In this study, we numerically solve a class of two-point boundary-value-problems with a Riemann-Liouville-Caputo fractional derivative, where the solution might contain a weak singularity. Using the shooting technique based on the secant iterative approach, the boundary value problem is first transformed into an initial value problem, and the initial value problem is then converted into an analogous integral equation. The functions contained in the fractional integral are finally approximated using linear interpolation. An adaptive mesh is produced by equidistributing a monitor function in order to capture the singularity of the solution. A modified Gronwall inequality is used to establish the stability of the numerical scheme. To show the effectiveness of the suggested approach over an equidistributed grid, two numerical examples are provided.
[1] Q. M. Al-Mdallal, M. I. Syam, and M. N. Anwar, A collocation-shooting method for solving fractional boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), pp. 3814–3822.
[2] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water resources research, 36 (2000), pp. 1403–1412.
[3] Z. Cen, J. Huang, and A. Xu, An efficient numerical method for a two-point boundary value problem with a Caputo fractional derivative, J. Comput. Appl. Math., 336 (2018), pp. 1–7.
[4] Z. Cen, J. Huang, A. Xu, and A. Le, A modified integral discretization scheme for a two-point boundary value problem with a Caputo fractional derivative, J. Comput. Appl. Math., 367 (2020), pp. 112465, 10.
[5] Z. Cen, L.-B. Liu, and J. Huang, A posteriori error estimation in maximum norm for a two-point boundary value problem with a Riemann-Liouville fractional derivative, Appl. Math. Lett., 102 (2020), pp. 106086, 8.
[6] D. del Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas, 13 (2006), pp. 082308, 16.
[7] K. Diethelm, and N. J. Ford, Volterra integral equations and fractional calculus: do neighboring solutions intersect?, J. Integral Equations Appl., 24 (2012), pp. 25–37.
[8] K. Diethelm, and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms, 16 (1997), pp. 231–253.
[9] H. Fazli, F. Bahrami, and S. Shahmorad, Extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions, Computational Methods for Differential Equations, 11 (2023), pp. 32–41.
[10] J. L. Gracia, E. O’Riordan, and M. Stynes, Convergence analysis of a finite difference scheme for a two-point boundary value problem with a Riemann-Liouville-Caputo fractional derivative, BIT, 60 (2020), pp. 411–439.
[11] J. Huang, Z. Cen, L.-B. Liu, and J. Zhao, An efficient numerical method for a Riemann-Liouville two-point boundary value problem, Appl. Math. Lett., 103 (2020), pp. 106201, 8.
[12] A. Jaishankar, and G. H. McKinley, Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), pp. 20120284, 18.
[13] L. Jia, H. Chen, and V. J. Ervin, Existence and regularity of solutions to 1-D fractional order diffusion equations, Electron. J. Differential Equations, (2019), pp. 93, 21.
[14] J. F. Kelly, H. Sankaranarayanan, and M. M. Meerschaert, Boundary conditions for two-sided fractional diffusion, J. Comput. Phys., 376 (2019), pp. 1089–1107.
[15] R. L. Magin, Fractional calculus in bioengineering, vol. 2, Begell House Redding, 2006.
[16] M. N. Oqielat, T. Eriqat, Z. Al-Zhour, A. El-Ajou, and S. Momani, Numerical solutions of time-fractional nonlinear water wave partial differential equation via caputo fractional derivative: an effective analytical method and some applications, Applied and Computational Mathematics, 21 (2022), pp. 207–222.
[17] 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), pp. 2065– 2082.
[18] P. Patie, and T. Simon, Intertwining certain fractional derivatives, Potential Anal., 36 (2012), pp. 569–587.
[19] 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., 68 (2022), pp. 3545–3563.
[20] M. A. Shallal, A. H. Taqi, H. N. Jabbar, H. Rezazadeh, B. F. Jumaa, A. Korkmaz, and A. Bekir, A numerical technique of the time fractional gas dynamics equation using finite element approach with cubic hermit element, Applied and Computational Mathematics, 21 (2022), pp. 269–278.
[21] M. K. Shkhanukov, On the convergence of difference schemes for differential equations with a fractional derivative, Dokl. Akad. Nauk, 348 (1996), pp. 746–748.
[22] M. K. Shkhanukov, A. A. Kerefov, and A. A. Berezovski˘ı, Boundary value problems for the heat equation with a fractional derivative in the boundary conditions, and difference methods for their numerical realization, Ukra¨ın. Mat. Zh., 45 (1993), pp. 1289–1298.
[23] M. Waurick, Homogenization in fractional elasticity, SIAM J. Math. Anal., 46 (2014), pp. 1551–1576.
Maji, S., & Natesan, S. (2024). Adaptive-grid technique for the numerical solution of a class of fractional boundary-value-problems. Computational Methods for Differential Equations, 12(2), 338-349. doi: 10.22034/cmde.2023.55266.2296
MLA
Sandip Maji; Srinivasan Natesan. "Adaptive-grid technique for the numerical solution of a class of fractional boundary-value-problems". Computational Methods for Differential Equations, 12, 2, 2024, 338-349. doi: 10.22034/cmde.2023.55266.2296
HARVARD
Maji, S., Natesan, S. (2024). 'Adaptive-grid technique for the numerical solution of a class of fractional boundary-value-problems', Computational Methods for Differential Equations, 12(2), pp. 338-349. doi: 10.22034/cmde.2023.55266.2296
VANCOUVER
Maji, S., Natesan, S. Adaptive-grid technique for the numerical solution of a class of fractional boundary-value-problems. Computational Methods for Differential Equations, 2024; 12(2): 338-349. doi: 10.22034/cmde.2023.55266.2296