A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition

• Main Authors: Adiguzel Dosiyev, Rifat Reis
• Format: Article
• Language: English
• Published: SpringerOpen 2019-08-01
• Series: Advances in Difference Equations
• Subjects: Finite difference method; Nonlocal integral boundary condition; Laplace’s equation; Uniform error
• Online Access: http://link.springer.com/article/10.1186/s13662-019-2282-2

Abstract A new constructive method for the finite-difference solution of the Laplace equation with the integral boundary condition is proposed and justified. In this method, the approximate solution of the given problem is defined as a sequence of 9-point solutions of the local Dirichlet problems. It is proved that when the exact solution u(x,y) $u(x,y)$ belongs to the Hölder calsses C4,λ $C^{4,\lambda }$, 0<λ<1 $0<\lambda <1$, on the closed solution domain, the uniform estimate of the error of the approximate solution is of order O(h4) $O(h^{4})$, where h is the mesh step. Numerical experiments are given to support analysis made.