10.1 Section 2 Background .1 Cosmology .11 Math Section 10.3

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

Section 2 Dirichlet Problem

The first boundary value problem or Dirichlet problem for Poisson's equation

(2.1)

is stated as follows: Find the function u(x,y) which satisfying inside a certain domain G Equation (2.1) and on its boundary the boundary condition

(2.2)

where j(x,y) is a given continuous function.

Choosing h and l along x and y, respectively, construct the grid

and at each interior mesh point (xi, yk) replace the derivatives by finite-differences relation (10.1.3) of Section 10.1 and Equation (2.1) into the difference equation

(2.3)

where fik = f(xi, yk).

Equations (2.3) together with the values of uik at the boundary form a system of linear algebraic equations in the values of the function u(x,y) at the points (xi, yk). In the case of a rectangle with h = l, this system is the simplest:

(2.4)

also, the values at the boundary equal those of the boundary function. If f(x, y) = 0, Equation (2.1) is called Laplace's equation and the corresponding finite-difference equations become

(2.5)

The grid points for the last two sets of equations are shown in the figure below on the left hand side, in which only the grid subscripts are shown. At times, the scheme of the figure on the right hand side is more convenient.

Figure 5

Figure 6

Then. the finite difference scheme of Laplace's equation is

(2.6)

and of Poisson's equation

The error incurred by the replacement of a differential equation by a difference equation, i. e., the remainder term Rik for Laplace's Equation is estimated by

where

The error of the approximate difference solution comprises errors due to

1. replacement of the differential equation by the difference equation,
2. approximation of the
boundary conditions,
3. the fact that the system of difference equations is solved by an
approximate method.

Example 1: Consider the problem of a steady distribution of heat in an insulated flat square plate with unit sides, if its boundary is kept at constant temperature.

It is known ([52]) that the function u(x,y), determining the temperature distribution, is the solution of Laplace's equation

for the appropriate boundary conditions, displayed in the next figure.

Solution: The preceding figure shows the grid with h = 1/4 and nine internal points. By virtue of the symmetry of the boundary conditions:

(2.7)

and the number of unknown values of the function u at the internal points is reduced to 6, i.e., it is unnecessary to write out the difference equations for the grid points (3,1), (3, 2) and (3, 3). For the remaining 6 internal points (1,1), (2,1), (1,2), (2,2) and (1,3), the six equations are:

(2.8)

The 12 boundary values of the function are included in these equations. By the boundary condition they are

(2.9)

Note that the boundary conditions do not involve the remaining grid points! Finally, taking into account (2.7) and (2.9), one arrives at the system of equations

Gaussian elimination yields the solution

Example 2: The problem of elastic deformation of a square plate under the action of a constant force reduces to Poisson's equation ([42])

with zero boundary values. Solve this problem for the unit square and step length h = 1/4.

Solution: In this case, the values of the unknown function are completely symmetrical, since all boundary conditions are zero and the function f(x,y) is constant, whence it is sufficient to set up finite-difference equations for a quarter of the square, i.e.. for the points (1,1), (2,1), (1,2) and (2,2) (cf. the last figure). Taking into account the zero boundary conditions, one arrives at the system of the equations:

which, due to its symmetry, reduces to the 3 equations:

with the solution

Exercises

1. Solve Laplace's equation at the points p, q, r and s of a square with the boundary conditions in the figure below for

a = 0.9 + 0.1k, k = 0, 1, 2, b = 1.01 + 0.01n, n = 0, 1, 2, 3, 4.

Figure 8

2. Solve Laplace's equation with h = 1/4 in a square with the vertices A(0,0), B(0,1), C(1,1) and D(1,0) for the boundary conditions, given in the table:

Table 100 - Boundary conditions.