to 0.9inRead:Sections 11.8, 12.8, 12.9 to 0.9 inProblems:page 707 #1,3,7; page 711 #1, 5, 7, 8; to 0.9inpage 717 #1, 2, 14, 16; This page #1, 2; to 0.9inpage 768 #1; page 778 # 1, 13, 14, 15, 16, 17

Fall, 1996Chemical Engineering 535, Assignment 7page Problems: This page

1.

Show that the Green's function, , satisfies the end conditions and

where is the Dirac delta function, (See page 218) and that, therefore, , as a function of x for fixed , can be found by solving the problem

with the end conditions satisfied. Use this to find the Green's function for

2.

Consider the problem

where L is a Sturm-Liouville operator, L(u)=(ru')'+pu. Find f(x) and a Sturm-Liouville operator so that the problem becomes equivalent to finding a solution v(x) to

(i.e. Convert a problem with non-homogeneous boundary conditions to one in standard form, i.e. with homogeneous boundary conditions.) Apply this method to find a solution to

Correction: In Assignment 6 (on page 2) the definition of the inner product for its use in the Statements of the theorems should read

i.e. r(x) should be replaced by q(x). Please make this correction.

Let the Sturm-Liouville operator, L(u) be defined by

with end conditions given by

Then consider the problem, L(u)=f(x) with u satisfying the end conditions. For given q(x), one can find the eigenvalues, and the eigenfunctions, of the problem , with u satisfying the end conditions. Then, if none of the eigenvalues is zero (Theorem 1 , page 715 addresses the case where an eigenvalue is zero) the solution is given in equations (7) and (8), page 713, which we paraphrase as

It appears, then, that the solution can be given in terms of a Green's function: Let so that is a normalized eigenfunction: . Then

and the solution to L(u)=f(x), with the end conditions satisfied, is

Laplace's equation in more than one dimension:

The Laplacian operator, , is defined by

In cylindrical coordinates,

For spherical coordinates, see problem 8, page 778. Laplace's equation, namely , governs steady state distribution of temperature, T.

In the plane, we may use separation of variables to solve Laplace's equation in a rectangle where the unknown, T, is given on one of the edges of the rectangle and is zero on the others. Adding the solutions of four of such problems will solve the problem where T is given on all four edges.

In polar coordiates, separation of variables will lead to a solution inside a disk, where T is assigned on the edge of the disk. This will give rise to Fourier series.

In Example 1, page 773, there is solved the problem of LaPlace's equation in a cylinder, , . Boundary conditions consist of setting T=0 at the ends, z=0 and , and assigning the value of T on the lateral face: . The separation of variables method then leads to a solution of the form

(See equation (13), page 777.) See the rest of page 777 for the evaluation of the coefficients and .

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