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As in the case of the heat conduction equation, there are various generalizations of the wave equation (5). One important equation is known as the telegraph equation and has the form
utt + cut + ku = a2uxx + F (x, t), (7)
where c and k are nonnegative constants. The terms cut, ku, and F(x, t) arise from a viscous damping force, an elastic restoring force, and an external force, respectively. Note the similarity of Eq. (7), except for the term a2uxx, with the equation for the spring-mass system derived in Section 3.8; the additional term a2uxx arises from a consideration of internal elastic forces.
For a vibrating system with more than one significant space coordinate, it may be necessary to consider the wave equation in two dimensions,
a (uxx + uyy) = utt, (8)
or in three dimensions,
a2(uxx + uyy + uzz) = utt¦
Chapter 10. Partial Differential Equations and Fourier Series
REFERENCES The following books contain additional information on Fourier series:
Buck, R. C., and Buck, E. F., Advanced Calculus (3rd ed.) (New York: McGraw-Hill, 1978). Carslaw, H. S., Introduction to the Theory of Fourier’s Series and Integrals (3rd ed.) (Cambridge: Cambridge University Press, 1930; New York: Dover, 1952).
Courant, R., and John, F., Introduction to Calculus and Analysis (New York: Wiley-Interscience, 1965; reprinted by Springer-Verlag, New York, 1989).
Kaplan, W., Advanced Calculus (4th ed.) (Reading, MA: Addison-Wesley, 1991).
A brief biography of Fourier and an annotated copy of his 1807 paper are contained in: Grattan-Guinness, I., Joseph Fourier 1768-1830 (Cambridge, MA: MIT Press, 1973).
Useful references on partial differential equations and the method of separation of variables include the following:
Churchill, R. V, and Brown, J. W., Fourier Series and Boundary Value Problems (5th ed.) (New York: McGraw-Hill, 1993).
Haberman, R., Elementary Applied Partial Differential Equations (3rd ed.) (Englewood Cliffs, NJ: Prentice Hall, 1998).
Pinsky, M. A., Partial Differential Equations and Boundary Value Problems with Applications (3rd ed.) (Boston: WCB/McGraw-Hill, 1998).
Powers, D. L., Boundary Value Problems (4th ed.) (San Diego: Academic Press, 1999).
Strauss, W. A., Partial Differential Equations, an Introduction (New York: Wiley, 1992). Weinberger, H. F., A First Course in Partial Differential Equations (New York: Wiley, 1965; New York: Dover, 1995).
Boundary Value Problems and Sturm-Liouville Theory
As a result of separating variables in a partial differential equation in Chapter 10 we repeatedly encountered the differential equation
X" + êX = 0, 0 < x < L
with the boundary conditions
X (0) = 0, X (L) = 0.
This boundary value problem is the prototype of a large class of problems that are important in applied mathematics. These problems are known as Sturm-Liouville boundary value problems. In this chapter we discuss the major properties of Sturm-Liouville problems and their solutions; in the process we are able to generalize somewhat the method of separation of variables for partial differential equations.
11.1 The Occurrence of Two-Point Boundary Value Problems
In Chapter 10 we described the method of separation of variables as a means of solving certain problems involving partial differential equations. The heat conduction problem consisting of the partial differential equation
a2uxx = ut, 0 < x < L, t > 0 (1)
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
subject to the boundary conditions
u(0, t) = 0, u(L, t) = 0, t > 0 (2)
and the initial condition
u(x, 0) = f (x), 0 < x < L, (3)
is typical of the problems considered there. A crucial part of the process of solving such problems is to find the eigenvalues and eigenfunctions of the differential equation
X" + XX = 0, 0 < x < L (4)
with the boundary conditions
X (0) = 0, X (L) = 0 (5)