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Boundary Effect in Accuracy Estimate of the Grid Method for Solving Fractional Differential Equations

  • V. L. MakarovEmail author
  • N. V. Mayko
Article

Abstract

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.

Keywords

differential equation Dirichlet boundary condition fractional derivative grid solution error estimate boundary effect 

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsNational Academy of Sciences of UkraineKyivUkraine
  2. 2.Taras Shevchenko National University of KyivKyivUkraine

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