Consider the following heat equation Uxx(x, t) – ut(x, t) = 0, (0 < x < 10, 0

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Consider The Following Heat Equation Uxx X T Ut X T 0 0 X 10 0 T 0 Eq Q4 1 With The Given Boundary Co 1 (149.1 KiB) Viewed 22 times

Consider the following heat equation Uxx(x, t) – ut(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 2 t u(x, t) = 2n=1 Kn sin ηπχ e 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) 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) = 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).