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

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Eq Q4 1 Question 4 25 Points Consider The Following Heat Equation Uxx X T Uz X T 0 0 X 10 0 T With Th 1 (51.07 KiB) Viewed 26 times

Eq.(Q4-1) Question 4 (25 points) Consider the following heat equation Uxx(x, t) – uz(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 u(x,t) = En=1 Kn sin "IX 10 where Kr's are some constants satisfying Eq.(Q4-1) and boundary conditions. Find an expression for K, 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 K. (C). (10 points) Now, if the original boundary conditions Eq.(Q4-2) are replaced by the following new ones: ux(0,t) = uz (10,t) = 0, (0 <t<.) Eq.(Q4-5) Find the solution u(x, t) satisfying boundary conditions Eq. (Q4-5) and initial condition Eq.(Q4-3).