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# Y λy 0 x 1 y 0 y 0 0 y 1 0 this is not exactly the

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y ′′ + λy = 0 , 0 < x < 1 , y (0) + y (0) = 0 , y (1) = 0 . (This is not exactly the same problem as the preceding one!) 5. Compute all eigenfunctions and eigenvalues of the following regular Sturm-Liouville problem. y ′′ + λy = 0 , 0 < x < 1 , y (0) + 1 2 y (0) = 0 , y (1) = 0 . (Even more interesting!) III. Separation of variables. The equations seen in this course were: The heat equation in an interval (bar), a rectangle, a disk. Mostly in an interval. The wave equation in an interval (finite string), in a rectangle (rectangular membrane), and a disk (circular membrane). Laplace’s equation in a rectangle or disk. Disk problems for the wave equation (also for the heat equation) involve Bessel functions. For practice: Review (do again or for the first time) the exercises from homeworks 5, 6, 7, and 8. As mentioned above, expect at least two, perhaps three, exercises of this type in the final exam. Looking (doing) again the first three exercises of Exam 2 is also a good idea. IV. Equilibrium Solutions. Here one considers problems of the form u t ( x, t ) = ku xx ( x, t ) + q ( x ) , 0 < x < L, t > 0 with boundary conditions of one of the following forms valid for all t > 0: u (0 , t ) = α, u ( L, t ) = β, or u x (0 , t ) = α, u ( L, t ) = β, or u x (0 , t ) = α, u x ( L, t ) = β, with an initial condition of the form u ( x, 0) = f ( x ), 0 < x < L . The question is: Determine the equilibrium solution if it exists; determine conditions under which an equilibrium solution exists. For practice, here is the 4th exercise from Exam 2, followed by a more conceptual exercise:

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3 1. Consider the ICBC problem u t ( x, t ) = 4 u xx ( x, t ) + γx, 0 < x < 2 , t > 0 , u x (0 , t ) = 5 , u x (2 , t ) = 3 , t > 0 , u ( x, 0) = - 1 2 x 2 + 5 x, 0 < x < 2 , Determine the value of γ for which there is an equilibrium solution (10 points) and find it (10 points). 2. Suppose u ( x, t ) is a solution of the problem u t ( x, t ) = ku xx ( x, t ) + q ( x ) , 0 < x < L, t > 0 , u x (0 , t ) = α, u x ( L, t ) = β, t > 0 u ( x, 0) = f ( x ) , 0 < x < L. Show that L 0 u ( x, t ) dx = ( k ( β - α ) + L 0 q ( x ) dx ) t + C, where C is a constant. In addition: (a) Use this to explain why L 0 q ( x ) dx = - k ( β - α ) is a necessary condition for equilibrium.
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