Eq.(Q4-1) Consider the following heat equation Uxx(x, t) - ut(x, t) = 0, (0 < x < 10, 0

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Eq Q4 1 Consider The Following Heat Equation Uxx X T Ut X T 0 0 X 10 0 T 0 With The Given Boundary Con 1 (49.77 KiB) Viewed 17 times

Eq.(Q4-1) Consider the following heat equation Uxx(x, t) - ut(x, t) = 0, (0 < x < 10, 0<t<0) with the given boundary conditions u(0,t) = 0, u(10,t) = 0, (0 <t<00) 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 te()。 u(x, t) = En=1 Kn sin ηπα 10 where K'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, $5x<10 x, 0 < 5 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) = 4,(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).