Abstract
In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.
Highlights
This paper is devoted to the numerical computation of the three dimensional elliptic and hyperbolic PDEs
The BVMs are a class of linear multistep methods (LMM) with step number k and whose k additional conditions are imposed at the beginning of the integration process and at the end so that they form a discrete analog of the continuous boundary value problems
In this paper, we have developed a highly accurate 3D problem solver. This has been achieved by the discretization of two of the spatial variables and the construction of a continuous BVM via the interpolation and collocation approach for solving the resulting semidiscretized system
Summary
This paper is devoted to the numerical computation of the three dimensional elliptic and hyperbolic PDEs. The BVMs are a class of linear multistep methods (LMM) with step number k and whose k additional conditions are imposed at the beginning of the integration process and at the end so that they form a discrete analog of the continuous boundary value problems. They are used for the numerical approximation of both initial and boundary value problems. Convergence analysis We discuss the convergence of the BVMs in the following theorem Theorem 2 Let U be an approximation of the solution vector U for the system obtained on a partition πL := {L1 = x0 < x1 < .
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