SPLINE SOLUTIONS OF LINEAR FRACTIONAL BVPS WITH TWO CAPUTOS APPROACHES

Year 2018, Volume: 8 Issue: 2, 399 - 410, 01.12.2018

H. Tariq G. Akram

Abstract

In this paper, an ecient numerical methods based on cubic polynomial spline func- tions are proposed for the linear fractional boundary value problems FBVPs with Caputos left and right fractional operator. In computing the approximation to the solutions of FBVPs, consistency relations have been derived with the help of spline functions. For convergence analysis of this method, it is assumed that the exact solu- tion of FBVP belongs to a class of C6-functions. Numerical examples are considered to illustrate the accuracy and eciency of this method and compare the results with other methods developed by Akram and Tariq in [18] and Zahra and Elkholy in [28-30].

Keywords

Linear FBVPs, Cubic Spline Function, Caputos Fractional Operators, Error Bound.

References

• Ciesielski, M., Leszczynski, J., (2003), Numerical simulations of anomalous diffusion, In: Computer Methods Mech, Conference Gliwice Wisla Poland.
• Metzler, R., Klafter, J., (2000) The random walks guide to anomalous diffusion: a fractional dynamics aproach, Physics Reports, 339, pp.1-77.
• Moghaddam, B. P., Mostaghim, Z. S., (2013), A numerical method based on finite differ- ence for solving fractional delay differential equations, Journal of Taibah University for Science, 7, pp.120-127.
• Aleroev, T. S.,(1982), The Sturm-Loiuville Problem for a Second Order Ordinary Differential Equation with Fractional Derivatives in the Lower Terms (in Russian), Differentialnye Uravneniya, 18(2), pp.341-342.
• Jafari H., Daftardar-Gejji, V., (2006), Positive Solutions of Nonlinear Fractional Bound- ary Value Problems using Adomian Decomposition Method, Applied Mathematics and Computation, 180, pp.700-706.
• Odibat, Z. M., Momani, S., (2006), Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order, International Journal Nonlinear Sciences and Numerical Simulation, 7, pp.27- 34.
• Abu Arqub, O., El-Ajou, A., Momani, S., (2015), Constructing and predicting solitary pat- tern solutions for nonlinear time-fractional dispersive partial differential equations, J. Comput. Phys. 293, pp.385399.
• Abu Arqub, O.,El-Ajou, A. , Bataineh, A., Hashim, I., (2013), A representation of the exact solution of generalized Lane-Emden equations using a new analytical method, Abstr. Appl. Anal. Volume 2013, Article ID 378593, 10 pages.
• El-Ajou, A., Abu Arqub, O., Momani, S., (2015), Approximate analytical solution of the non- linear fractional KdV-Burgers equation: A new iterative algorithm, Journal of Com- putational Physics 293, pp.81-95.
• El-Ajou, A., Abu Arqub, O., Momani, S., Baleanu, D., Alsaedi, A., (2015), A novel expansion iterative method for solving linear partial differential equations of fractional order, Applied Mathematics and Compu- tation 257, pp.119-133.
• Abu Arqub, O., Maayah, B., (2016), Solutions of Bagley-Torvik and Painlev equations of frac- tional order using iterative reproducing kernel algorithm, Neural Computing and Ap- plications, 2016. DOI 10.1007/s00521- 016-2484-4.
• Ersoy, O., Korkmaz, A., Dag, I., (2016), Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves, Mediterranean Journal of Mathematics, 13(6), pp.4975-4994.
• Siddiqi, S. S., Arshed, S., (2015), Numerical solution of time-fractional fourth-order partial differential equa- tions, Int. J. Comput. Math. 92(7), pp.14961518.
• Korkmaz, A., Dag, I., (2016), Quartic and quintic B-spline methods for advection-diffusion equa- tion, Applied Mathematics and Computation, 274, pp.208219.
• Siddiqi, S. S., Akram, G., Nazeer, S., (2007), Quintic Spline Solution of Linear Sixth-Order Boundary Value Problems, International Journal of Computer Mathematics, 84(3), pp.347-368.
• Siddiqi S. S., Akram, G., (2008), Solution of eighth-order boundary value problems using the non-polynomial spline technique, Journal of Computational and Applied Mathematics, 215, pp.288-301.
• Akram G., Siddiqi, S. S., (2007), Solution of Tenth-Order Boundary Value Problems using Eleventh Degree Spline, Applied Mathematics and Computation, 185, pp.115-127.
• G. Akram, H. Tariq, (2016), An Exponential Spline Technique for Solving Fractional Boundary Value Prob- lem, Calcolo, 53(4), pp. 545-558.
• G. Akram, H. Tariq, (2017), Cubic polynomial Spline Scheme for Fractional Boundary Value Problems with Left and Right Fractional Operators, International Journal of Applied and Computational Mathematics, 3(2), pp.937-946.
• Akram, G., Tariq, H., (2017), Quintic spline collocation method for fractional boundary value problems, Journal of the Association of Arab Universities for Basic and Applied Sci- ences, 23, pp.57-65
• Tariq, H. , Akram, G., (2017), Quintic spline technique for time fractional fourth-order partial differential equation. Numer. Methods Partial Differential Eq., 33(2), pp.445-466.
• Ahmad, B., Nieto, J. J., (2009), Existence of solutions for nonlocal boundary value problems of higher- order nonlinear fractional differential equations, Abstr. Appl. Anal., vol. 2009, Article ID 494720, pages doi:10.1155/2009/494720.
• Shuqin, Z., (2006), Existence of solution for boundary value problem of fractional order, Acta Math. Sci. 26, pp.220228.
• Bai, Z., (2010), On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. 72, pp.916924.
• Taukenova, F. I., Shkhanukov-Lafishev, M. Kh., (2006), Difference methods for solving boundary value problems for fractional differential equations, Comput. Math. Math. Phys, 46, pp.1785-1795.
• Chen, W., Sun, H., Zhang, X., Korosak, D., (2010), Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl, 59, pp.1754-1758.
• Su,X., Liu, L., (2007), Existence of solution for boundary value problem of nonlinear fractional differential equation, Appl. Math. J. Chinese Univ. Ser. B 22 (3), pp.291-298.
• Zahra, W. K., Elkholy, S. M., (2012), Quadratic spline solution for boundary value problem of fractional order, Numerical Algorithms, 59, pp.373-391.
• Zahra, W. K., Elkholy, S. M., (2013), Cubic Spline Solution of Fractional Bagley-Torvik Equation, Electronic Journal of Mathematical Analysis and Applications, 1(2), pp.230-241.
• Zahra, W. K., Elkholy, S. M., (2012), The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations, International Journal of Mathematics and Mathematical Sciences, Volume 2012, Article ID 638026, 16 pages.
• Podlubny, I., (1999), Fractional Differential Equation, Academic, San Diego.
• Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., (2006) Theory of Application of Fractional Differential Equations, 1st edn. Belarus.
• Kosmatov, N., (2009), Integral equations and initial value problems for nonlinear differential, Nonlinear Analysis, 70, pp.25212529.
• Miller K. S., Ross, B., (1993), An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York.
• Usmani, R.A., (1978), Discrete variable methods for a boundary value problem with engineering applications, Math. Comput. 32, pp.1087-1096.
• Ramadan, M.A., Lashien, I.F., Zahra, W.K., (2007), Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems, Applied Mathematics and computation, 184(2), pp.476484.