Symbolic Solutions of Algebraic Differential Equations (ADE-solve)

In this project we are concerned with mathematical algorithms for determining sym- bolic solution formulas for algebraic differential equations (ADEs). An ADE is a poly- nomial relation between a function, some of its derivatives, and possibly the variables in which this function is defined. The solution of differential equations is one of the most important problems in mathematics, and has an enormous area of applications in sci- ence, engineering, finance, etc. The study of differential equations started in the 18th century with the work of Euler, dAlembert, Lagrange and Laplace as a central tool in the description of the mechanics of continua and, more generally, as the principal mode of analytical study of models in the physical sciences. The analysis of physical models has remained to the present day one of the fundamental concerns of the development of differential equations. However, beginning in the middle of the 19th century, particularly with the work of Riemann, differential equations also became an essential tool in other branches of mathematics. Today they play an important role in the modeling of financial markets. We take a computer algebra point of view; i.e. we typically want to compute a formula for the function solving the given ADE, not simply numerical values at particular points. Our algebro-geometric approach involves concepts from differential algebra, algebraic ge- ometry, and computer algebra. To a given algebraic ordinary differential equation (i.e., we are seeking a function in one variable) we relate a hypersurface, called the solution hy- persurface. A polynomial/rational/radical/algebraic solution of the equation generates a polynomial/rational/radical/algebraic parametric curve on the solution hypersurface. We try to decide the existence of such parametric curves, and in the positive case, actually determine them. In particular, for ordinary ADEs of order 1 we aim at finding all polynomial and rational solutions, and also find power series solutions. Also we intend to extend our algebro-geometric method to algebraic partial differential equations; i.e. where we are seeking multivariate functions. Finally, we will implement our approach in a computer algebra system such as Maple or Mathematica. We are convinced that this project will be able to extend the frontier of symbolic solutions to differential equations, thereby providing valuable tools for science.

Summary for public relations purposes Project ADE-solve ({P31327-N32}) Franz Winkler In this project we were concerned with mathematical algorithms for determining symbolic solution formulas for algebraic differential equations (ADEs). An ADE is a polynomial relation between a function, some of its derivatives, and possibly the variable in which this function is defined. Solving differential equations is one of the most important problems in mathematics, and it has an enormous area of applications in science, engineering, finance, etc. The study of differential equations started in the 18th century with the work of Newton, Leibniz, Euler, d'Alembert, Lagrange and Laplace as a central tool in the description of the mechanics of continua and, more generally, as the principal mode of analytical study of models in the physical sciences. The analysis of physical models has remained to the present day one of the fundamental concerns of the development of differential equations. However, beginning in the middle of the 19th century, particularly with the work of Riemann, differential equations also became an essential tool in other branches of mathematics. Today they play an important role in the modeling of financial markets. In our approach to differential equations we take a computer algebra point of view; i.e., we typically want to compute a formula for the function solving the given ADE, not simply numerical values at particular points. Our algebro-geometric approach involves concepts from differential algebra, algebraic geometry, and computer algebra. To a given algebraic ordinary differential equation (AODE) we relate a hypersurface (curve of surface). From a parametrization of this hypersurface we then derive a solution of the given AODE if possible. We are able to decide whether the given AODE has a rational solution, an algebraic solution with given order of field extension, and also power series solution. Moreover we can determine general solutions containing a parameter and describing (almost) all solutions of a given type. We have implemented our results in the computer algebra system \maple, thus making them available for applications in other fields of mathematics and the sciences.