Chemical Engineering 535
Fall, 1996
B. Bernstein, Instructor
Assignment 8

 
Read:Sections 4.1, 4.2, 4.3, 4.5

Problems:page 198 # 2, 7, 15, 17, 27, 43, 45, 71, 75

		page 212 # 1, 3, 11, 17, 21, 25, 30, 31, 32, 37, 45

		page 225 # 1, 3, 9, 11;page 237 # 1, 3, 7, 11, 14

		This page, # 1

Problems: This page

1. The Green's function for the problem u''(x)=f(x), 0<x<1, u(0)=u(1)=0 is given by , for , and for . Expand this Green's function into a Fourier sine series and show that you get

   

   where the are the eigenvalues and the are the eigenfunctions of

   

The Laplace Transform transforms a function f(t) into another function, F(s), by the transformation

 

The inverse, , of the Laplace transform is the rule that reverses it, so that equation 1 is equivalent to

 

The power of the Laplace transform stems from its ability to turn certain differential equations or certain integral equations into algebraic equations. Although the solution of these algebraic equations is often not too difficult, the completion of the task of solving the original differential or integral equations requires the inversion of a Laplace transform. For this reason, we must know the properties of the Laplace transform so that we have some hope of completing the task.

The Laplace transform is a linear transformation (Theorem 3, page 193) as its its inverse (page 194):

 

A table of Laplace transforms of some functions is given on page 195. We mention just a few of them

The following theorems are useful in manipulating Laplace Transforms:

Theorem 3: page 197 (First Shifting Theorem:)

Theorem 1: page 200 (Laplace Transform of the Derivative:)

 

which, of course, may be repeatedly applied, e.g.

These can be used to solve an initial value problem for a differential equation. for example (Example 5, page 203), after taking the Laplace transform of

and solving for , one gets

Theorem 3: page 208 (Differentiation of the Laplace Transform:)

The Heaviside unit step function, H(t) is defined by

Theorem 1: page 215 (Second Shifting Theorem)

In particular,

The Dirac delta function, is not really a function, but, combined with an integral, is an operator: It satisfies

Since, then

one can regard to be the derivative of H(t). Indeed, since , it is consistent with equation 4 that the Laplace transform of can be regarded as the Laplace transform of the derivative of H(t). It is readily verified that (see the first shifting theorem)

If f(t) is a periodic function of period , i.e.  , then

Theorem 2: page 220

The convolution of two functions, f(t) and g(t), is defined as the function

Theorem 1: page 232 (Convolution Theorem)

i.e. the Laplace transform of the convolution of two functions is the product of their Laplace transforms. This can be useful in solving Volterra integral equations, such as equation (8, page 235): Solve for x(t) the equation

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