Abstract
In this paper, explicit and implicit Crandall's formulas are applied for finding the solution of the one-dimensional heat equation with nonlinear nonlocal boundary conditions. The integrals in the boundary equations are approximated by the composite Simpson quadrature rule. Here nonlinear terms are approximated by Richtmyer's linearization method. Finally, some numerical examples are given to show the effectiveness of the proposed method.
Highlights
Nonlocal nonlinear conditions; explicit Crandall’s scheme (ECS); implicit Crandall’s scheme (ICS)
This paper is concerned with the numerical solution of the heat equation
We still have to determinates two unknowns u0 and uM+1, for this we approximate integrals in (1.3) and (1.4) numerically by the composite Simpson quadrature formula (We have chosen this approximation since it is of the same, fourth-order of accuracy in space as the methods used for the interior part of the problem) which requires M to be even
Summary
Nonlocal nonlinear conditions; explicit Crandall’s scheme (ECS); implicit Crandall’s scheme (ICS). Much less effort is given to the problem with nonlocal nonlinear type boundary conditions (3) and (4). Proposed the implicit difference scheme for the solution of the heat equation with nonlinear nonlocal boundary condition.
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