INTRODUCTION

The two integral transform pairs considered in this section are a bit different from those considered in Section 15.4. The finite Fourier cosine and sine transforms are based not on the Fourier integral, but rather on the Fourier cosine and sine series discussed in Section 12.3. The transform of a function f is an integral, whereas the inverse transform is a Fourier series.

Finite Transforms

We begin with the definitions of the finite Fourier cosine and sine transforms. Before reading Definition 15.5.1, we suggest that you review Definition 12.3.1. Let f be a function defined on the interval

Operational Properties

Like the Fourier cosine and sine transforms discussed in the preceding section, the finite Fourier cosine and sine transforms are not appropriate for transforming an odd-order derivative of a function. So if f and are continuous and is piecewise continuous on the interval then the finite cosine transform of follows from integration by parts:

Using the notation we have

(5)

Similarly, for the finite sine transform we have

(6)

where

The transforms given in (5) and (6) are the determining condition in deciding which transform to use in solving a boundary-value problem. If the value of the derivative is given at 0 and p, we use the finite cosine transform, whereas if the function u is specified at 0 and p, we use the finite sine transform.

A word of caution when using the finite cosine transform. Consider the cases and separately. The solution of the ODE obtained by transforming the partial differential equation for may not include the solution of that ODE when See Problem 4 in Exercises 15.5. Also, in Problem 7, consider the cases and

EXAMPLE 1 Finite Sine Transform

Find the finite sine transform of on Find the inverse transform.

SOLUTION

With and integration by parts we see from (3) that the finite sine transform of f is

Then from (4) the inverse transform is

≡

EXAMPLE 2 Using the Finite Cosine Transform

Solve

SOLUTION

In view of the fact that the two boundary conditions involve the first partial derivative of we use the finite cosine transform with From (1) we have

(7)

and so, in view of (5),

As usual, we assume that we can interchange integration and differentiation:

Now in the case the transform of the partial differential equation is

This ordinary first-order differential equation is linear and has the solution Using integration by parts, the transform of the initial condition is

Hence (8)

Now in the case the transform of the partial differential equation is

From (7) we see that the transform of the initial condition is

Hence (9)

From (2) the form of the solution is

Finally, using (8) and (9) the inverse transform is

≡

15.5 Exercises Answers to selected odd-numbered problems begin on page ANS-39.

In Problems 1 and 2, find the finite sine transform of the function f. Find the inverse transform.

In Problems 3 and 4, find the finite cosine transform of the function f. Find the inverse transform.

In Problems 5–14, use a finite Fourier transform to solve the given boundary-value problem. The symbol is a constant.

5. 
   
6. 
   
7. 
   
8. 
   
9. 
   
10. 
    
11. 
    
12. 
    
13. 
    
14. 
    
15. Use a finite Fourier transform to find the steady-state temperature in the plate shown in FIGURE 15.5.1.

    

16. Use the substitution to change the boundary-value problem in Problem 15 into a boundary-value problem for Then use a finite Fourier transform to solve this second problem for Finally, determine Resolve any differences between the solution of Problem 15 with the solution of this problem.
17. Use the finite cosine transform of the fourth derivative
    
    to show that a solution of the boundary-value problem
    
    is
18. Use the finite sine transform of the fourth derivative
    
    to solve the boundary-value problem