Boundary Effect in Accuracy Estimate of the Grid Method for Solving Fractional Differential Equations
We construct and analyze grid methods for solving the first boundary-value problem for an ordinary differential equation with the Riemann–Liouville fractional derivative. Using Green’s function, we reduce the boundary-value problem to the Fredholm integral equation, which is then discretized by means of the Lagrange interpolation polynomials. We prove the weighted error estimates of grid problems, which take into account the impact of the Dirichlet boundary condition. All the results give us clear evidence that the accuracy order of the grid scheme is higher near the endpoints of the line segment than at the inner points of the mesh set. We provide a numerical example to support the theory.
Keywordsdifferential equation Dirichlet boundary condition fractional derivative grid solution error estimate boundary effect
Unable to display preview. Download preview PDF.
- 2.E. F. Galba, “On the order of accuracy of the difference scheme for the Poisson equation with mixed boundary condition,” A Collection of Papers “Optimization of software algorithms,” V. M. Glushkov Inst. of Cybernetics AS UkrSSR (1985), pp. 30–34.Google Scholar
- 6.V. L. Makarov and L. I. Demkiv, “Improved error estimates of traditional difference schemes for parabolic equations,” Proc. Ukr. Math. Congress (2001), pp. 31–42.Google Scholar
- 7.V. L. Makarov and L. I. Demkiv, “Accuracy estimates of difference schemes for parabolic equations that take into account initial–boundary effect,” Dopov. Nac. Akad. Nauk Ukrainy, No. 2, 26–32 (2003).Google Scholar
- 9.J. A. Machado, A. M. S. F. Galhano, and J. J. Trujillo, “On development of fractional calculus during the last fifty years,” Scientometrics, Vol. 98, Iss. 1, 577–582 (2014).Google Scholar
- 11.V. V. Vasil’ev and L. A. Simak, Fractional Calculus and Approximation Methods in Modeling of Dynamic Systems [in Ukrainian], NAN Ukrainy, Kyiv (2008).Google Scholar
- 14.V. L. Makarov and N. V. Mayko, “The boundary effect in the accuracy estimate for the grid solution of the fractional differential equation,” Computational Methods in Applied Mathematics, Vol. 20, Iss. 10 (2018). DOI: https://doi.org/10.1515/cmam-2018-0002.
- 16.A. A. Samarskii, R. D. Lazarov, and V. L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions [in Russian], Vysshaya Shkola, Moscow (1987).Google Scholar
- 18.A. V. Bitsadze, Some Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981).Google Scholar
- 19.I. P. Gavrilyuk and V. L. Makarov, Computing Techniques, Pt. 1 [in Ukrainian], Vyshcha Shkola, Kyiv (1995).Google Scholar