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by: Eunice Schoen


Eunice Schoen
GPA 3.61


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This 2 page Study Guide was uploaded by Eunice Schoen on Saturday September 26, 2015. The Study Guide belongs to MATH 440 at James Madison University taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/214036/math-440-james-madison-university in Mathematics (M) at James Madison University.




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Date Created: 09/26/15
M440 Possible Final Exam Questions iDisclaimer Some may be worded di erently some may only include selected parts some problems or parts of problems may be combined or maybe not Fall 2006 Additional problems related to solving 1st order pdes will also be on the final you guys botched these on Exam 1 These problems are PLEDGED You may discuss generalities with classmatesl but no specificsI no looking at others solutions a 1 m I Let ex Let C be the Fourier Cosine Series Coe icientsfor e e 5C0 2C COS quotIE quot1 a Determine the C without actually computing the coe icient integrals by properly di 2rentiating the above Fourier Cosine Series for f x ex twice b Justify why your first di erentiation yields the Fourier Sine Series of ex c Use your work in part a to give the Fourier Sine Series of ex again without needing to compute the coe icient integral 0 lt x lt 1 t gt 0 u ku 2 Solve the heat equation 3 VP u0t u1t 0 t gt 0 Where is this model incompatible xx ux0x 0ltxlt1 3 Repeat exercise 2 with the following changes a For insulated boundaries b For insulated boundaries and initial temperature distribution 9 0 3 COS 475x c For insulated boundaries and initial temperature distribution 9 0 1 d For zero prescribed temperature at the boundaries and initial temperature distribution 9 0 1 2 When the boundary and initial conditions are as in part d and when there is a heat source in the rod directly proportional to position along the rod That is the PDE becomes u kuDC ax a gt 0 Hint Solve by the method of eigenfunction expansion and appropriately represent the nonhomogeneous source term as a Fourier Series u uu 4u 0ltxltltgt0 4 COHSider the heat equation IBVP 110 711 0 t gt 0 which corresponds to a 1D rod either with ux0 fx 0 lt x ltl heat loss through the lateral sides with outside temperature 0 or with an insulated lateral surface and heat source of minus twice the temperature in the rod at any pointx and any time t a Determine the equilibrium temperature distribution ux by considering u gt 0 b Solve the PDE 3 VP for uxt by the method of separation of variables c Justify why your infinite series solution in part b is valid d Use your solution in part b to examine the limit as I gt ofuxi e Compare parts a and d u um4u 0ltxltltgt0 5 Consider the heat equation IBVP u0t ult 0 t gt 0 Solve the PDEIB VPfor uxi by the ux0 fx 0 lt x ltl method of eigenfunction expansion Then use your solution to show each of the following a If4lt7L39l2 then u gt0 as t gt b1f47rl2 then u ACSin as t gt cIf4gt7rl2 u gt as t gt d What does the above suggest that the critical length of the rod is That is for what length rod does the long time temperature distribution change character Discuss in words physically what the three casesaic predict 6 a Solve the eigenvalue problem X x IXx X0 X27r X 0 X 27L39 b Discuss a scenario both physical and mathematical IBVP for the heat equation where this problem arises in the separation of variables c Give the solution ofyour I BVP in b 7 Determine the equilibrium temperature distribution for the thin circular ring problem in each of the following ways a By solving the ODE for ux from the equilibrium problem by considering u gt 0 b By computing Fight 0 8 Exercise 5 6 9 Exercise 5 7 10 Derive the formula for the Laplacian A L7 r 9 in polar coordinates by properly converting the 2 partials of 9 y in the Cartesian expression A 9 y to polar form You may want to get help from yourM23 7 or other text on polar coordinate transformations and multidimensional chainrules utkuu0ltxltltgt0 I a Determine an series solution to the heat equation I BVP MAOJ 711 0 t gt 0 ux0 fx 0 lt x ltl b Use your work in a to give the solution of the I B VP when fx x Derive any orthogonality results you need to complete your solution i e to determine all constants c Determine the equilibrium temperature distribution for your problem in b d Discussphysically the I BVP ofb and the result ofc u kuu 0ltxltltgt0 12 Let kgt0 and consider the IBVP u0t ult 0 t gt 0 This PDE is called the backwards heat ux0 fx 0 lt x ltl equation a From your knowledge of the solution of the above 3 VP when the PDE is the regular heat equation PDE u klDC give an series solution to this backwards heat equation 3 VP Hint No work needed here b Show that the above 3 VP has a unique solution Hint Prove by contradiction c The above 3 VP is not stable Explain in words what this means What does this imply about the backwards heat equation 3 VP problem posed above 13 Justi that our 0quot series solution 96 I 2 H X I to the Neumann I BVPfor the 1 D Wave n0 Equation is valid


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