Question 4 (25 points) Consider the following heat equation Uxx(x, t) – uz(x, t) = 0, (0 < x < 10, 0

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Question 4 25 Points Consider The Following Heat Equation Uxx X T Uz X T 0 0 X 10 0 T 0 Eq Q4 1 Wit 1 (93.36 KiB) Viewed 19 times

Question 4 (25 points) Consider the following heat equation Uxx(x, t) – uz(x, t) = 0, (0 < x < 10, 0 <t<0) Eq.(Q4-1) with the given boundary conditions u(0,t) = 0, u(10,t) = 0, (0 < t < 0) Eq.(Q4-2) and initial condition u(x,0) = f(x), (0 < x < 10) Eq.(Q4-3) (a). (5 points) After calculations, u(x, t) can be expressed by the following series ппх t u(x,t) = 2n=1 Kn sin e-(1) 1 10 where Kr’s are some constants satisfying Eq.(Q4-1) and boundary conditions. Find an expression for Kn such that u(x,t) also satisfies the initial condition. (b). (10 points) For f(x) x, 0 < x < 5, (10 - X, 5 < x < 10 Eq.(Q4-4) Find Kn. (c). (10 points) Now, if the original boundary conditions Eq.(Q4-2) are replaced by the following new ones: Uz(0,t) = Ux(10, t) = 0, (0 <t<oo) Eq.(Q4-5) Find the solution u(x, t) satisfying boundary conditions Eq.(Q4-5) and initial condition Eq.(Q4-3).