Another method for finding particular solutions of equations of mathematical physics

M. L. Zaytsev ; V. B. Akkerman

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 119-127

Résumé

Differential equations of Mathematical Physics and their partial solutions are analyzed by means of analytical endeavors. While a possibility of reducing the dimension of the overdetermined systems of ordinary differential equations (ODEs) was shown in our previous studies, such that consistent reduction of these equations yielded partial solutions of the original system of partial differential equations (PDEs), here we consider an arbitrary overdetermined system of $1^{st}$-oder PDEs, with the number of equations exceeding that of unknown variables. Specifically, these equations are differentiated with respect to all the variables certain times. As a result, a new set of implicit equations is obtained, with the number of equations exceeding the number of unknown variables, and with these unknown variables being all the unknown functions of the previous system, as well as their derivatives. This new equations are solved, and it is demonstrated that such a solution appears also either the solution to the initially-determined system of equations or the corresponding partial derivatives of this solution unless the Jacobian is not identically zero. If an original overdetermination of the system of differential equations is performed by means of a Cauchy problem, then the solution to this Cauchy problem is among the solution of the system. This method allows finding the analytical solution of the system as long as the system is overdetermined for any sufficiently smooth solution. It is shown that the number of reduced implicit equations can be very large. A number of these reduced implicit equations, in which the minimum cost of computing power is required, is estimated. In particular, this method is employed to the oscillator equations. In summary, a potential computational realization of this method is discussed, including the need to produce a large number of symbol operations, and comparing this method with the method of differential links.

Détail

Keywords: overdetermined systems of differential equations, dimension of differential equations, particular and analytical solutions, partial differential equations.
Mots-clés : ODE