Eigenvalues and solutions for laminar flow heat transfer with CHEBFUN

Eigenvalues and solutions for laminar flow heat transfer with CHEBFUN. This exercise demonstrates how to find the eigenfunctions using CHEBFUN. The procedure is the same as that for the transient conduction prolbem. The first step is to declare ξ as a CHEBFUN and write Eq. (12.44) as a CHEBOP operator with an LHS and RHS such that the RHS is the term with λ2. Please fill in the code details.

Now we can use the eig funciton in MATLAB directly, but in our case this is an overloaded operator and computes the eigenvalues of the differential equation rather than for a set of of algebraic equations. But the syntax is the same. Thus [ F, L ] = eig (LHS, RHS) gives the first six eigenvalues and the corresponding eigenfunctions. These functions can be manipulated in a symbolic manner. Apply this to the Graetz problem to find the eigenfunctions and eigenvalues. The series coefficient and the complete solution can be coded, and the method provides a simple and efficient method to solve convective transport problems in internal flow.