Ch 11.1: The Occurrence of Two-Point Boundary Value Problems As a result of separating variables in a partial differential equation in Chapter 10, we repeatedly encountered This boundary value problem is the prototype of a larger 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.

Heat Conduction Problem In Chapter 10 we used the method of separation of variables to solve certain problems involving partial differential equations. The heat conduction problem is typical of the problems considered here.

Eigenvalue Problem In using the separation of variables method to solve such problems, a crucial part of the process is to find eigenvalues and eigenvectors of the differential equation with boundary conditions or perhaps The sine or cosine functions that result from solving this boundary value problem are used to expand the initial temperature distribution f(x) in a Fourier series.

First Generalization In this chapter we extend the results of Chapter 10. Our main goal is to show how the method of separation of variables can be used to solve more general problems. We are interested in three types of generalizations. First, we consider more general partial differential equations. For example, the equation can arise in the study of heat conduction in a bar of variable material properties in the presence of heat sources. If p and r are constants, and if the source terms qu and F are zero, then this equation reduces to the heat equation.

Second Generalization Secondly, we allow more general boundary conditions. In particular, we consider boundary conditions of the form Such conditions occur when the rate of heat flow through an end of the bar is proportional to the temperature there. Usually, h1 and h2 are nonnegative constants, but in some cases they may be negative or depend on t. The boundary conditions for the previously mentioned heat conduction problem are obtained as h1  , and h2  . Also, h1 = 0 and h2 = 0 correspond to insulated ends of bar.

Third Generalization The third generalization discussed in this chapter concerns the geometry of the region in which the problem is posed. The results of Chapter 10 are adequate only for a restricted class of problems, mainly those in which the region of interest is rectangular, or in a few cases, circular. Later in this chapter we consider certain problems posed in a few other geometrical regions.

Generalized Heat Conduction Problem (1 of 3) Consider the generalized heat conduction problem To solve this problem, we assume u(x,t) = X(x)T(t). Substituting this into our differential equation, we obtain or where  is a constant.

Boundary Conditions (2 of 3) Thus From the boundary conditions we have and hence

Eigenvalue Problem (3 of 3) To proceed further, we must solve Although this is a more general linear homogeneous two-point boundary value problem than the solutions behave in very much the same way. For certain values of , called eigenvalues, our current problem above has nontrivial solutions, called eigenfunctions. These eigenfunctions for the basis for series solutions of a variety of problems in partial differential equations, including our generalized heat conduction problem.

Form of Differential Equation In this chapter we discuss properties of solutions of two-point boundary value problems for second order linear equations. Sometimes we consider the general linear homogeneous equation investigated in Chapter 3, However, for most purposes it is better to discuss equations in which the first and second derivative terms are related as in It is always possible to transform from the general equation (1) above so that the derivatives appear as in Equation (2) above.

Number of Boundary Conditions Boundary value problems with higher order differential equations can also occur. In them the number of boundary conditions must equal the order of the equation. As a rule, the order of the differential equation is even, and half the boundary conditions are given at each end of interval. It is also possible for a single boundary condition to involve values of the solution and/or its derivatives at both boundary points. For example,

Example 1: Boundary Value Problem (1 of 6) Consider the boundary value problem One place where this problem occurs is in the heat conduction problem in a bar of unit length. The boundary condition at x = 0 corresponds to a zero temperature there. The boundary condition at x = 1 corresponds to a rate of heat flow that is proportional to the temperature there, and the units are chosen so that the constant of proportionality is 1 (see Appendix A in Chapter 10).

Example 1: Boundary Conditions (2 of 6) Our boundary value problem is If  = 0, then the general solution is y = c1x + c2. The boundary conditions require c1 = c2 = 0, and hence  = 0 is not an eigenvalue. If  > 0, then the general solution is The boundary condition at x = 0 requires c2 = 0, and the boundary condition at x = 1 implies For a nontrivial solution, we assume c1  0.

Example 1: Eigenvalue Equation for  > 0 (3 of 6) We have Note that and the above equation is not satisfied. Thus we assume the cosine term is nonzero and rewrite our equation as We can solve this equation numerically for the eigenvalues , or find the points of intersections of

Example 1: Eigenvalues for  > 0 (4 of 6) Given below are the graphs of The intersection points are given with reasonable accuracy by The accuracy improves as n increases. Hence the eigenvalues are The corresponding eigenfunctions are

Example 1: Boundary Conditions for  < 0 (5 of 6) Recall our boundary value problem If  < 0, let  = - where  > 0. The general solution is Proceeding as in the previous case, we find that Given below are the graphs of Since there are no intersection points for  > 0,  cannot be negative.

Example 1: Complex  (6 of 6) Finally, it can be shown by direction calculation that has no complex eigenvalues. However, in Section 11.2 we consider in more detail a large class of problems that includes this example. One characteristic of these problems is that they have only real eigenvalues. Therefore we omit the discussion of the nonexistence of complex eigenvalues here.