions for Bratu-type equations, Comput Math Appl. 59 (2010), 2802-2808.
2. Aksoy Y., Pakdemirli M., Abbasbandy S. and Boyaci H., New perturbation-iteration solutions for nonlinear heat transfer equations, Int J Heat Fluid Fl. 22 (2012), 814-828.
3. Arikoglu A. and Ozkol I., Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Soliton Fract. 40 (2009), 521-529.
4. Baskonus, H. M. and Bulut H., On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method, Open Math, 13(1) (2015), 547-556.
5. Baskonus, H. M., Mekkaoui, T., Hammouch, Z. and Bulut, H., Active control of a chaotic fractional order economic system, Entropy, 17(8) (2015), 5771-5783.
6. Biala T.A., Afolabi Y.O. and Asim O.O., Laplace variational iteration method for integro-differential equations of fractional order, Int J Pure Appl Math. 95.3 (2014), 413-426.
7. Cavlak E. and Bayram M., An approximate solution of fractional cable equation by homotopy analysis method, Bound. Value Probl., 2014(1), 58.
8. Cooper K. and Mickens R. E., Generalized harmonic balance/numerical method for determining analytical approximations to the periodic solutions of the x^(4/3) potential, J. Sound Vibr. 250 (2002), 951-954.

9. Dolapci İ. T., Şenol M. and Pakdemirli M., New perturbation iteration solutions for Fredholm and Volterra integral equations, J Appl Math. (2013).
10. El-Sayed A., Nour H., Raslan W. and El-Shazly E., A study of projectile motion in a quadratic resistant medium via fractional differential transform method, Appl Math Model. 39.10 (2015), 2829-2835.
11. Esen A. and Tasbozan O., Numerical solution of time fractional nonlinear Schrodinger equation arising in quantum mechanics by cubic B-spline finite elements, Malaya J. Mat., 3(4) (2015), 387-397.
12. Esen A. and Tasbozan O., An approach to time fractional gas dynamics equation: Quadratic B-spline Galerkin method, Appl. Mat. Comput., 261 (2015), 330-336.
13. G. von Groll and Ewins D.J., The harmonic balance method with arc-length continuation in rotor/stator contact problems, J. Sound Vibr. 241 (2001), 223-233.
14. Heaviside O., Electromagnetic theory, Cosimo Inc.2008.
15. He J. H., Iteration Perturbation Method for Strongly Nonlinear Oscillators, J. Sound Vibr. 7 (2001), 631-642.
16. He J. H., Homotopy perturbation method with an auxiliary term, Abst Appl Anal. 2012.
17. Hou J., Qin B. and Yang C., Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid Functions and the Collocation Method, J Appl Math. 2012.
18. Hu H. and Xiong Z.G., Oscillations in an x^((2m+2)/(2n+1)) potential, J. Sound Vibr. 259 (2003), 977-980.
19. İbiş B. and Bayram M., Approximate solution of time-fractional advection-dispersion equation via fractional variational iteration method, The Scientific World Journal, 2014.
20. İbiş B. and Bayram M., Numerical comparison of methods for solving fractional differential–algebraic equations (FDAEs), Comput Math Appl., 62(8) (2011), 3270-3278.
21. Iqbal S and Javed A., Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation, Appl Math and Comput. 217 (2011), 7753-7761.
22. Jordan D.W. and Smith P., Nonlinear ordinary differential equations: An introduction to dynamical systems, Vol. 2, Oxford University Press, USA, (1999).
23. Kurt A. and Tasbozan O., Approximate analytical solution of the time fractional Whitham-Broer-Kaup equation using the Homotopy Analysis Method, International Journal of Pure and Applied Mathematics, 98(4) (2015), 503-510.
24. Luchko Y.F. and Srivastava H.M., The exact solution of certain differential equations of fractional order by using operational calculus, Comput Math Appl. 29.8 (1995), 73-85.
25. Mainardi F., Fractals and fractional calculus in continuum mechanics, Springer Verlag, 1997.
26. Mickens R. E., Iteration procedure for determining approximate solutions to non-linear oscillator equations, J. Sound Vibr. 116 (1987), 185-187.
27. Mickens R. E., A generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators, J. Sound Vibr. 287 (2005), 1045-1051.
28. Mickens R. E., Iteration method solutions for conservative and limit-cycle x^(1/3) force oscillators, J. Sound Vibr. 292 (2006), 964-968.
29. Momani S., Odibat Z. and Erturk V. S., Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation, Phys Lett A. 370 (2007), 379-387.
30. Nayfeh A. H, Perturbation methods, John Wiley & Sons, 2008.
31. Oldham K. B., Fractional differential equations in electrochemistry, Adv Eng Softw. 41 (2010), 9-12.
32. Pakdemirli M., Aksoy Y. and Boyacı H., A New Perturbation-Iteration Approach for First Order Differential Equations, Math Comput Appl. 16 (2011), 890-899.
33. Podlubny I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press, 1998.
34. Senol B., Ates A., Alagoz B.B. and Yeroglu C., A numerical investigation for robust stability of fractional-order uncertain systems, Isa T. 53 (2014), 189-198.
35. Şenol M., Dolapci I. T., Aksoy Y. and Pakdemirli M., Perturbation-Iteration Method for First-Order Differential Equations and Systems, Abstr Appl Anal. 2013.
36. Şenol M. and Dolapci I. T., On the Perturbation-Iteration Algorithm for fractional differential equations, J King Saud Univ Sci. 28.1 (2016), 69-74.
37. Skorokhod A. V., Hoppensteadt F.C. and Salehi H.D., Random perturbation methods with applications in science and engineering, Springer Science & Business Media, 2002.
38. Toyoda M. and Watanabe T., Existence and uniqueness theorem for fractional order differential equations with boundary conditions and two fractional order, J. Nonlinear Convex Anal. 17.2 (2016), 267-273.
39. Wang S. Q. and He J. H., Nonlinear oscillator with discontinuity by parameter-expansion method, Chaos Soliton Fract. 35 (2008), 688-691.
40. Yakar A. and Koksal M. E., Existence results for solutions of nonlinear fractional differential equations, Abstr Appl Anal. 2012.
41. Yu Z.S. and Lin J. Z., Numerical research on the coherent structure in the viscoelastic second-order mixing layers, Appl Math Mech-Engl. 19 (1998), 717-723.