Solving linear and nonlinear fractional differential equations using spline functions. (English) Zbl 1235.65015
Summary: Suitable spline functions of polynomial form are derived and used to solve linear and nonlinear fractional differential equations. The proposed method is applicable for \(0 < \alpha \leq 1\) and \(\alpha \geq 1\), where \(\alpha\) denotes the order of the fractional derivative in the Caputo sense. The results obtained are in good agreement with the exact analytical solutions and the numerical results presented elsewhere. Results also show that the technique introduced here is robust and easy to apply.
MSC:
| 65D07 | Numerical computation using splines |
| 65L99 | Numerical methods for ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
Keywords:
spline functions; linear fractional differential equations; nonlinear fractional differential equationsReferences:
| [1] | J. S. Leszczynski, An Itorduction to Fractional Mechanics, Czestochowa University of Technology, Cz\cestochowa, Poland, 2011. |
| [2] | K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons Inc., New York, NY, USA, 1993. · Zbl 0789.26002 |
| [3] | A. Oustaloup, La Derivation Non Entiere, Hermes, Paris, France, 1995. |
| [4] | J. A. T. Machado, “Analysis and design of fractional-order digital control systems,” Systems Analysis Modelling Simulation, vol. 27, no. 2-3, pp. 107-122, 1997. · Zbl 0875.93154 |
| [5] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 |
| [6] | J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140-1153, 2011. · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027 |
| [7] | K. Deithelm, N. J. Ford, et al., “Analysis of fractional differential equation,” Numerical Analysis Report 377, The University of Manchester, 2003. |
| [8] | R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1-10, 2007. · Zbl 1123.34302 · doi:10.1016/j.jmaa.2006.12.036 |
| [9] | I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press Inc., San Diego, Calif, USA, 1999. · Zbl 0924.34008 |
| [10] | K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York. NY, USA, 1974. · Zbl 0292.26011 |
| [11] | S. Momani, “Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 459-472, 2005. · Zbl 1070.65105 · doi:10.1016/j.amc.2004.06.025 |
| [12] | S. Momani, Z. Odibat, and A. Alawneh, “Variational iteration method for solving the space- and time-fractional KdV equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 1, pp. 262-271, 2008. · Zbl 1130.65132 · doi:10.1002/num.20247 |
| [13] | Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,” Computers & Mathematics with Applications. An International Journal, vol. 58, no. 11-12, pp. 2199-2208, 2009. · Zbl 1189.65254 · doi:10.1016/j.camwa.2009.03.009 |
| [14] | K. Deithelm and A. D. Freed, “The FracPECE subroutine for the numerical solution of differential equations of fractions order,” in Forschung und Wiessenschaftliches Rechnen 1998, S. Heinzel and T. Plesser, Eds., no. 52, pp. 57-71, GWDG-Bericht, Gesellschaftfur Wiessenschaftliches Datenverabeitung, Göttingen, Germany, 1999. |
| [15] | K. Deithelm and A. D. Freed, “On the of solution of nonlinear differential equations used in the modeling of viscoplasticity,” in Scientific Computing in chemical Engineering-II. Computational Fluid Dynamics Reaction Engineering, and Molecular Properties, F. Keil, W. Mackens, H. Vob, and J. Werther, Eds., pp. 217-224, Springer, Heidelberg, Gemany, 1999. |
| [16] | K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional order,” Electronic Transactions on Numerical Analysis, vol. 5, pp. 1-6, 1997. · Zbl 0890.65071 |
| [17] | N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26-33, 2007. · Zbl 1209.65116 · doi:10.1016/j.physleta.2007.06.016 |
| [18] | G. Micula, T. Fawzy, and Z. Ramadan, “A polynomial spline approximation method for solving system of ordinary differential equations,” Babes-Bolyai Cluj-Napoca. Mathematica, vol. 32, no. 4, pp. 55-60, 1987. · Zbl 0652.65054 |
| [19] | M. A. Ramadan, “Spline solutions of first order delay differential equations,” Journal of the Egyptian Mathematical Society, vol. 13, no. 1, pp. 7-18, 2005. · Zbl 1084.34069 |
| [20] | M. A. Ramadan, T. S. El-Danaf, and M. N. Sherif, “Numerical solution of fractional differential eqautions using polynomial spline functions,” submitted. |
| [21] | Z. Ramadan, “On the numerical solution of a system of third order ordinary differential equations by spline functions,” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, vol. 19, pp. 155-167, 2000. · Zbl 0981.65077 |
| [22] | J. Munkhammar, “Riemann-Liouville fractional derivatives and the Taylor-Riemann series,” U.U.D.M. Project Report 7, Department of Mathematics, Uppsala University, 2004. |
| [23] | S. Momani, O. K. Jaradat, and R. Ibrahim, “Numerical approximations of a dynamic system containing fractional derivatives,” Journal of Applied Sciences, vol. 8, no. 6, pp. 1079-1084, 2008. |
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