Trial solution methods to solve the hyperbolic heat conduction equation

Trial solution methods combined with Laplace transformation technique are used to present an analytic approximate solution for the hyperbolic heat conduction (HHC) equation. The trial solution methods used in this work are weighted residual methods and Ritz variational method. The weighted residual methods involves the application of different optimizing criteria, which are the collocation, subdomain, least square and the Galrekin optimizing methods. Trial solution procedures are carried out after transforming the HHC equation from the time domain into the Laplace domain. The solution of the transformed equation is expanded in the form of a shape function. The shape function is a function of space and undetermined coefficients. In this work, two shape functions are used: polynomial and hyperbolic. Applying the trial solution methods yields a system of algebraic equations that is solved symbolically using a commercial computerized symbolic code. Finally, the solution in time domain is obtained by inverting the solution of the transformed equation. It is found that the trial solution methods using polynomial approximate functions up to fourth order are not able, to capture the sharp gradient in the vicinity of the heat wave. Whereas, the hyperbolic shape function mimic the exact solution for all methods.