10.6 Section 2 Background .1 Cosmology .11 Math Section 10.8

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

Section 7 Hyperbolic Equations

Consider the mixed boundary value problem for the equation of a vibrating string

(7.1)

with the initial conditions

(7.2)

and the boundary conditions

(7.3)

Since the change of variable t = at reduces (7.1) to

(7.4)

set in (7.1) a = 1.

Construct for t ? 0, 0 ? x ? s the two families of parallel straight lines

and replace the derivatives in (7.4) by finite differences. Using the symmetry of these formulae, one obtains

(7.5)

With these equations become

(7.6)

It has been shown ([2]) that for a ? 1, this equation is stable.

For a = 1, (7.6) assumes its simplest form

(7.7)

The error estimate ([2]) of the solution of (7.6) for 0 ? x ? s, 0 < t ? T is

(7.8)

where u is the exact solution and

Note that in writing down (7.6), the grid points shown in the preceding figure have been used, which is an explicit diagram, since (7.6) allows to find the values of u(x,t) at tj+1, if the values at the two preceding layers are known. In order to solve Problem (7.1) - (7.3), the values on two initial layers must be known. They can be found from the intial conditions by one of the two methods:

First method: In the initial condition (7.2), replace the derivative ut(x,0) by the difference relation

For the value of u(x, t) on the layers j = 0 and j = 1 this yields

(7.9)

The error estimate of the values ui1 has now the form ([2])

(7.10)

where

Second method: Replace the derivative ut(x,0) by

where ui,-1 are the values on the first layer j = -1. Then, by the initial conditions (7.2),

(7.11)

Next, write down the difference equation (7.7) for the layer j = 0:

(7.12)

Eliminate from (7.11) and (7.12) the values ui,-1 and obtain

(7.13)

The error estimate for ui,1 is ([2])

(7.14)

where

This method of computing the initial values is used in Example 1.

Third method: If f(x) has a finite second derivative, then the values of ut,1 can be determined with the aid of the Taylor expansion

(7.15)

By (7.4) and the initial conditions (7.2), one finds

Equation (7.15) now yields

(7.16

The error in ui,1, given by this formula, is of O(l 3). This method of obtaining initial values is considered in Example 2.

Note: The method can also be used to solve a mixed boundary value problem for the non-homogeneous equation

,

with the diffcrence equation

Example 1: Find the solution of the problem

(7.17)

Solution: Construct a square grid with h = l = 0.05. Find the values of u(x, t) on the two initial layers by the second method. Taking into account that

one finds

(7.18)

Filling in the table:

Table 113 - Solution to problem (17).

1. Enter the values of ui0 = f(xi) for xi = ih into the first row of the table above.(it corresponds to the value t0 = 0). Since the problem is symmetric, fill in the table for ) ? x ? 0.5. The first column, which corresponds to x0 = 0, contains the boundary values..

2. Find ui1 from (7.18), using ui0 in the first row and enter the results in the second row..
3. Compute uij on the next layers, using (7.7). For j = 1, the numbers are

Compute in the same manner the values for j = 2, 3, ..., 10. The last row of the table contains the values of the exact solution at t = 0.5.

Example 2: Solve the problem

(7.19)

Solution: Use h = l = p/18. Find u(x,t) on the first two layers by the third method, using a Taylor expansion.

Filling in of the table:

1. Compute ui0 = fi = xi(p - x), i = 0, 1, . . . ,18 and enter them into the first row of the next table.

Table 114 - Solution of problem (19).

Since the problem is symmetric, fill in the table for 0 for 0 ? x ? p/2. Enter the boundary values into the first column of the table.

2. Determine ui1. Since Fi = 0, fi"= -2, by (7.16),

whence follow the ui1.to be entered into the second row of the table.

3. Compute ui,j+1, j = 1, 2, 3, 4, 5 from (7.7). For j = 2,

etc.

Exercises

In Problems 1 to 3 solve the equation

for the initial and boundary conditions

for 0 ? t ? 0.5, 0 ? x ? 1 with h = 0.

1.

Compute ui,1 by the third method above.

2. When   the function f(x) is given by the table

3. When the functions are given by the table below and a = 1.1, 1.2, . . . , 1.5.

4. Solve the equation

with

Compute ui,1 by the first method above.

5. Solve the equation

with h = l = 0.1 for

:

Compute ui,1 using the first method above.