Kesirli mertebeden integro-diferansiyel denklemlerin çözümü için sayısal bir yöntem

Bu çalışmada, sinc sıralama yöntemi kesirli mertebeden Volterra integro-diferansiyel denklemleri yaklaşık olarak çözmek için geliştirilmiştir. Kesirli türev, kesirli analizde sıkça kullanılan Caputo anlamında tanımlanmıştır. Elde edilen sonuçlar iki yeni teorem ile verilmiştir. Bazı sayısal örnekleri teorik sonuçları göstermek için sunulmuştur.

A numerical method for solution of integro-differential equations of fractional order

In this study, sinc-collocation method is introduced for solving Volterra integro-differential equations of fractionalorder. Fractional derivative is described in the Caputo sense often used in fractional calculus. Obtained results aregiven to literature as two new theorems. Some numerical examples are presented to demonstrate the theoretical results.

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  • Momani, S., Noor, M. A. (2006). Numerical methods for fourth-order fractional integrodifferential equations. Applied Mathematics and Computation, 182(1), 754-760.
  • Momani, S., Qaralleh, R. (2006). An e cient method for solving systems of fractional integrodifferential equations. Computers & Mathematics with Applications, 52(3), 459-470.
  • Huang, L., Li, X. F., Zhao, Y., Duan, X. Y. (2011). Approximate solution of fractional integro-differential equations by Taylor expansion method. Computers & Mathematics with Applications, 62(3), 1127-1134.
  • Nazari, D., Shahmorad, S. (2010). Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions. Journal of Compu-tational and Applied Mathematics, 234(3), 883-891.
  • Arikoglu, A., Ozkol, I. (2009). Solution of fractional integro-differential equations by using fractional differential transform method. Chaos, Solitons & Fractals, 40(2), 521-529.
  • Saeed, R. K., Sdeq, H. M., (2010). Solving a system of linear fredholm fractional integrodifferential equations using homotopy perturbation method. Australian Journal of Basic and Applied Sciences, 4(4), 633-638.
  • Nawaz, Y. (2011). Variational iteration method and homotopy perturbation method for fourthorder fractional integro-differential equations. Computers & Mathematics with Applications, 61(8), 2330-2341.
  • Alkan, S., (2015).A new solution method for nonlinear fractional integro-differential equations, Dis-crete and Continuous Dynamical Systems - Series S, 8(6), 1065-1077.
  • Alkan, S., Yildirim, K., Secer, A. (2016). An e cient algorithm for solving fractional differential equations with boundary conditions, Open Physics, 14(1), 6-14.
  • Alkan, S., Secer, A. (2015). Solution of nonlinear fractional boundary value problems with nonhomo-geneous boundary conditions, Applied and Computational Mathematics, 14(3),284-295.
  • Secer, A., Alkan, S., Akinlar, M. A., Bayram, M. (2013). Sinc-Galerkin method for approximate solutions of fractional order boundary value problems. Boundary Value Problems, 2013(1), 281.
  • Zarebnia, M., Nikpour, Z. (2009). Solution of linear Volterra integro-differential equations via Sinc functions. International Journal of Applied Mathematics and Computation, 2(1), 001-010.
  • Rawashdeh, E. A. (2011). Legendre wavelets method for fractional integro-differential equations. Ap-plied Mathematical Sciences, 5(2), 2467-2474.
  • Ma, X., Huang, C. (2014). Spectral collocation method for linear fractional integro-differential equa-tions. Applied Mathematical Modelling, 38(4), 1434-1448.
  • Zhu, L., Fan, Q. (2012). Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Communications in Nonlinear Science and Numerical Simulation, 17(6), 2333-2341.
  • Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198. Academic press, 1998.
  • Lund, J., Bowers, K. L., Sinc methods for quadrature and differential equations, SIAM, 1992.
  • Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional integrals and derivatives. Theory and Appli-cations, Gordon and Breach, Yverdon, 1993.
  • Stenger, F., Handbook of Sinc numerical methods, CRC Press, 2010.
  • Zhao, J., Xiao, J., Ford, N. J. (2014). Collocation methods for fractional integro-differential equations with weakly singular kernels. Numerical Algorithms, 65(4), 723-743.
  • Rawashdeh, E. A. (2006). Numerical solution of fractional integro-differential equations by collocation method. Applied Mathematics and Computation, 176(1), 1-6.
  • Aziz, I., Fayyaz, M. (2013). A new approach for numerical solution of integro-differential equations via Haar wavelets. International Journal of Computer Mathematics, 90(9), 1971-1989.
  • Mittal, R. C., Nigam, R. (2008). Solution of fractional integro-differential equations by Adomian decomposition method. The International Journal of Applied Mathematics and Mechanics, 4(2), 87-94.
  • Sweilam, N. H., Khader, M. M. (2010). A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations. The ANZIAM Journal, 51(04), 464-475.
  • Abbasbandy, S., Hashemi, M. S., Hashim, I. (2013). On convergence of homotopy analysis method and its application to fractional integrodifferential equations. Quaestiones Mathematicae, 36(1), 93-105.
  • Zhang, X., Tang, B., He, Y. (2011). Homotopy analysis method for higher-order fractional integro-differential equations. Computers & Mathematics with Applications, 62(8), 3194- 3203.
  • Nawaz, Y. (2011). Variational iteration method and homotopy perturbation method for fourthorder fractional integro-differential equations. Computers & Mathematics with Applications, 61(8), 2330-2341.
  • Khader, M. M., Sweilam, N. H. (2013). On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method. Applied Mathematical Modelling, 37(24), 9819-9828.
  • Yuanlu, L. (2010). Solving a nonlinear fractional di erential equation using Chebyshev wavelets. Com-munications in Nonlinear Science and Numerical Simulation, 15(9), 2284-2292.
  • Atangana, A., Koca, I. (2016). On the new fractional derivative and application to nonlinear Baggs and Freedman model. J. Nonlinear Sci. Appl., 9(2016), 2467-2480.
  • Koca, I., Atangana, A. (2016). Analysis of a nonlinear model of interpersonal relationships with time fractional derivative. Journal of Mathematical Analysis, 7(2), 1-11.
  • Atangana, A. (2016). On the new fractional derivative and application to nonlinear Fishers reaction-di usion equation. Applied Mathematics and Computation, 273, 948-956.
  • Atangana, A., Koca, I. (2016). Chaos in a simple nonlinear system with Atangana Baleanu derivatives with fractional order. Chaos, Solitons and Fractals, 1-8.
  • Carpinteri, A., Cornetti, P., Sapora, A. (2014). Nonlocal elasticity: an approach based on fractional calculus. Meccanica, 49(11), 2551- 2569.
  • Tejado, I., Valrio, D., Valrio, N. (2015). Fractional Calculus in Economic Growth Modelling: The Span-ish Case. In CONTROLO2014Proceedings of the 11th Portuguese Conference on Automatic Control, Springer International Publishing, 449-458.
  • Meilanov, R. P., Magomedov, R. A. (2014). Thermodynamics in Fractional Calculus. Journal of En-gineering Physics and Thermophysics, 87(6), 1521-1531.
  • Mohsen, A., El-Gamel, M. (2008). On the Galerkin and collocation methods for two-point boundary value problems using sinc bases. Computers & Mathematics with Applications, 56(4), 930-941.
Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi-Cover
  • ISSN: 1301-4048
  • Yayın Aralığı: Yılda 6 Sayı
  • Başlangıç: 1997
  • Yayıncı: Sakarya Üniversitesi Fen Bilimleri Enstitüsü