On the numerical solution of nonlinear fractional-integro differential equations

Yıl 2017, Cilt: 5 Sayı: 3, 118 - 127, 01.07.2017

Mehmet Senol , Hamed Daei Kasmaei

Öz

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. 

Anahtar Kelimeler

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

Kaynakça

• Aksoy Y. and Pakdemirli M., New perturbation-iteration solutions for Bratu-type equations, Comput Math Appl. 59 (2010), 2802-2808.
• 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.
• Arikoglu A. and Ozkol I., Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Soliton Fract. 40 (2009), 521-529.
• 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.
• 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.
• 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.
• Cavlak E. and Bayram M., An approximate solution of fractional cable equation by homotopy analysis method, Bound. Value Probl., 2014(1), 58.
• 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.
• Dolapci İ. T., Şenol M. and Pakdemirli M., New perturbation iteration solutions for Fredholm and Volterra integral equations, J Appl Math. (2013).
• 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.
• 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.
• 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.
• 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.
• Heaviside O., Electromagnetic theory, Cosimo Inc.2008.
• He J. H., Iteration Perturbation Method for Strongly Nonlinear Oscillators, J. Sound Vibr. 7 (2001), 631-642.
• He J. H., Homotopy perturbation method with an auxiliary term, Abst Appl Anal. 2012.
• 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.
• Hu H. and Xiong Z.G., Oscillations in an x^((2m+2)/(2n+1)) potential, J. Sound Vibr. 259 (2003), 977-980.
• İbiş B. and Bayram M., Approximate solution of time-fractional advection-dispersion equation via fractional variational iteration method, The Scientific World Journal, 2014.
• İbiş B. and Bayram M., Numerical comparison of methods for solving fractional differential–algebraic equations (FDAEs), Comput Math Appl., 62(8) (2011), 3270-3278.
• 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.
• Jordan D.W. and Smith P., Nonlinear ordinary differential equations: An introduction to dynamical systems, Vol. 2, Oxford University Press, USA, (1999).
• 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.
• 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.
• Mainardi F., Fractals and fractional calculus in continuum mechanics, Springer Verlag, 1997.
• Mickens R. E., Iteration procedure for determining approximate solutions to non-linear oscillator equations, J. Sound Vibr. 116 (1987), 185-187.
• Mickens R. E., A generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators, J. Sound Vibr. 287 (2005), 1045-1051.
• Mickens R. E., Iteration method solutions for conservative and limit-cycle x^(1/3) force oscillators, J. Sound Vibr. 292 (2006), 964-968.
• 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.
• Nayfeh A. H, Perturbation methods, John Wiley & Sons, 2008.
• Oldham K. B., Fractional differential equations in electrochemistry, Adv Eng Softw. 41 (2010), 9-12.
• Pakdemirli M., Aksoy Y. and Boyacı H., A New Perturbation-Iteration Approach for First Order Differential Equations, Math Comput Appl. 16 (2011), 890-899.
• 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.
• 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.
• Şenol M., Dolapci I. T., Aksoy Y. and Pakdemirli M., Perturbation-Iteration Method for First-Order Differential Equations and Systems, Abstr Appl Anal. 2013.
• 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.
• Skorokhod A. V., Hoppensteadt F.C. and Salehi H.D., Random perturbation methods with applications in science and engineering, Springer Science & Business Media, 2002.
• 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.
• Wang S. Q. and He J. H., Nonlinear oscillator with discontinuity by parameter-expansion method, Chaos Soliton Fract. 35 (2008), 688-691.
• Yakar A. and Koksal M. E., Existence results for solutions of nonlinear fractional differential equations, Abstr Appl Anal. 2012.
• 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.