A collocation method for solving boundary value problems of fractional order

Yıl 2018, , 1601 - 1608, 01.12.2018

Sertan Alkan , Aydin Seçer

https://doi.org/10.16984/saufenbilder.352088

Cited By: 2

Öz

In this paper, the sinc collocation method is used to obtain the solution of the second-order fractional boundary value problems based on the conformable fractional derivative. For this purpose a theorem is proved to represent the terms having fractional derivatives in terms of sinc basis functions. To show the efficiency and accuracy of the present method, some problems are solved and the obtained solutions are compared with the approximate solutions obtained by using the other numerical methods as well as the exact solutions of the problems.

Anahtar Kelimeler

Differential equations of fractional order, boundary value problems

Kesirli mertebeden sınır değer problemlerini çözmek için bir sıralama yöntemi

Öz

Bu makalede sinc sıralama yöntemi, uyumlu kesirli türev içeren ikinci mertebeden kesirli sınır değer problemlerinin çözümünü elde etmek için kullanıldı. Bu amaçla sinc baz fonksiyonlarının kesirli türevlerini içeren terimleri ifade etmek için bir teorem ispat edildi. Yöntemin etkinliğini ve doğruluğunu göstermek için bazı problemler çözüldü ve elde edilen çözümler diğer sayısal yöntemler kullanılarak elde edilen yaklaşık çözümler ve problemlerin tam çözümleri ile karşılaştırıldı.

Anahtar Kelimeler

Kesirli mertebeden diferansiyel denklemler

Kaynakça

• S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993.
• K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, NewYork: Wiley, 1993.
• K. B. Oldham, J. Spanier, The fractional calculus, Academic Press, NewYork and London, 1974.
• I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
• R. Herrmann, Fractional Calculus:An Introduction for Physicists, World Scientific, Singapore, 2014.
• J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007.
• S. Alkan, V. Hatipoglu, Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order, Tbilisi Mathematical Journal, (2017),10(2), pp. 1-13.
• R. P. Meilanov, R. A. Magomedov, Thermodynamics in Fractional Calculus, Journal of Engineering Physics and Thermophysics, (2014), 87(6):1521-1531.
• V. F. Hatipoglu, , S. Alkan, A. Secer, An efficient scheme for solving a system of fractional differential equations with boundary conditions, Advances in Difference Equations, 2017.1 (2017): 204.
• Khalil, R., Al Horani, M., Yousef, A. and Sababeh, M. A new definition of fractional derivative, J. Comput. Appl. Math., (2014), 264:65-70.
• Thabet Abdeljawad, On the conformable fractional calculus, Journal of Computational and Applied Mathematics, (2015), 279:57-66.
• N. Benkhettou, S. Hassani, D.F.M. Torres, A conformable fractional calculus on arbitrary time scales, J. King Saud Univ.-Sci., (2016), 28(1):93-98.
• W.S. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math., (2015), 290:150-158.
• H. Batarfi, J. Losada, J.J. Nieto, W. Shammakh, Three-point boundary value problems for conformable fractional differential equations, J. Funct. Space, (2015), 6.
• E. Hesameddini, E. Asadollahifard, Numerical solution of multi-order fractional differential equations via the sinc collocation method, Iranian Journal Of Numerical Analysis And Optimization, (2015), 5(1):37-48.
• G. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett. A., (2010), 374:2506-2509.
• V. Daftardar-Gejji, H. Jafari, Solving a multi-order fractional differential equation using adomian decomposition, Appl. Math. Comput., (2007), 189:541-548.
• O. Abdulaziz, I. Hashim, S. Momani, Solving systems of fractional differential equations by homotopy-perturbation method, Phys. Lett. A., (2008), 372:451-459.
• I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., (2009), 14:674-684.
• M. U. Rehman, R. A. Khan, A numerical method for solving boundary value problems for fractional differential equations, Appl. Math. Model., (2012), 36(3):894-907.
• F. Stenger, Approximations via Whittaker's cardinal function, J. Approx. Theory, (1976), 17(3):222-240.
• F. Stenger, A sinc-Galerkin method of solution of boundary value problems, Math. Comput., (1979), 33(145):85-109.
• E. T. Whittaker, On the functions which are represented by the expansions of the interpolation theory, Proc. R. Soc. Edinb., (1915), 35:181-194.
• J. M. Whittaker, Interpolation Function Theory, Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, London, 1935.
• S. Alkan, A new solution method for nonlinear fractional integro-differential equations, DCDS-S, (2015), 8(6):1065-1077.
• S. Alkan, A. Secer, Solution of nonlinear fractional boundary value problems with nonhomogeneous boundary conditions, Appl. Comput. Math.,(2015), 14(3):284-295.
• Y. Wang, H. Song, D. Li, Solving two-point boundary value problems using combined homotopy perturbation method and Greens function method, Appl. Math. Comput., (2009), 212(2): 366-376.