Boyce/DiPrima 9 th ed, Ch 11.3: Non- Homogeneous Boundary Value Problems Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and Richard C. DiPrima, ©2009 by John Wiley & Sons, Inc. In this section we discuss how to solve nonhomogeneous boundary value problems for both ordinary and partial differential equations. We focus on problems in which the differential equation is nonhomogeneous and the boundary conditions are homogeneous. We assume that the solution can be expanded in a series of eigenfunctions of a related homogeneous problem, and then we determine the coefficients in this series so that the nonhomogeneous problem is satisfied.

Nonhomogeneous Sturm-Liouville Problems (1 of 9) Consider the boundary value problem consisting of the nonhomogeneous differential equation where  is a given constant and f a given function on 0  x  1, and the boundary conditions We assume that the functions p, p', q and r are continuous on 0  x  1, and further, that p(x) > 0 and r(x) > 0 on 0  x  1. We solve this problem by making use of the eigenfunctions of the corresponding homogeneous differential equation and the above boundary conditions.

Eigenfunctions for Homogeneous Case (2 of 9) Let 1 < 2 < … < n < … be the eigenvalues of this homogeneous problem, and let  1,  2, …,  n, … be the corresponding normalized eigenfunctions. We now assume that the solution y =  (x) of our original nonhomogeneous problem can be expressed as a series of the form

Eigenfunction Expansion (3 of 9) Thus we assume solution y =  (x) of our nonhomogeneous problem can be expressed as a series of the form where each  n satisfies the homogeneous differential equation and boundary conditions. However since we do not know  (x), we cannot use the above coefficient formula to find b n. Instead, we will try to determine b n so the nonhomogeneous problem is satisfied. Note that  (x) as given by the above series will satisfy the boundary conditions, since each  n does.

Substituting Series into Equation (4 of 9) Consider the differential equation that  (x) must satisfy: We substitute the series into the differential equation and attempt to determine the b n so that the differential equation is satisfied. We thus have assuming we can interchange summation and differentiation.

Rewriting Equation (5 of 9) We thus have Note that the function r appears on both sides of this equation. This suggests that we write the last term as r(x)[ f (x)/r(x)]. If f /r satisfies the conditions of Theorem , then It follows that

Coefficient Equations For   n (6 of 9) After collecting terms and canceling the nonzero r(x), we have For this equation to hold for 0  x  1, it can be shown that the coefficient on  n (x) must be zero, and hence ( n -  )b n – c n = 0, n = 1, 2, …. Suppose that   n for n = 1, 2, 3, …. That is,  is not equal to any eigenvalue of the corresponding homogeneous problem. Then b n = c n /( n -  ) for n = 1, 2, …., and

Formal Solution (7 of 9) We note that is a formal solution of the nonhomogeneous boundary value problem. That is, we have not shown that the series converges. However, any solution of this problem satisfies the conditions of Theorem , and in fact the more stringent conditions in the discussion immediately following this theorem. Thus it is reasonable to expect that this series does converge at each point x. This is the case, for example, provided f is continuous.

Coefficients when  = m (8 of 9) Next, recall ( n -  )b n – c n = 0, n = 1, 2, …. Suppose now that  = m for some m. That is,  equals one of the eigenvalues of the corresponding homogeneous problem. If  = m and c m  0, then it is impossible to solve for b m, and the nonhomogeneous problem has no solution. If  = m and c m = 0, then the above equation is satisfied regardless of b m, and hence b m is arbitrary. In this case the boundary value problem does have a solution, but it is not unique, since it contains an arbitrary multiple of the eigenfunction  m.

Orthogonality Condition (9 of 9) In the case  = m and c m = 0, it follows that Thus if  = m, the nonhomogeneous boundary value problem can be solved only if f is orthogonal to the eigenfunction  m corresponding to the eigenvalue m.

Theorem The nonhomogeneous boundary value problem has a unique solution for each continuous f whenever  is different from all the eigenvalues of the corresponding homogeneous problem. The solution is given by and converges for each x in 0  x  1. If  is equal to an eigenvalue m of the homogeneous problem, then the nonhomogeneous problem has no solution unless f is orthogonal to  m. In this case the solution is not unique and contains an arbitrary multiple of  m.

Theorem (Fredholm Alternative) For a given value of , the nonhomogeneous boundary value problem has a unique solution for each continuous f (if  is not equal to any eigenvalue m of the corresponding homogeneous problem), or else the homogeneous problem has a nontrivial solution (the eigenfunction  m corresponding to m ).

Example 1: Boundary Value Problem (1 of 4) Consider the boundary value problem This problem can be solved directly in an elementary way, and has the solution However, we will use method of eigenfunction expansions. Rewriting the differential equation, we have and hence  = 2, r(x) = 1, and f(x) = x.

Example 1: Orthonormal Eigenfunctions (2 of 4) We seek the solution of the given problem as a series of normalized eigenfunctions  n of the corresponding homogeneous problem These orthonormal eigenfunctions were found in Example 2 of Section 11.2 and are where the eigenvalues satisfy

Example 1: Coefficients (3 of 4) Recall that in Example 1 of Section 11.1 we found that Note  = 2  n for n = 1, 2,…. Thus we assume that the solution to the nonhomogeneous problem is given by The c n were found in Example 3 of Section 11.2, with

Example 1: Nonhomogeneous Solution (4 of 4) Thus our solution to the nonhomogeneous problem is By Theorem , this solution is unique, and hence must be equivalent to the solution mentioned earlier, The equivalence of these two solutions can also be shown by expanding this solution in terms of the eigenfunctions  n. Note that in other problems we may not be able to obtain a solution except by series or numerical methods.

Nonhomogeneous Heat Conduction Problems (1 of 8) To show how eigenfunction expansions can be used to solve nonhomogeneous problems for partial differential equations, consider the generalized heat conduction problem We assume that the functions p, p', q and r are continuous on 0  x  1, and further, that p(x) > 0 and r(x) > 0 on [0,1]. As in Section 11.1, we assume u(x,t) = X(x)T(t) and arrive at

Sturm-Liouville Problem (2 of 8) With our assumptions on p, q, r, the boundary value problem is a Sturm-Liouville problem. Thus there exists a sequence of eigenvalues 1 < 2 < …, and corresponding normalized eigenfunctions  1,  2 …. We will solve the nonhomogeneous problem by assuming and then showing how to determine the coefficients b n (t). Note that the boundary conditions are satisfied for u(x,t) of this form, as each  n satisfies the boundary conditions.

Differential Equation: Right Side (3 of 8) We next substitute u(x,t) into the differential equation From the first two terms on the right side, we formally obtain Thus

Differential Equation: Left Side and Nonhomogeneous Term (4 of 8) Consider again our differential equation On the left side, we obtain For the nonhomogeneous term F(x,t), we once again look at the ratio F(x,t)/r(x), obtaining Since F(x,t) is given, we can consider  n (t) to be known.

Differential Equation (5 of 8) Thus our differential equation becomes or For this equation to hold for 0 < x < 1, it can be shown that the coefficient on  n (x) must be zero, and hence

Differential Equation for b n (t) (6 of 8) Thus b n (t) is a solution of the first order differential equation where To determine b n (t) completely, we need an initial condition This is obtained from the initial condition Thus

Coefficents b n (t) (7 of 8) Thus b n (t) is a solution of the first order differential equation subject to the initial condition Using the method of integrating factors from Section 2.1, The first term on the right side depends on the function f through the coefficients B n, while the second term depends on the nonhomogeneous term F through the coefficients  n (s).

Solution (8 of 8) Thus solution of the generalized heat conduction problem is given by

Example 2: Heat Conduction Problem (1 of 5) Consider the heat conduction problem To solve, we use the normalized eigenfunctions  n (x) of the corresponding homogeneous problem where we recall from Example 2 of Section 11.2 that

Example 2: Form of Solution (2 of 5) We thus assume the following: where c n, as in Example 3 of Section 11.2, are given by

Example 2: Coefficents b n (t) (3 of 5) From the previous slide, we have It follows that

Example 2: Solution (4 of 5) From the previous slides, we have Thus the solution of our heat conduction problem is given by

Example 2: Speed of Convergence (5 of 5) Our solution can be written as Recall from Example 1 of Section 11.1 that and hence n is nearly proportional to n 2. Since the trigonometric terms are bounded as n  , the first series converges similar to the series  ( n ) -2   n -4, and hence at most two or three terms are required to obtain an excellent approximation to this part of the solution. The second series converges even more rapidly, because of the additional factor of e - nt.

Time-Independent Nonhomogeneous Boundary Conditions Eigenfunction expansions can be used to solve a much greater variety of problems than the preceding discussion and examples may suggest. For example, time-independent nonhomogeneous boundary conditions can be handled much as in Section To reduce the problem to one with homogeneous boundary conditions, subtract from u a function v that is chosen to satisfy the given boundary conditions. The difference w = u – v satisfies a problem with homogeneous boundary conditions, but with a modified forcing term and initial condition. This problem can be solved by the methods of this section.

Eigenfunctions of a Simpler Problem (1 of 2) One potential difficulty in using eigenfunction expansions is that the normalized eigenfunctions  n (x) of the corresponding homogeneous problem must be found. For a differential equation with variable coefficients this may be difficult, if not impossible. In such a case it is sometimes possible to use other functions, such as eigenfunctions of a simpler problem, that satisfy the same boundary conditions. For example, if u(0,t) = 0 and u(1,t) = 0, then we may want to replace the functions  n (x) by sin(n  x). These functions at least satisfy the boundary conditions, although in general they are not solutions of the corresponding homogeneous equation.

Eigenfunctions of a Simpler Problem (2 of 2) Next, we expand the nonhomogeneous term F(x,t) in a series of the form F(x,t)/r(x) =  n (t)  n (x), where again  n (x) is replaced by sin(n  x). These series for u(x,t) and F(x,t) are then substituted into the differential equation. Upon collecting the coefficients of sin(n  x) for each n, we have an infinite set of linear first order differential equations from which to determine b 1 (t), b 2 (t),…. The equations here for b n (t) are coupled, and cannot be solved one by one, as before, but must be solved simultaneously. In practice, the infinite system is replaced by an approximating finite system, from which approximations to a finite number of coefficients are calculated.

Higher Order Boundary Value Problems, Formal Solutions Boundary value problems for equations of higher than second order can also often be solved by eigenfunction expansions. In some cases the procedure parallels almost exactly that for second order problems. However, a variety of complications can also arise. Finally, we emphasize that the discussion in this section has been purely formal. Separate and sometimes elaborate arguments must be used to establish convergence by eigenfunction expansions or to justify some of the steps used, such as term-by-term differentiation of eigenfunction series.

Green’s Functions There are also other, altogether different methods for solving nonhomogeneous boundary value problems. One of these leads to a solution expressed as a definite integral rather than as an infinite series. This approach involves certain functions known as Green’s functions. See Problems 28 – 36 in the exercises.