Obtaining exact analytical solutions for nonstationary heat conduction problems using orthogonal methods

Through the interplay of orthogonal methods by L. V. Kantorovich, Bubnov–Galerkin and a heat balance integral method there have been obtained an exact analytical solution of a nonstationary heat conduction problem for an infinite plate under the symmetrical first-type boundary conditions. It was possible to obtain an exact solution through the employment of approximate methods due to the appliance of trigonometric coordinate functions, possessing the property of orthogonality. They enable us to determine eigenvalues not through the solution of the Sturm–Liouville boundary value problem, which supposes the second-order differential equation integration, but through the solution of a differential equation for an unknown function on time, which is the first-order equation. Due to the property of coordinate functions mentioned above, while determining constants of integration out of initial conditions it is possible to avoid solving large systems of algebraic linear equations with ill-conditioned matrix of coefficients. Thus, it simplifies both the process of obtaining a solution and its final formula and provides an opportunity to find not only an approximate, but also an exact analytical solution, represented by an infinite series.