ON THE CLOSED-FORM SOLUTIONS OF THE WAVE, DIFFUSION AND BURGERS EQUATIONS IN FLUID-MECHANICS

In this paper the closed-form solutions of the first-order wave equation with source terms u(t) + g(u)u(x) = f(u), and of the diffusion equation of the general form u(t) + g(u)u(x) = vu(xx), are constructed. Both equations are considered for smooth initial and boundary conditions, namely u(0, x) = phi(x) and u(t, x(0)) = f(t), respectively. Furthermore, the case of Burgers equation with source terms u(t) + uu(x) = vu(xx) - lambda u, appearing in the aerodynamic theory, is investigated. The developed solution techniques and the obtained closed-form solutions may be proved powerful in applications.