![]() ![]() In our approach, for the situation where the completion process leads to a consistent system, we solve the latter by imitating what one would do with pen and paper: Solve one equation, substitute it into the next equation, and continue the process until the equations of the system are exhausted. the orders of the individual equations in the system can be different). Recently, using machinery from differential geometry, Kruglikov and Lychagin have extended the Bour –Mayer method to systems of PDEs in several dependent and independent variables of mixed orders (i.e. When applicable, it has the advantage of being easy to implement and efficient. This differs from the traditional approach, which uses differential Gr öbner bases to discover compatibility conditions. obstructions to the integrability) of the underlying system of PDEs and to iteratively prepend these compatibility conditions to the system until a consistent or an inconsistent system is found. The approach we adopt uses the Bour –Mayer method to find compatibility conditions (i.e. Our ultimate goal is to automate the search of general symbolic solutions of these systems. In this article, we focus solely on the integration of simultaneous systems of scalar first-order PDEs that is, our systems have at least two equations, one dependent variable (the unknown function) and several independent variables. These problems comprise the computation of discrete symmetries of differential equations, the calculation of differential invariants and the determination of generalized Casimir operators of a finite-dimensional Lie algebra. ![]() The search of solutions of many problems leads to overdetermined systems of partial differential equations (PDEs). The method we employ for assessing the consistency of the underlying system differs from the traditional use of differential Gr öbner bases, yet seems more efficient and straightforward to implement. ![]() We solve compatible systems recursively by imitating what one would do with pen and paper: Solve one equation, substitute its solution into the remaining equations, and iterate the process until the equations of the system are exhausted. Our approach relies on the Bour –Mayer method to determine compatibility conditions via Jacobi –Mayer brackets. We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. ![]()
0 Comments
Leave a Reply. |