Solving fractional differential equations by the ultraspherical integration method
DOI:
https://doi.org/10.5269/bspm.45992Abstract
In this paper, we present a numerical method to solve a linear fractional differential equations. This new investigation is based on ultraspherical integration matrix to approximate the highest order derivatives to the lower order derivatives. By this approximation the problem is reduced to a constrained optimization problem which can be solved by using the penalty quadratic interpolation method. Numerical examples are included to confirm the efficiency and accuracy of the proposed method.
References
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22. A. Saadatmandi and M. Dehghan, A new operational Matrix for solving fractional-order differential equations, Computers and Mathematics with Applications, 59 (2010) 1326-1336. https://doi.org/10.1016/j.camwa.2009.07.006
23. G. Szego, Orthogonal polynomials, American mathematical society, 1976.
2. Q. M. Al-Mdallal, M. A. Hajji and A. S. Abu Omer Fractional-order Legender -collocation method for solving fractional initial value problems, Applied Mathematics and Computation. 321 (2018)74-84. https://doi.org/10.1016/j.amc.2017.10.012
3. Q. M. Al-Mdallal and M. A. Hajji, The Chebyshev collocation -path following method for solving sixth-order SturmLiouville problems, Applied Mathematics and Computation. 232 (2014) 391-398. https://doi.org/10.1016/j.amc.2014.01.083
4. Q. M. Al-Mdallal and M. A. Hajji, A convergent algorithm for solving higher-order nonlinear fractional boundary value problems, Fractional Calculus and Applied Analysis. 18(6) (2015) 1423-1440. https://doi.org/10.1515/fca-2015-0082
5. Q. M. Al-Mdallal and M. I. Syam, An efficient for solving non-linear singularly perturbed two points boundary- value problems of fractional order, Communications in Nonlinear Science and Numerical Simultion. 17(6) (2012) 2299-2308. https://doi.org/10.1016/j.cnsns.2011.10.003
6. Q. M. Al-Mdallal, Q. N. Syam and M. N. Answar, A collocation-shooting method for solving fractional boundary value problems, Commun Nonlinear Sci. NumerSimol. 15 (2010) 3814-3822. https://doi.org/10.1016/j.cnsns.2010.01.020
7. L. Blank, Numerical treatment of differential equations of fractional order, Numerical analysis Report 287, Manchester Center for Numerical Computational Mathematics, (1996).
8. H. Beyer and H. Kempfle, Definition of physically consistent damping laws with fractional derivatives, Zeitschrift for Anqweandte Mathematich and Mechanich. 75 (1995) 623-635. https://doi.org/10.1002/zamm.19950750820
9. A. Carpinteri and F. Mainardi, Fractals and fractional calculus in continuum mechanics, New York, Springer verlger wien (1997). https://doi.org/10.1007/978-3-7091-2664-6
10. V. Daftardar-Gejji and H. Jafari, Analysis of a system of nonautonomous fractional equations in volving Caputo derivatives, J. Math. Aval. Appl. 328 (2007) 26-33. https://doi.org/10.1016/j.jmaa.2006.06.007
11. V. Daftardar-Gejji and H. Jafari, Adomian decompositions a tool for solving a system of fractional differential equations, J. Math. Aval. Appe. 301 (2005) 508-518. https://doi.org/10.1016/j.jmaa.2004.07.039
12. S. E. El-Gendi, Chebyshev solutions of Differential Integral and Integro-Differential Equations, Computers Journal. 12 (1969) 282-287. https://doi.org/10.1093/comjnl/12.3.282
13. T. M. El-Gendy and M. S. Salim, Penalty functions with partial quadratic interpolation technique in the constrained optimization problems, Journal of Institute of Math. Computer Science. 3 (1990) 85-90.
14. H. M. El-Hawary, M. S. Salim and H. S. Hussein, Legendre spectral method for solving integral and integro-differential equations, International Journal of Computer Mathematics. 76 (2000) 1-2. https://doi.org/10.1080/00207160008805021
15. H. M. El-Hawary, M. S. Salim and H. S. Hussein, An optimal Ultraspherical Approximation of Integrals, International Journal of Computer Mathematics. 76 (2000) 219-237. https://doi.org/10.1080/00207160008805021
16. M. M. El-kady, H. S. Hussein and A. Mohamed Ibrahim, Ultraspherical spectral in integration method for solving linear integro-differential equations, World Academy of Science Engineering and Technology. 33 (2009) 880-887.
17. S. Irandoust-Pakchin, S. Abdi-Mazraeh Exact solutions for some of the fractional differential equations by using modification of He's variational iteration method, Mathematical sciences Journal. 5 (2011) 151-160.
18. M. Lakestani, M. Dehghan and S. Irandoust-Pakchin, The construction of operational matirx of fractional derivatives using B-Spline functions, Communications in Non linear Science and Numerical Simulation. 17 (2012) 1149-1162. https://doi.org/10.1016/j.cnsns.2011.07.018
19. Liu SK, Liv SD, Special functions, Chinese, Meteorology Press, (2003).
20. V. E. Lynch, B. A. Carreras, D. Del-Castillo-Negrete, K. M. Ferrera-Mejias and H. R. Hicks Numerical methods for the solution of partial differential equations of fractional order, J. Comput. phys. 192 (2003) 406-421. https://doi.org/10.1016/j.jcp.2003.07.008
21. F. Mainardi, Fractional calculus some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum Mechanics, New York, springer. (1997) 291-348. https://doi.org/10.1007/978-3-7091-2664-6_7
22. A. Saadatmandi and M. Dehghan, A new operational Matrix for solving fractional-order differential equations, Computers and Mathematics with Applications, 59 (2010) 1326-1336. https://doi.org/10.1016/j.camwa.2009.07.006
23. G. Szego, Orthogonal polynomials, American mathematical society, 1976.
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2021-12-17
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