Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay
-
Ali Konuralp
and
Sercan Öner
Abstract
In this study, a method combined with both Euler polynomials and the collocation method is proposed for solving linear fractional differential equations with delay. The proposed method yields an approximate series solution expressed in the truncated series form in which terms are constituted of unknown coefficients that are to be determined according to Euler polynomials. The matrix method developed for the linear fractional differential equations is improved to the case of having delay terms. Furthermore, while putting the effect of conditions into the algebraic system written in the augmented form in which the coefficients of Euler polynomials are unknowns, the condition matrix scans the rows one by one. Thus, by using our program written in Mathematica there can be obtained more than one semi-analytic solutions that approach to exact solutions. Some numerical examples are given to demonstrate the efficiency of the proposed method.
References
[1] M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre Pseudospectral Method, Int. J. Pure Appl. Math. 74(3) (2012), 287–297.Search in Google Scholar
[2] M. L. Morgado, N. J. Ford and P. Lima, Analysis and numerical methods for fractional differential equations with delay, J. Comput. Appl. Math. 252 (2013), 159–168.10.1016/j.cam.2012.06.034Search in Google Scholar
[3] M. M. Khader, The use of generalized Laguerre polynomials in spectral methods for solving fractional delay differential equations, J. Comput. Nonlinear Dyn. 8(4) (2013), 41018.10.1115/1.4024852Search in Google Scholar PubMed PubMed Central
[4] Z. Wang, A numerical method for delayed fractional-order differential equations, J. Appl. Math. 2013 (2013), 1–7. Article ID 256071.10.1155/2013/256071Search in Google Scholar
[5] U. Saeed and M. Rehman, Hermite wavelet method for fractional delay differential equations, J. Diff. Equations 2014 (2014), 8. Article ID 359093.10.1155/2014/359093Search in Google Scholar
[6] H. Sokhanvar and Askari-Hemmat, A numerical method for solving delay fractional differential and integro-differential equations, J. Mahani Math. Res. Cent. 4(1–2) (2015), 11–24.Search in Google Scholar
[7] V. Daftardar-Gejji, Y. Sukale and S. Bhalekar, Solving fractional delay differential equations: A new approach, Fract. Calc. Appl. Anal. 16 (2015), 400–418.10.1515/fca-2015-0026Search in Google Scholar
[8] M. Q. Xu and Y. Z. Lin, Simplified reproducing kernel method for fractional differential equations with delay, Appl. Math. Lett. 52 (2016), 156–161.10.1016/j.aml.2015.09.004Search in Google Scholar
[9] H. Singh, R. K. Pandey and D. Baleanu, Stable numerical approach for fractional delay differential equations, Few-Body Syst. 58 (2017), 156, 1–18.10.1007/s00601-017-1319-xSearch in Google Scholar
[10] M. Caputo, Linear model of dissipation whose Q is almost frequency independent. II, Geophys. J. Int. 13(5) (1967), 529–539.10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar
[11] K. B. Oldham and J. Spanier, 1974. The Fractional Calculus.Search in Google Scholar
[12] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, 1993. New York: Wiley.Search in Google Scholar
[13] M. M. Khader, The use of generalized Laguerre polynomial sin spectral methods for solving fractional delay differential equations, J. Comput. Nonlinear Dyn. 8(4) (2013), 041018.10.1115/1.4024852Search in Google Scholar
[14] R. K. Pandey, S. Sharma and K. Kumar, Collocation method for generalized Abel’s integral equations, J. Comput. Appl. Math. 302 (2016), 118–128.10.1016/j.cam.2016.01.036Search in Google Scholar
[15] R. K. Pandey, N. Kumar and R. N. Mohaptra, An approximate method for solving fractional delay differential equations, Int J. Appl. Comput. Math. 3(2) (2017), 1395–1405.10.1007/s40819-016-0186-3Search in Google Scholar
[16] S. Sharma, R. K. Pandey and K. Kumar, Galerkin and collocation methods for weakly singular fractional integro-differential equations, Iran. J. Of Sci. Technol., Trans. A: Sci. 43 (2019), 1649.10.1007/s40995-018-0608-7Search in Google Scholar
[17] M. A. Balcı and M. Sezer, Hybrid Euler–Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations, Appl. Math. Comput. 273 (2016), 33–41.10.1016/j.amc.2015.09.085Search in Google Scholar
[18] B. İbis and M. Bayram, A new collocation method based on Euler polynomials for solution of generalized pantograph equations, New Trends Mathl. Sci. 4(4) (2016), 285–294.10.20852/ntmsci.2016.115Search in Google Scholar
[19] A. Konuralp and S. Öner, Improved Euler-Taylor Matrix method for generalized functional integro-differential equations, International Students Science Conference, 5–6 May 2017, İzmir Katip Çelebi University, Izmir – Turkey, 2017.Search in Google Scholar
[20] A. Konuralp and S. Oner, An approximate solution for linear fractional differential equations by using Euler polynomials, Manisa Celal Bayar University, Üsitem 1. International University Industry Cooperation, r&d and innovation congress, 18–19 December 2017, 2017.Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Nonlinear Forced Vibration Analysis of FG Cylindrical Nanopanels Based on Mindlin’s Strain Gradient Theory and 3D Elasticity
- Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay
- A Remanufacturing Duopoly Game Based on a Piecewise Nonlinear Map: Analysis and Investigations
- Development of a Momentum Transfer Coefficient in Particle Horizontal Channel Flow
- Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations
- On the Dynamics and Control of Fractional Chaotic Maps with Sine Terms
- A stable class of modified Newton-like methods for multiple roots and their dynamics
- Experimental and numerical studies on failure and energy absorption of composite thin-walled square tubes under quasi-static compression loading
- Existence of periodic solutions with minimal period for fourth-order discrete systems via variational methods
- Derivative free iterative methods with memory having higher R-order of convergence
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Nonlinear Forced Vibration Analysis of FG Cylindrical Nanopanels Based on Mindlin’s Strain Gradient Theory and 3D Elasticity
- Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay
- A Remanufacturing Duopoly Game Based on a Piecewise Nonlinear Map: Analysis and Investigations
- Development of a Momentum Transfer Coefficient in Particle Horizontal Channel Flow
- Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations
- On the Dynamics and Control of Fractional Chaotic Maps with Sine Terms
- A stable class of modified Newton-like methods for multiple roots and their dynamics
- Experimental and numerical studies on failure and energy absorption of composite thin-walled square tubes under quasi-static compression loading
- Existence of periodic solutions with minimal period for fourth-order discrete systems via variational methods
- Derivative free iterative methods with memory having higher R-order of convergence
