Monday, February 1, 2010

Elliptic Partial Differential Equations

As examples of elliptic partial differential equations, we consider the Laplace equation, Poisson equation, and Helmholtz equation. Recall that the Laplacian of the function u(x,y) is

[Graphics:Images/EllipticPDEMod_gr_1.gif].

With this notation, we can write the Laplace, Poisson, and Helmholtz equations in the following forms:

[Graphics:Images/EllipticPDEMod_gr_2.gif]

[Graphics:Images/EllipticPDEMod_gr_3.gif]

[Graphics:Images/EllipticPDEMod_gr_4.gif]

[Graphics:Images/EllipticPDEMod_gr_5.gif]

[Graphics:Images/EllipticPDEMod_gr_6.gif]

[Graphics:Images/EllipticPDEMod_gr_7.gif]


It is often the case that the boundary values for the function
u(x,y) are known at all points on the sides of a rectangular region R in the plane. In this case, each of these equations can be solved by the numerical technique known as the finite-difference method.

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Introductory Methods of Numerical Analysis