Symbolic and Algebraic Mehtods for LPDOs (DIFFOP)

The solution of Partial Differential Equations (PDEs) is one of the most important problems of mathematics, and has an enormous area of applications. As is the case for many other types of mathematical problems, solution methods for PDEs can be classified into symbolic (or analytical) and numerical methods. Of course, an analytical solution is to be preferred. Indeed, using an analytical solution, one can compute a numerical solution to any precision and on any segment of the domain, analyze the solution`s behavior at infinity and at extremal points, explore dependence on parameters, etc. Whereas some simple Ordinary Differential Equations (ODEs) can be solved analytically, this happens more and more rarely as the complexity of the equations increases. One of the methods for extending the range of analytically solvable PDEs consists in transformations of PDEs and the corresponding transformations of their solutions. Thus, based on the fact that a second-order equation can be solved if one of its factorizations is known, the famous method of Laplace Transformations suggests a certain sequence of transformations of a given equation. Then, if at a certain step in this transformation process an equation becomes factorizable, an analytical solution of this transformed equation - and then of the initial one - can be found. Nowadays the search for analytical solutions of PDEs can benefit tremendously from the use of modern software systems in computer algebra and symbolic computation. The aim of this project is the further development and generalization of analytical approaches to the solution of PDEs and the corresponding algebraic theory of differential operators. In our previous work we have introduced the notion of obstacle for the factorization of a differential operator, i.e. conditions preventing a given operator from being factorizable. These obstacles give rise to a ring of obstacles and furthermore to a classification of operators w.r.t. to their factorization properties. From obstacles we can also get (Laplace) invariants of operators w.r.t. to certain (gauge) transformations. We have shown how such systems of invariants can be extended to full systems of invariants for certain low order operators. Another related problem is the description of the structure of families of factorizations. For operators of order 3 it has been shown that a family of factorizations depends on at most 3 or 2 parameters, each of these parameters being a function on one variable. In this project we plan to generalize the idea of obstacles to the case of systems of PDEs, and study their properties. At least for the case of linear partial differential operators (LPDOs) with coprime factors of the symbol, this seems to be achievable. For partial operators of order 4 the description of the structure of corresponding families of factorizations remains open. Generalizations to LPDOs with arbitrary symbols (without the complete factorization assumption), to high order LPDOs, and to those in multiple-dimensional space are of interest also. We will also work on methods for the determination of invariants for operators of order 3 or more. This should make it possible to classify operators in terms of full systems of invariants. The classical Laplace method for transforming PDEs of order 2 has been generalized in various ways: to nonlinear PDEs, to PDEs of higher order, etc. We plan to continue this work, which has obvious important applications for the analytical solution of PDEs. The theoretical results achieved in this project will be implemented in a symbolic computation program package.

In Mathematics, and in particular in Algebra, we are concerned with the solution of equations. Equations come in many varieties: linear equations, algebraic or polynomial equations, non-algebraic equations, differential equations are some important examples. A function in a variable quantity typically exhibits variable growth, which is described by the derivation of this function. The derivation in turn is again a function, and its derivation is called the second derivation of the original function; etc. Many interesting phenomena in science and technology, but also in economics, admit descriptions as relations between certain functions and their derivations. Such a relation is called a differential equation. Of course there are also differential equations depending on several variables, so- called partial differential equations; and also differential equations in several unknown functions. In the project DIFFOP we investigated algebraically desribable differential equations; i.e. polynomial relations between the unknown function and its derivatives. Specifically, in DIFFOP we have achieved results in the following areas: Often differential equations (as also other types of equations) can be described by an operator, which should yield the value zero when applied to the function in question. So we have to determine the null space of this operator. If we can represent a differential operator as the composition of two (simpler) operators, i.e. if we can factor the operator, then the problem of solving is reduced to the problem of solving the two simpler factor operators. The question of factoring of linear partial differential operators (LPDO) is closely related to invariants, or characterizations which stay the same under transformations. We could extend the applicability of existing methods for factorization, and we could derive a complete system of invariants for LPDOs of order 3. When we are given several LPDOs or also difference equations, we could make substantial progress by deriving consequences depending only on fewer variables. For treating this kind of elimination problems we were able to generalize the well-known method of Gröbner bases. We could derive a complete algebraic method for determining general rational solutions for so-called algebraic differential equations in one variable. First we neglect the differential structure of the equation, and regard it simply as an algebraic relation between the function and its derivation (or derivations). In this way we get an algebraic variety, and a rational solution generates a rational curve on this variety. Our method is based crucially on parametrization of algebraic curves and surfaces, which we have investigated intensively in previous research projects. So the project DIFFOP has contributed decisively towards the development of new methods for the analysis and solution of differential equations.