Adapted formal algorithms for the solution of differential equations (Q2706061)

This paper is devoted to creating new algorithms with computer realization for numerical solving linear first order partial differential equations. The author introduces a new method based on the operator analog of the given partial differential equation. For instance the equation NEWLINE\[NEWLINEy\partial u(x,y)/\partial x-x\partial u(x,y)/\partial y=\widehat u(x,y)NEWLINE\]NEWLINE is considered with unknown function \( u(x,y)\) and given \(\widehat u(x,y)\). It is shown that there exists an operator analog NEWLINE\[NEWLINE(\overline YD_x-\overline XD_y)f(x,y)=\widehat f(x,y),NEWLINE\]NEWLINE where \(f\) and \(\widehat f\) belong to the set of formal series \(F_{xy}\equiv \{\sum_{k,l=0}^{\infty }c_{kl}x^ky^l c_{kl}\in C\}\) (\(\overline X\), \(\overline Y\), \(D_x\), \(D_y\) are linear operators). Hence, the partial differential equation can be solved by solving the operator equation.