Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations. (English) Zbl 1533.65256
Summary: A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time \(t=0\). A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45J05 | Integro-ordinary differential equations |
| 45D05 | Volterra integral equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
Keywords:
Volterra integro-differential equation; weakly singular kernel; fractional collocation method; error analysis; superconvergenceReferences:
| [1] | Atkinson, KL, An Introduction to Numerical Analysis (1989), New York: Wiley, New York · Zbl 0718.65001 |
| [2] | Atkinson, KE, The Numerical Solution of Integral Equations of the Second Kind. Cambridge Monographs on Applied and Computational Mathematics (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0899.65077 · doi:10.1017/CBO9780511626340 |
| [3] | Brunner, H.; Tang, T., Polynomial spline collocation methods for the nonlinear Basset equation, Comput. Math. Appl., 18, 5, 449-457 (1989) · Zbl 0687.65130 · doi:10.1016/0898-1221(89)90239-3 |
| [4] | Brunner, H., Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels, IMA J. Numer. Anal., 6, 2, 221-239 (1986) · Zbl 0634.65142 · doi:10.1093/imanum/6.2.221 |
| [5] | Brunner, H., Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge Monographs on Applied and Computational Mathematics (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1059.65122 |
| [6] | Brunner, H.; Pedas, A.; Vainikko, G., Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39, 3, 957-982 (2001) · Zbl 0998.65134 · doi:10.1137/S0036142900376560 |
| [7] | Brunner, H.; Schötzau, D., \(hp\)-discontinuous Galerkin time-stepping for Volterra integrodifferential equations, SIAM J. Numer. Anal., 44, 1, 224-245 (2006) · Zbl 1116.65127 · doi:10.1137/040619314 |
| [8] | Diogo, T.; Lima, PM; Pedas, A.; Vainikko, G., Smoothing transformation and spline collocation for weakly singular Volterra integro-differential equations, Appl. Numer. Math., 114, 63-76 (2017) · Zbl 1357.65316 · doi:10.1016/j.apnum.2016.08.009 |
| [9] | Hou, D.; Chuanju, X., A fractional spectral method with applications to some singular problems, Adv. Comput. Math., 43, 5, 911-944 (2017) · Zbl 1382.65467 · doi:10.1007/s10444-016-9511-y |
| [10] | Hu, Q., Stieltjes derivatives and \(\beta \)-polynomial spline collocation for Volterra integrodifferential equations with singularities, SIAM J. Numer. Anal., 33, 1, 208-220 (1996) · Zbl 0851.65098 · doi:10.1137/0733012 |
| [11] | Hu, Q., Geometric meshes and their application to Volterra integro-differential equations with singularities, IMA J. Numer. Anal., 18, 1, 151-164 (1998) · Zbl 0907.65142 · doi:10.1093/imanum/18.1.151 |
| [12] | Jones, S.; Jumarhon, B.; McKee, S.; Scott, JA, A mathematical model of a biosensor, J. Engrg. Math., 30, 3, 321-337 (1996) · Zbl 0856.92008 · doi:10.1007/BF00042754 |
| [13] | Kangro, R.; Parts, I., Superconvergence in the maximum norm of a class of piecewise polynomial collocation methods for solving linear weakly singular Volterra integro-differential equations, J. Integral Equ. Appl., 15, 4, 403-427 (2003) · Zbl 1065.65148 · doi:10.1216/jiea/1181074984 |
| [14] | Ma, Z.; Huang, C., An \(hp\)-version fractional collocation method for Volterra integro-differential equations with weakly singular kernels, Numer. Algorithms, 92, 4, 2377-2404 (2023) · Zbl 1511.65061 · doi:10.1007/s11075-022-01394-9 |
| [15] | Mustapha, K., A superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels, Math. Comput., 82, 284, 1987-2005 (2013) · Zbl 1273.65198 · doi:10.1090/S0025-5718-2013-02689-0 |
| [16] | Renardy, M.; Hrusa, WJ; Nohel, JA, Mathematical Problems in Viscoelasticity, volume 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics (1987), Harlow: Longman Scientific & Technical, Harlow · Zbl 0719.73013 |
| [17] | Shi, X.; Wei, Y.; Huang, F., Spectral collocation methods for nonlinear weakly singular Volterra integro-differential equations, Numer. Methods Partial Differ. Equ., 35, 2, 576-596 (2019) · Zbl 1416.65236 · doi:10.1002/num.22314 |
| [18] | Tang, T., Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61, 3, 373-382 (1992) · Zbl 0741.65110 · doi:10.1007/BF01385515 |
| [19] | Tang, T., A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 13, 1, 93-99 (1993) · Zbl 0765.65126 · doi:10.1093/imanum/13.1.93 |
| [20] | Vainikko, G., Multidimensional Weakly Singular Integral Equations. Lecture Notes in Mathematics (1993), Berlin: Springer, Berlin · Zbl 0789.65097 · doi:10.1007/BFb0088979 |
| [21] | Wang, Z.; Guo, Y.; Yi, L., An \(hp\)-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Comput., 86, 307, 2285-2324 (2017) · Zbl 1364.65299 · doi:10.1090/mcom/3183 |
| [22] | Yi, L.; Guo, B., An \(h-p\) version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels, SIAM J. Numer. Anal., 53, 6, 2677-2704 (2015) · Zbl 1330.65206 · doi:10.1137/15M1006489 |
| [23] | Zhou, Y.; Stynes, M., Block boundary value methods for linear weakly singular Volterra integro-differential equations, BIT, 61, 2, 691-720 (2021) · Zbl 1472.65170 · doi:10.1007/s10543-020-00840-1 |
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