Mathematics 3363 Review for Examination I Fall 2015 1. Study the PDE help les on Dr. Walker's web site. 2. Review suggested homework problems and Graded Homework 1. 3. Derive the heat diusion equation in one space dimension and in two space dimensions. 4. Find the function such that 00 () = 1 + for 0 2, (0) = 1, and (2) = 4. 5. Find the value of for which the following problem has an equilibrium temperature distribution. 2 ( ) + for 0 and 0 ( ) = 2 ( 0) = () for 0 (0 ) = 1, and ( ) = for 0 Let be the equilibrium solution so that () = lim ( ) for 0 Find a formula for () that does not contain any undetermined constants. 1 6. Show that all Sturm-Liouville problems are self-adjoint by showing that 0 1 0 1 () = () 1 0 1 0 for all such problems. 7. Consider the following two-point boundary value problem in which is a positive number. (i) 00 () = () for 0 , (ii) 0 (0) = 0, and (iii) () = 0. Use the Rayleigh Quotient to show that all eigenvalues are non negative. How do you know that 0 is not an eigenvalue? 8. Find 2 2 matrices and so that conditions (i) and (ii) given in the previous problem are equivalent to (0) () 0 + = 0 (0) 0 () 0 9. For the two-point boundary value problem given in problem 7, nd the matrix () and the determinant () in the case where 0. 10. For the two-point boundary value problem given in problem 7, nd a proper listing of eigenvalues and eigenfunctions. 11. Suppose that { } is orthogonal on [ ] and 6= 0 for =1 P = 1 . Suppose that = =1 . Derive a formula that gives in terms of , , and the inner product. Suggestion: Note that X = =1 so = X = =1 X =1 2 12. Derive the solution to ( ) (0 ) ( ) ( 0) = = = = 2 2 ( ) for 0 and 0 , 0 for 0 0 for 0 and () for where and is a positive number. 13. Derive the solution to ( ) (0 ) ( ) ( 0) = 2 ( ) for 0 and 0 , 2 = 0 for 0 = 0 for 0 and = () for where and is a positive number. 14. Derive the solution to ( ) (0 ) ( ) ( 0) = 2 ( ) for 0 and 0 , 2 = 0 for 0 = 0 for 0 and = () for 0 where each of and is a positive number. 15. Find the solution to 2 ( ) = 2 ( ) for 0 and 0 1, (0 ) = (1 ) = 0 for 0, and ( 0) = sin for 0 1. 3 with 0 and where = tan 16. Sketch the graphs where = 2 with 0 on the same set of axes and explain how to nd numerical approximations to the rst two numbers such that 2 sin cos = 0 17. Suppose that each of and is a positive number. Find the function that satises 00 () = () for 0 , (0) = 0, and () = 1. This is not an eigenvalue problem. 18. Suppose that each of and is a positive number. Derive the solution to 2 2 ( ) + 2 ( ) = 0 for 0 and 0 , 2 (0 ) = ( ) = 0 for 0 , ( 0) = (), and ( ) = 0 for 0 . 19. Suppose that each of and is a positive number. Derive the solution to 2 2 ( ) + 2 ( ) = 0 for 0 and 0 , 2 (0 ) = ( ) = 0 for 0 , ( ) = (), and ( 0) = 0 for 0 . 20. Suppose that each of and is a positive number. Derive the solution to 2 2 ( ) + 2 ( ) = 0 for 0 and 0 , 2 (0 ) = () and ( ) = 0 for 0 ( ) = 0 and ( 0) = 0 for 0 . 4 21. Suppose that each of and is a positive number. Derive the solution to 2 2 ( ) + 2 ( ) = 0 for 0 and 0 , 2 (0 ) = 0 and ( ) = () for 0 , ( ) = 0 and ( 0) = 0 for 0 . 22. Find the function of the form ( ) = + + + such that (0 0) (2 0) (2 4) (0 4) = = = = 1 3 4 and 2 23. Suppose that each of and is a positive number. Derive the solution to 2 2 ( ) + 2 ( ) = 0 for 0 2 and 0 1, 2 (0 ) = 2 + + 4 and (2 ) = 8 + 6 for 0 1, ( ) = 4 + 6 and ( 0) = 2 + 4 for 0 2. In order to improve the convergence of the series solution, do this by rst nding a function of the form ( ) = + + + that agrees with the given boundary conditions at the four corners of the rectangle. Then let ( ) = ( ) ( ) for all ( )in the rectangle. Calculate the boundary conditions for ( will be zero at the four corners ) and noting that is also a solution to Laplace's equation nd the function Find by noting that = + . 5 24. Suppose that each of and is a positive number. Derive the solution to 2 ( ) 2 (0 ) ( ) ( 0) ( 0) 2 ( ) for 0 and all in R, 2 = 0 for all in R, = 0 for all in R, = () for 0 , and = 2 = () for 0 . 25. Suppose that each of and is a positive number. Derive the solution to 2 ( ) 2 (0 ) ( ) ( 0) ( 0) 2 ( ) for 0 and all in R, 2 = 0 for all in R, = 2 = 0 for all in R, = () for 0 , and = () for 0 . 6