# Irregular linear systems of PDEs with the conditions on solutions' projections

## TL;DR

This paper develops a theoretical framework using generalized Jordan sets to address boundary condition selection in irregular linear PDEs with irreversible operators, enabling existence, uniqueness, and continuous dependence results.

## Contribution

It introduces a novel approach combining Jordan structure, skeleton decomposition, and Lyapunov methods to solve boundary condition problems for singular PDEs.

## Key findings

- Unified method for boundary condition selection in irregular PDEs
- Proved existence and uniqueness theorems for solutions
- Applicable to integral-differential equations with partial derivatives

## Abstract

The theory of complete generalized Jordan sets is employed to reduce the PDE with the irreversible linear operator $B$ of finite index to the regular problems. It is demonstrated how the question of the choice of boundary conditions is connected with the $B$-Jordan structure of coefficients of PDE. The various approaches shows the combination alternative Lyapunov method, Jordan structure coefficients and skeleton decomposition of irreversible linear operator from the main part equation are among the most powerfull methods to attack such challenging problem. On this base the complex problem of the {\it correct choice of boundary conditions} for the wide class of the singular PDE can be solved. Aggregated existence and uniqueness theorems can be proved, solution may continuously depend on the function determied from the experiments. Such theory can be applied to the integral--differential equations with partial derivatives.