Solve the parabolic equation (6.24) with K =1, subject to the following boundary conditions: u(0,t)=0, u(1,t)=10, u(x,0)=0 for all x except x=1.

Solve the parabolic equation (6.24) with K =1, subject to the following boundary conditions: u(0,t)=0, u(1,t)=10, u(x,0)=0 for all x except x=1. When x=1, u(1,0)=10. Use the function heat to determine the solution for t =0 to 0.5 in steps of 0.01 with 20 divisions of x. You should plot the solution for ease of visualization. Solve the wave equation, (6.29) with c=1, subject to the following boundary and initial conditions: u(t,0) = u(t,1) = 0, u(0,x)= sin(πx) +2sin(2πx), and ut(0,x)= 0, where the subscript t denotes partial differentiation with respect to t. Use the function fwave to determine the solution for t =0 to 4.5 in steps of 0.05, and use 20 divisions of x. Plot your results and compare with a plot of the exact solution, which is given by u = sin (πx) cos (πt) + 2 sin (2πx) cos (2πt).

 

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