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Numerical Solution of Partial Differential Equations
Published in Ramin S. Esfandiari, Numerical Methods for Engineers and Scientists Using MATLAB®, 2017
A Dirichlet problem refers to a problem that involves solving an elliptic equation in a specific region in the xy-plane, where the unknown function is prescribed along the boundary of the region. On the other hand, a Neumann problem refers to a boundary-value problem where the normal derivative of u, that is, un = ∂u/∂n, is given on the boundary of the region. Note that along a vertical edge of a region, un is simply ux = ∂u/∂x, and along a horizontal edge it is uy = ∂u/∂y. The mixed problem refers to the situation where u is specified on certain parts of the boundary, and un on the others.
Partial differential equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
as illustrated in Figure 11.2. A problem conssting of Laplace’s equation on a region in the plane when the values of unknown function are specified on its boundary is called a Dirichlet problem or the first boundary value problem. Thus, Eq. (11.4.1 together with the boundary conditions (11.4.2 is a Dirichlet problem for the Laplace equation over a rectangle.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
as illustrated in Figure 11.2. A problem consisting of Laplace's equation on a region in the plane when the values of unknown function are specified on its boundary is called a Dirichlet problem or the first boundary problem. Thus, Eq. (11.4.1) together with the boundary conditions (11.4.2) is a Dirichlet problem for the Laplace equation on a rectangle.
Mathematical problems of dynamical interaction of fluids and multiferroic solids
Published in Applicable Analysis, 2023
George Chkadua, David Natroshvili
Keeping in mind the homogeneous initial conditions (19)–(21) of problem , we can find the functions and for by the fourth and fifth equations of the dynamical system (1) for t = 0 and by the Dirichlet boundary conditions (16)–(17). Indeed, the pair is a solution to the following Dirichlet boundary value problem for the strongly elliptic system of partial differential equations: If and , then the above formulated Dirichlet problem is uniquely solvable in the space .