Eq.(Q4-1) Question 4 (25 points) Consider the following heat equation Uxx(x,t) – Up(x,t) = 0, (0

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Eq Q4 1 Question 4 25 Points Consider The Following Heat Equation Uxx X T Up X T 0 0 X 10 0 T With The G 1 (51.72 KiB) Viewed 23 times

Eq.(Q4-1) Question 4 (25 points) Consider the following heat equation Uxx(x,t) – Up(x,t) = 0, (0<x< 10, 0<t<) with the given boundary conditions u(0,t) = 0, u(10,t) = 0, (0 <t<) and initial condition u(x,0) = f(x), (0 < x < 10) Eq.(Q4-2) Eq.(Q4-3) (a). (5 points) After calculations, u(x,t) can be expressed by the following series NT u(x, t) = &n=1 Kn sin плх e 10 where Ki's are some constants satisfying Eq.(Q4-1) and boundary conditions. Find an expression for K, such that u(x,1) also satisfies the initial condition. (b). (10 points) For 0<x<5, f(x) = {i} (10 - x, 5 < x < 10 Find K. Eq.(Q4-4) (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<0) Eq.(Q4-5) Find the solution u(x, t) satisfying boundary conditions Eq.(Q4-5) and initial condition Eq.(Q4-3).