Крайовий ефект в оцінці точності сіткового методу для розв'язування диференціального рівняння з дробовою похідною / Макаров В. Л., Майко Н. В. (2019)
Ukrainian

English  Cybernetics and Systems Analysis   /     Issue (2019, 55 (1))

Makarov V.L., Mayko N.V.
Boundary effect in error estimate of the grid method for solving a fractional differential equation

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

Keywords: boundary effect, differential equation, Dirichlet boundary condition, error estimate, fractional derivative, grid solution, Boundary value problems, Differential equations, Integral equations, Ordinary differential equations, Boundary effects, Dirichlet boundary condition, Error estimates, Fractional derivatives, Fractional differential equations, Fredholm integral equations, Grid problems, Lagrange interpolations, Boundary conditions


Cite:
Makarov V.L., Mayko N.V. (2019). Boundary effect in error estimate of the grid method for solving a fractional differential equation. Cybernetics and Systems Analysis, 55 (1), 80-95. doi: https://doi.org/10.1007/s10559-019-00113-y http://jnas.nbuv.gov.ua/article/UJRN-0000947557 [In Ukrainian].


 

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