Heat Conduction in an Insulated Bar

We consider the unsteady heat conduction in an insulated bar. The thermal energy balance can be transformed into the dimensionless form 

Since we have not used the symmetry boundary condition that dTfdy = 0 at y = 1/2, we will solve the problem by performing an orthogonal collocation in the spatial domain using the shifted Legendre polynomials as the basis function. This will reduce the problem to a set of initial-value ordinary differential equations that can be solved using IMSL ordinary differential equation routines. As discussed by Cooper et al. (1986), seven internal collocation points accurately describe the solution to the partial differential equations. Therefore, we use equation (8.12.14) to approximate the second spatial derivative. This reduces the original partial differential equation of (8.12.15) to 

Since the boundary conditions are iilways satisfied, dT l)/dt = dT 9)/dt = 0, and T(l) = T(9) = 0, we only need to solve for seven differential equations 

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