Abstract
This important numerical method is given for the numerical solution of singularly perturbed convection-diffusion nonlocal boundary value problem. First, the behavior of the exact solution is analyzed, which is needed for analysis of the numerical solution in later sections. Next, uniformly convergent finite difference scheme on a Shishkin mesh is established, which is based on the method of integral identities with the use exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that the method is first order accurate expect for a logarithmic factor, in the discrete maximum norm. Finally, the numerical results are presented in table and graphs, and these results reveal the validity of the theoretical results of our method.
Highlights
In this work, we consider singularly perturbed convection-di¤usion problem with nonlocal boundary value "u00(x) + a(x)u0(x) + b (x) u (x) = f (x) ; 0 < x < 1; (1.1) u (0) = A; (1.2) mX2 u (1) ciu = B; (1.3)i=1 where 0 < " 0; and a(x), b(x) and f (x) are assumed to be su¢ ciently continuously di¤erentiable functions in [0; 1].Received by the editors: February 05, 2018; Accepted: July 25, 2018. 2010 Mathematics Subject Classi...cation. 65L10, 65L11, 65L12, 65L15, 65L20, 65L70, 34B10
Uniformly convergent ...nite di¤erence scheme on a Shishkin mesh is established, which is based on the method of integral identities with the use exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form
The plan of the study is as follows: We evaluate that for the numerical solution of the nonlocal problem (1.1)-(1.3), this method is uniformly convergent of ...rst order on Shishkin mesh, in discrete maximum norm, independently of singular perturbation parameter "
Summary
Singular perturbation, ...nite di¤erence scheme, Shishkin mesh, uniformly convergence, nonlocal condition. Perturbed di¤erential equations with nonlocal boundary value have been studied by many authors. A ...nite di¤erence scheme on an uniform mesh for solving linear (nonlinear) singularly perturbed problem with nonlocal condition have been found in [1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 15]. The plan of the study is as follows: We evaluate that for the numerical solution of the nonlocal problem (1.1)-(1.3), this method is uniformly convergent of ...rst order on Shishkin mesh, in discrete maximum norm, independently of singular perturbation parameter ". Throughout the paper, C will mean a positive constant independent of " and the mesh parameter
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