Eigenvalues without pain: CHEBFUN code. Eigenfunctions can be derived using the CHEBFUN with MATLAB since it has an overloaded eig function. The following code solves for the eigenfunctions of the Robin. The code can also readily evaluate the series coefficient. Thus the whole procedure of the method of separation of variables can be automated with a code similar to this.

On running the code you should get the following results for Biot \(=1\).

The eigenvalues are \(0.8603,3.4256,6.4373,9.5293,12.6453,15.7713\), and 18.9024 .

The series coefficients for Biot \(=1\) can be calculated as \(1.1191,-0.1517,0.0466,-0.0217\), 0.0124 , and -0.0080 .

It is only a matter of summing the series for any chosen time to find the temperature profiles. You can write a small piece of code and generate time-temperature plots for any given Biot number.

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xi = A = chebfun('xi', [0,1]); chebop (0,1); A. op (xi,u) diff(u,2); = % A.1bc % A. rbc - - = 'neumann'; 'Robin'; A.lbc (u) diff(u, 1) % neumann Biot 1.0 A.rbc= Q(u) diff(u, 1) + Biot u = (0,1); B chebop(0, 1); B.opQ(xi,u) -u; %% %, Then we find the eigenvalues with eigs. [F,L] eigs (A,B) omega- sqrt (diag(L)) % eigenvalues. % series coefficient can be evaluated i = 1; for i = 1:6 center F(0,1); N1 center int (F(:,1)); N2 B1 (1) A1 (i) int (F(:,1). - F(:,1) int (F(i) )/center = N1/N2 % Answer C1 = 1.119 end