Collocation based approximations for a class of fractional boundary value problems
Abstract
A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented.
Keyword : fractional weakly singular integro-differential equation, Caputo derivative, boundary value problem, collocation method, graded grid
How to Cite
Soots, H. B., Lätt, K., & Pedas, A. (2023). Collocation based approximations for a class of fractional boundary value problems. Mathematical Modelling and Analysis, 28(2), 218–236. https://doi.org/10.3846/mma.2023.16359
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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G. Vainikko. Multidimensional Weakly Singular Integral Equations. SpringerVerlag, Berlin, Heidelberg, 1993. https://doi.org/10.1007/BFb0088979
G. Vainikko. Which functions are fractionally differentiable? Zeitschrift fu¨r Analysis und ihre Anwendungen, 35(4):465–487, 2016. https://doi.org/10.4171/ZAA/1574
M.P. Velasco, D. Usero, S. Jiménez, L. Vázquez, J.L. Vázquez-Poletti and M. Mortazavi. About some possible implementations of the fractional calculus. Mathematics, 8(6):893, 2020. https://doi.org/10.3390/math8060893
M. Vikerpuur. Two collocation type methods for fractional differential equations with non-local boundary conditions. Mathematical Modelling and Analysis, 22(5):654–670, 2017. https://doi.org/10.3846/13926292.2017.1355339
J. Zhao, J. Xiao and N.J. Ford. Collocation methods for fractional integrodifferential equations with weakly singular kernels. Numerical Algorithms, 65(4):723–743, 2014. https://doi.org/10.1007/s11075-013-9710-2
K. Atkinson and W. Han. Theoretical Numerical Analysis: A Functional Analysis Framework. Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-0-387-21526-6
D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo. Fractional Calculus: Models and Numerical Methods. World Scientific, Boston, 2016. https://doi.org/10.1142/10044
H. Brunner, H. Han and D. Yin. The maximum principle for time-fractional diffusion equations and its application. Numerical Functional Analysis and Optimization, 36(10):1307–1321, 2015. https://doi.org/10.1080/01630563.2015.1065887
H. Brunner, A. Pedas and G. Vainikko. Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM Journal on Numerical Analysis, 39(3):957–982, 2001. https://doi.org/10.1137/S0036142900376560
H. Brunner and P.J. van der Houwen. The Numerical Solution of Volterra Equations. North-Holland, Amsterdam, 1986.
Z. Cen, A. Le and A. Xu. A posteriori error analysis for a fractional differential equation. International Journal of Computer Mathematics, 94(6):1185–1195, 2017. https://doi.org/10.1080/00207160.2016.1184263
K. Diethelm. The Analysis of Fractional Differential Equations: An ApplicationOriented Exposition Using Differential Operators of Caputo Type. SpringerVerlag, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-14574-2
N.J. Ford, M.L. Morgado and M. Rebelo. A nonpolynomial collocation method for fractional terminal value problems. Journal of Computational and Applied Mathematics, 275:392–402, 2015. https://doi.org/10.1016/j.cam.2014.06.013
R. Garrappa. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics, 6(2):16, 2018. https://doi.org/10.3390/math6020016
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
N. Kinash and J. Janno. An inverse problem for a generalized fractional derivative with an application in reconstruction of time- and space-dependent sources in fractional diffusion and wave equations. Mathematics, 7(12):1138, 2019. https://doi.org/10.3390/math7121138
M. Kolk, A. Pedas and E. Tamme. Smoothing transformation and spline collocation for linear fractional boundary value problems. Applied Mathematics and Computation, 283:234–250, 2016. https://doi.org/10.1016/j.amc.2016.02.044
N. Kopteva and M. Stynes. An efficient collocation method for a Caputo twopoint boundary value problem. BIT Numerical Mathematics, 55(4):1105–1123, 2015. https://doi.org/10.1007/s10543-014-0539-4
H. Liang and M. Stynes. Collocation methods for general Caputo two-point boundary value problems. Journal of Scientific Computing, 76:390–425, 2018. https://doi.org/10.1007/s10915-017-0622-5
Z. Navickas, T. Telksnys, I. Timofejeva, R. Marcinkevičius and M. Ragulskis. An operator-based approach for the construction of closed-form solutions to fractional differential equations. Mathematical Modelling and Analysis, 23(4):665– 685, 2018. https://doi.org/10.3846/mma.2018.040
A. Pedas, E. Tamme and M. Vikerpuur. Spline collocation for fractional weakly singular integro-differential equations. Applied Numerical Mathematics, 110:204–214, 2016. https://doi.org/10.1016/j.apnum.2016.07.011
A. Pedas, E. Tamme and M. Vikerpuur. Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems. Journal of Computational and Applied Mathematics, 317:1–16, 2017. https://doi.org/10.1016/j.cam.2016.11.022
A. Pedas, E. Tamme and M. Vikerpuur. Numerical solution of linear fractional weakly singular integro-differential equations with integral boundary conditions. Applied Numerical Mathematics, 149:124–140, 2020. https://doi.org/10.1016/j.apnum.2019.07.014
A. Pedas and M. Vikerpuur. Spline collocation for multi-term fractional integrodifferential equations with weakly singular kernels. Fractal and Fractional, 5(3):90, 2021. https://doi.org/10.3390/fractalfract5030090
I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999.
S. Samko, A.A. Kilbas and O. Marichev. Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon, 1993.
H.G. Sun, Y. Zhang, D. Baleanu, W. Chen and Y.Q. Chen. A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation, 64:213–231, 2018. https://doi.org/10.1016/j.cnsns.2018.04.019
G. Vainikko. Multidimensional Weakly Singular Integral Equations. SpringerVerlag, Berlin, Heidelberg, 1993. https://doi.org/10.1007/BFb0088979
G. Vainikko. Which functions are fractionally differentiable? Zeitschrift fu¨r Analysis und ihre Anwendungen, 35(4):465–487, 2016. https://doi.org/10.4171/ZAA/1574
M.P. Velasco, D. Usero, S. Jiménez, L. Vázquez, J.L. Vázquez-Poletti and M. Mortazavi. About some possible implementations of the fractional calculus. Mathematics, 8(6):893, 2020. https://doi.org/10.3390/math8060893
M. Vikerpuur. Two collocation type methods for fractional differential equations with non-local boundary conditions. Mathematical Modelling and Analysis, 22(5):654–670, 2017. https://doi.org/10.3846/13926292.2017.1355339
J. Zhao, J. Xiao and N.J. Ford. Collocation methods for fractional integrodifferential equations with weakly singular kernels. Numerical Algorithms, 65(4):723–743, 2014. https://doi.org/10.1007/s11075-013-9710-2