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Homogeneous PDEs and BCs and Eigenvalues

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Please see the attached file for the fully formatted problems.

Consider the following homogeneous PDE and BCs:

u_t = u_xx in 0 < x < 1
u(x,0) = f(x)
u(0,t) = 0, u(1,t) + u_x(1,t) = 0

(a) Make the substitution u(x,t) = phi(x)G(t), separate variables, and find the equations for phi(x) and G(t). Be sure to include the voundary conditions appropriately.

(b) Show that only lambda > 0 produces non-trivial solutions. If lambda >0, find the equation satisfied by the eigenvalues. Unlike the previous examples that we have seen, you will not be able to solve for the eigenvalues explicity.

(c) Write the equation that the eigenvalues satisfy as

F(lambda) = 0.

Find approximate values for the first four eigenvalues by using the MATLAB function fzero to compute the roots of F(lambda). You will need to provide fzero with initial guesses; create a MATLAB plot of F(lambda) to get an idea where the roots are.

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A homogeneous partial differential equation with boundary conditions and eigenvalues is solved. The solution is detailed and well presented.

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