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#### Consider the unsteady heat equation in one-dimension: ?T at 22T + (712 – 1)e-2+ sin( 7x); ?r2 0 < x < l; t > 0 (2) with the initial and boundary conditions: T(0,t) = T(1,t) = 0; T(x,0) = sin(7x) (3) You are to solve this problem using Forward Euler for the time differencing and centered scheme for space differencing

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Consider the unsteady heat equation in one-dimension: ?T at 22T + (712 – 1)e-2+ sin( 7x); ?r2 0 < x < l; t > 0 (2) with the initial and boundary conditions: T(0,t) = T(1,t) = 0; T(x,0) = sin(7x) (3) You are to solve this problem using Forward Euler for the time differencing and centered scheme for space differencing. Discretize your domain into (N+1) equally spaced points. (a) Derive discrete equation for temperature. Use say 6 total points to derive the finite difference approximation. Use Ar for grid spacing and At for time step. (b) Choosing 21 grid points (including the boundary points), and At 0.001 solve for the temperature to steady state. Plot your temperature distribution T(x) for five different times on the same graph (starting with t = 0 to steady state) to show the evolution of the temperature distribution. Note: the five different step sizes should be at times that are approximately equally spaced between t 0 and steady state, rather than at consecutive time-steps. (c) Verify that the time-evolution of your solution is correct. To do this, you will have to reduce your step size and time-steps and rerun your program. Do this for 2 different finer grid resolutions and compare your solutions for the same time-level.