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

**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* = *h*^{2}/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 *a*_{i,j+1} and *b*_{i,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 *u*_{i,j+1}. Thus, the **explicit
method** allows to determine the values of *u*(*x, t*) at *t* = *t*_{j+1},
if its values at *t* = *t*_{j} are known.

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

.

**I****mplicit
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* = *h*^{2}/*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 *u*_{i0
}of Table 112 above the values of the initial function *f*(*x*_{i}) (*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 *a*_{i1}and *b*_{i1} of the above table.

**Implicit procedure:** By the boundary conditions,

Compute *u*_{i1}, (*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.