Fundamental Solutions of Homogeneous PDEs

Este artigo amplia o conjunto soluções da  EDP p-Laplace
(u_x)² u_{xx} + 2u_x u_y u_{xy} + (u_y)² u_{yy) = 0
apresentados em diversos artigos. Com esse intuito, utilizamos o método de Monge para equações diferenciais parciais uniformes, reduzindo esta EDP de segunda ordem no sistema de Monge, cuja solução resulta numa EDP de primeira ordem do tipo f(p; q) = 0. Na sequência aplicamos o nosso método para determinar a solução geral desta e portanto, uma solução generalizada com uma função arbitrária da EDP p-harmônica.

Purpose. The authors of known for us textbooks on quantum mechanics pay attention only to the first regular solution of the Schrödinger equation for the hydrogen atom. To exclude the second linearly independent solution from the general solution, different textbooks give various arguments such as invalid boundary conditions in the coordinate origin, the appearance of Dirac delta function, or divergence of the kinetic energy in the origin. Methods. Using the power series method, we obtained an exact analytic expression for the second independent solution of the Schrödinger equation for the hydrogen atom. Results. The solution consists of a sum of two parts, one of which increases indefinitely over long distances, while the other is limited and contains a logarithmic term. This feature is peculiar to all values of the orbital angular momentums. Conclusions. On the example of the hydrogen atom, we demonstrated the mathematically correct algorithm of the construction of the independent solutions for the power series method. In particular, this algorithm is important in the case of quantum systems with coupled channels which are described by two or more coupled Schrödinger equations.

A method for finding the general solution to the partial differential
equations: F(ux, uy) = 0; F(f(x) ux, uy) = 0 (or F(ux, h(y) uy) = 0)
is presented, founded on a Legendre like transformation and a theorem for Pfaffian differential forms. As the solution obtained depends on an arbitrary function, then it is a general solution. As an extension of the method it is obtained a general solution to PDE: F(f(x) ux, uy) = G(x), and then applied to unidimensional Hamilton-Jacobi equation.

In this paper, we construct the fundamental solution to a degenerate diffusion of Kol-mogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previously known results for the Fokker-Planck equation and the Kramers equation.

In this article we construct the fundamental solutions for the wave equation arising in the de Sitter model of the universe. We use the fundamental solutions to represent solutions of the Cauchy problem and to prove the $L^p-L^q$-decay estimates for the solutions of the equation with and without a source term.