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

**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 *G
*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 *f*_{ik} = *f*(*x*_{i}, *y*_{k}).

Equations (2.3) together with the values of *u*_{ik} at the boundary form a **system
of linear algebraic equations** in the values of the function *u*(*x*,*y*) at
the points (*x*_{i}, *y*_{k}). 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 *R*_{ik} 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
metho**d.

**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 + 0.1·*k*, *k = *0, 1, 2, *b
*= 1.01 + 0.01·*n, 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.