Using operational matrix for numerical solution of fractional differential equations

Authors
f mirzaee
e. hadadian nejad yosofi
10.29252/mmr.2.2.65
Abstract
In this article, we have discussed a new application of modification of hat functions on nonlinear multi-order fractional differential equations. The operational matrix of fractional integration is derived and used to transform the main equation to a system of algebraic equations. The method provides the solution in the form of a rapidly convergent series. Furthermore, error analysis of the proposed method is provided under several mild conditions. Three numerical examples are given to show the efficiency and accuracy of the method. Illustrative examples are included to demonstrate the validity, efficiency, and applicability of the method.
Keywords
Modification of hat functions
Fractional differential equations
Operational matrices
Fractional calculus
Caputo derivative
1. Diethelm K., Ford N.J., "Analysis of fractional differential equations", J. Math. Anal. Appl., 265 (2002) 229-248.

2. Diethelm K., Ford N.J., "Multi-order fractional differential equations and their numerical solution", Appl. Math. Comput., 154 (2004) 621-640.

3. Kiryakova V., "Generalized fractional calculus and applications", in: Pitman Res. Notes Math. Ser., 301, Longman-Wiley, New York (1994).

4. Podlubny I., "Fractional differential equations", Academic Press, San Diego (1999).

5. Samko S.G., Kilbas A.A., Marichev O.I., "Fractional integrals and derivatives", Theory and Applications, Gordon and Breach, Yverdon (1993).

6. Podlubny I., "Fractional differential equations: an introduction to fractional derivatives", fractional differential equations, to methods of their solution and some of their applications. New York: Academic Press (1999).

7. Gaul L., Klein P., Kemple S., "Damping description involving fractional operators", Mech. Syst. Signal. Pr., 5 (1991) 81-88.

8. Suarez L., Shokooh A., "An eigenvector expansion method for the solution of motion containing fractional derivatives", J. Appl. Mech., 64 (1997) 629-635.

9. Momani S., "An algorithm for solving the fractional convection-diffusion equation with nonlinear source term", Commun. Nonlinear Sci. Numer. Simul., 12 (7) (2007) 1283-1290.

10. Jafari H., Seifi S., "Solving a system of nonlinear fractional partial differential equations using homotopy analysis method", Commun. Nonlinear Sci. Numer. Simul., 14 (5) (2009) 1962-1969.

11. Sweilam N.H., Khader M.M., Al-Bar R.F., "Numerical studies for a multi-order fractional differential equation", Phys. Lett. A, 371(12) (2007) 26-33.

12. Das S., "Analytical solution of a fractional diffusion equation by variational iteration method, Comput", Math. Appl., 57 (3) (2009) 483-437.

13. Arikoglu A., Ozkol I., "Solution of fractional integro-differential equations by using fractional differential transform method", Chaos Solitons Fract., 40 (2) (2009) 521-529.

14. Erturk V.S., Momani S., Odibat Z., "Application of generalized differential transform method to multi-order fractional differential equations", Commun. Nonlinear Sci. Numer. Simul., 13 (8) (2008) 1642-1654.

15. Meerschaert M., Tadjeran C., "Finite difference approximations for two-sided space-fractional partial differential equations", Appl. Numer. Math., 56 (1) (2006) 80-90.

16. Pandey R.K., Singh O.P., Baranwal V.K., "An analytic algorithm for the space-time fractional advection-dispersion equation", Comput. Phys. Commun., 182 (2011) 1134-1144.

17. Saadatmandi A., Dehghan M., "A new operational matrix for solving fractional order differential equations", Comput. Math. Appl., 59 (2010) 1326-1336.

18. Odibat Z., Momani S., Erturk V.S., "Generalized differential transform method: application to differential equations of fractional order", Appl. Math. Comput., 197 (2008) 467-477.

19. Baranwal V.K., Pandey R.K., Tripathi M.P., Singh O.P., "Analytic solution of fractional-order heat and wave-like equations using generalized n-dimensional differential transform method", Z. Naturforsch, 66a (2011) 581-590.

20. Cuesta E., Lubich Ch., Palencia C., "Convolution quadrature time discretization of fractional diffusion-wave equation", Math. Comput., 75 (2006) 673-696.

21. Tripathi M.P., Baranwa V.K., Pandey R.K., Singh O.P., "A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun", Nonlinear Sci. Numer. Simulat., 18 (2013) 1327-1340.

22. Odibat Z., Shawagfeh N., "Generalized Taylors formula", Appl. Math. Comput., 186 (1) (2007) 286-293.

23. Li Y., "Solving a nonlinear fractional differential equation using Chebyshev wavelets", Commun. Nonlinear Sci. Numer, Simulat., 15 (2010) 2284-2292.

24. Arikoglu A., Ozkol I., "Solution of fractional differential equations by using differential transform method", Chao. Solitons Fract., 34 (2007) 1473-1481.

25. Rehman M.u., Ali Khan R., "The Legendre wavelet method for solving fractional differential equations", Commun, Nonlinear Sci. Numer, Simulat., 16 (2011) 4163-4173.

26. Li Y., Zhao W., "Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations", Appl. Math. Comput., 216 (2010) 2276-2285.

27. Li Y., Sun N., "Numerical solution of fractional differential equations using the generalized block pulse operational matrix", Comput. Math. Appl., 62 (2011) 1046-1054.

28. Wu J.L., "A wavelet operational method for solving fractional partial differential equations numerically", Appl. Math. Comput., 214 (1) (1009) 31-40.

29. Lepik Ü., "Solving fractional integral equations by the Haar wavelet method", Appl. Math. Comput., 214 (2) (2009) 468-478.

30. Delbosco D., Rodino L., "Existence and Uniqueness for a Nonlinear Fractional Differential Equation", J. Math. Anal. Appl., 204 (1996) 609-625.

31. Tenreiro Machado J.A., "Fractional derivatives: probability interpretation and frequency response of rational approximations", Commun. Nonlinear Sci. Numer. Simul., 14 (9-10) (2009) 3492-3497.

32. Atkinson K.E., "The numerical solution of integral equations of the second kind", Cambridge University Press, Cambridge (1997).

33. Mirzaee F., Hadadiyan E., "Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions", Appl. Math. Comput., 250 (2015) 805-816.

34. Mirzaee F., Hadadiyan E., "Approximation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions", J. Comput. Appl. Math., 302 (2016) 272–284.

35. Momani S., Odibat Z., "Numerical comparison of methods for solving linear differential equations of fractional order", Chao. Solitons Fract., 31 (2007) 1248-1255.

36. Odibat Z., Momani Sh., "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order", Chaos Solitons Fractals, 36 (2008) 167-174.

37. Khan N.A., Jamil M., Ara A., Das S., "Explicit solution of time-fractional batch reactor system", Int. J. Chem. React. Eng., 9 (2011) Article ID A91.

38. F-Batlle V., Perez R., Rodriguez L., "Fractional robust control of main irrigation canals with variable dynamic parameters", Control Eng. Pract., 15 (2007) 673-686.

39. Podlubny I., "Fractional-Order Systems and Controllers", IEEE Trans. Auto. Control, 44 (1999) 208-214.

40. Garrappa R., "On some explicit Adams multistep methods for fractional differential equations", J. Comput. Appl. Math., 229 (2009) 392-399.

41. Jamil M., Khan N.A., "Slip effects on fractional viscoelastic fluids", Int. J. Differ. Equ., 2011 (2011) Article ID 193813.

42. Mohammadi F., Hosseini M.M., "A comparative study of numerical methods for solving quadratic Riccati differential equation", J. Frank. Inst., 348 (2) (2011) 156-164.