10.5 Section 2 Background .1 Cosmology .11 Math Section 10.7

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

Section 6 Explicit method for Heat Conduction Equation

Let it be required to solve in the strip 0 <= x <= a, 0 <= t <= T the equation

(6.1)

for the conditions

(6.2)

Select the step lengths h and l for x and t, respectively, replace the derivatives by the finite difference relations (5.4) and (5.6) cat each interior pont and compute the values of f(x), j(t) and y(t) at the boundary points, set s = h2/2 and obtain the system


(6.3)
(6.4)
(6.5)
(6.6)

The explicit method ([2], [13], [31]) reduces (6.3) to

(6.7)

where the numbers ai,j+1 and bi,j+1 are determined consecutively by

(6.8)


(6.9)

The boundary condition (6.6) now yields

and successively determines by (6.7) the values of ui,j+1. Thus, the explicit method allows to determine the values of u(x, t) at t = tj+1, if its values at t = tj are known.

Explicit procedure: Using (6.5) to find from (6.8) and (6.9)

.

Implicit procedure. (6.7) yields

(6.10)

Example 1: Find by the explicit method the solution of

(6.11)

for the conditions

(6.12)

 

Solution: Set h = 0.1, l = 0.01, whence s = h2/l = 1. Find u(x, t) for t = 0.0l.

Explicit procedure: Find the values of u(x,t) on the layer t = 0.01, using (6.11) and (6.12).

Table 112
Solution of Problem (11), (12) by the "Passage" Method

Direct procedure. Enter in the row ui0 of Table 112 above the values of the initial function f(xi) (i = 0, 1, 2, ??? , 10, by (6.8) find the following numbers for j = 0:

Next, compute step by step from (6.9) for j = 0

etc.

Enter these results into the rows ai1and bi1 of the above table.

Implicit procedure: By the boundary conditions,

Compute ui1, (i = 9, 8, ??? , 1 from (6.10) for j = 0:

Exercises

Solve Problems 1 to 3 of Section V by this method and compare the results.