Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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NUMERICAL METHOD FOR A SYSTEM OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-LOCAL BOUNDARY CONDITIONS

  • S. Joe Christin Mary (Department of Mathematics Bharathidasan University) ;
  • Ayyadurai Tamilselvan (Department of Mathematics Bharathidasan University)
  • Received : 2021.07.19
  • Accepted : 2022.07.26
  • Published : 2023.01.31
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Abstract

A class of systems of Caputo fractional differential equations with integral boundary conditions is considered. A numerical method based on a finite difference scheme on a uniform mesh is proposed. Supremum norm is used to derive an error estimate which is of order κ − 1, 1 < κ < 2. Numerical examples are given which validate our theoretical results.

Keywords

Acknowledgement

The first author wishes to thank Bharathidasan University for its financial support under URF scheme. The authors wish to thank Department of Science and Technology, Government of India, for the computing facility under DST-PURSE phase II Scheme.

References

  1. M. Al-Refai, Basic results on nonlinear eigenvalue problems of fractional order, Electron. J. Differ. Equ. 2012 (2012), No. 191, 12 pp.
  2. D. Baffet and J. S. Hesthaven, High-order accurate local schemes for fractional differential equations, J. Sci. Comput. 70 (2017), no. 1, 355-385. https://doi.org/10.1007/s10915-015-0089-1
  3. D. Baleanu, A. K. Golmankhaneh, R. Nigmatullin, and A. K. Golmankhaneh, Fractional Newtonian mechanics, Cent. Eur. J. Phys. 8 (2010), 120-125.
  4. A. Carpinteri, P. Cornetti, and A. Sapora, Nonlocal elasticity: an approach based on fractional calculus, Meccanica 49 (2014), no. 11, 2551-2569. https://doi.org/10.1007/s11012-014-0044-5
  5. V. Daftardar-Gejji and H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl. 301 (2005), no. 2, 508-518. https://doi.org/10.1016/j.jmaa.2004.07.039
  6. K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, 2004, Springer-Verlag, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2
  7. S. Dixit, O. P. Singh, and S. Kumar, An analytic algorithm for solving system of fractional differential equations, J. Mod. Methods Numer. Math. 1 (2010), no. 1, 12-26.
  8. V. S. Erturk and S. Momani, Solving systems of fractional differential equations using differential transform method, J. Comput. Appl. Math. 215 (2008), no. 1, 142-151. https://doi.org/10.1016/j.cam.2007.03.029
  9. P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton), 16, Chapman & Hall/CRC, Boca Raton, FL, 2000.
  10. M. Fiedler, Special matrices and their applications in numerical mathematics, translated from the Czech by Petr Prikryl and Karel Segeth, Martinus Nijhoff Publishers, Dordrecht, 1986.
  11. J. L. Gracia and M. Stynes, Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math. 273 (2015), 103-115. https://doi.org/10.1016/j.cam.2014.05.025
  12. H. Jafari and V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math. 196 (2006), no. 2, 644-651. https://doi.org/10.1016/j.cam.2005.10.017
  13. H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 5, 1962-1969. https://doi.org/10.1016/j.cnsns.2008.06.019
  14. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  15. N. Kopteva and M. Stynes, An efficient collocation method for a Caputo two-point boundary value problem, BIT 55 (2015), no. 4, 1105-1123. https://doi.org/10.1007/s10543-014-0539-4
  16. S. J. C. Mary and A. Tamilselvan, Numerical method for a non-local boundary value problem with Caputo fractional order, J. Appl. Math. Comput. 67 (2021), no. 1-2, 671-687. https://doi.org/10.1007/s12190-021-01501-4
  17. R. P. Meilanov and R. A. Magomedov, Thermodynamics in fractional calculus, J. Eng. Phys. Thermophys 87 (2014), no. 6, 1521-1531. https://doi.org/10.1007/s10891-014-1158-2
  18. A. Panda, S. Santra, and J. Mohapatra, Adomian decomposition and homotopy perturbation method for the solution of time fractional partial integro-differential equations, J. Appl. Math. Comput. 68 (2022), no. 3, 2065-2082. https://doi.org/10.1007/s12190- 021-01613-x
  19. I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
  20. M. S. Rawashdeh and H. Al-Jammal, Numerical solutions for systems of nonlinear fractional ordinary differential equations using the FNDM, Mediterr. J. Math. 13 (2016), no. 6, 4661-4677. https://doi.org/10.1007/s00009-016-0768-7
  21. S. Santra and J. Mohapatra, Numerical analysis of Volterra integro-differential equations with Caputo fractional derivative, Iran. J. Sci. Technol. Trans. A Sci. 45 (2021), no. 5, 1815-1824. https://doi.org/10.1007/s40995-021-01180-7
  22. S. Santra and J. Mohapatra, A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type, J. Comput. Appl. Math. 400 (2022), Paper No. 113746, 13 pp. https://doi.org/10.1016/j.cam.2021. 113746
  23. S. Santra, A. Panda, and J. Mohapatra, A novel approach for solving multi-term time fractional Volterra-Fredholm partial integro-differential equations, J. Appl. Math. Comput. (2021) 1-19.
  24. E. Sousa, How to approximate the fractional derivative of order 1 < α ≤ 2, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), no. 4, 1250075, 13 pp. https://doi.org/10. 1142/S0218127412500757 https://doi.org/10.1142/S0218127412500757
  25. M. Stynes and J. L. Gracia, A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal. 35 (2015), no. 2, 698-721. https://doi.org/10.1093/imanum/dru011
  26. X.-J. Yang, D. Baleanu, Y. Khan, and S. T. Mohyud-Din, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian J. Phys. 59 (2014), no. 1-2, 36-48.
  27. M. A. Zaky and I. G. Ameen, A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions, Numer. Algorithms 84 (2020), no. 1, 63-89. https://doi.org/10.1007/s11075-019-00743-5