Systems of partial differential equations (PDEs) are usually solved by nu- merical methods. As opposed to this standard practice, in this project, we have been building tools that are capable of solving PDEs symbolically with the help of algebraic computations. The main motivation behind this endeav- our is the quest for exact solutions of PDEs. So far we have succeeded in ob- taining a symbolic computation algorithm for exponential solutions of linear constant coefficient PDEs, exact representation formulae for discrete partial difference equations. In the process of pursuing this project, we have also made important theoretical contributions in the field of systems described by PDEs, such as, state-space representation, l∞ /l2 -stability analysis, conic stability, Lyapunov theory for strongly autonomous systems.

The table below shows the notation we are going to use in this write-up.

Given a polynomial f(ξ) ∈ A the corresponding polynomial partial differential operator f(∂) acts on a smooth trajectory w ∈ W as f(∂)w ∈ W. Similarly, a row-vector of partial differential operators r(∂) = r1(∂) ··· rq(∂) acts on a column of trajectories w = col(w1,...,wq) as

                                                  r(∂)w = r1(∂)w1 + ··· + rq(∂)wq ∈ W.

This enables us to define the following set called the 1 behaviour of the system.

                                            B := {w ∈ Wq | r1(∂)w = r2(∂)w = ··· = rg(∂)w = 0}.

A set of central importance is the so called equation module of the system,

which is defined in the following manner:                                     

                                       R := {r(ξ) ∈ A1×q | r(ξ) = m i=1 αi(ξ)ri(ξ), αi(ξ) ∈ A} = r1,...,rm.

      When the space of trajectories W is that of smooth functions C∞(Rn, R) or its subspace of exponential functions then it is well-known that there is a one-to-one correspondence between behaviours and submodules of A1×q.

      We now elaborate on the algorithm for algebraically solving a system of linear constant coefficient PDEs. For the sake of simplicity, we consider only scalar systems, that is, q = 1. In this case, the role of the equation module is played by what is known as the equation ideal. Suppose we have an ordinary differential equation (ODE)

severe issues if one wants to extend this algorithm to PDEs. For example, what if there are multiple equations? Further, for ODEs, every remainder can be written uniquely as a polynomial of degree less than or equal to degree of g(ξ). How can we guarantee such uniqueness for PDEs? The solution comes from the algebraic notion of Gr¨obner basis. Every system of equations admits a special set of equations, given by a Gr¨obner basis of the system of equations. This special system is used to carry out division-with-remainder that gives a unique remainder for every polynomial. Every remainder is a 2 unique linear combination of standard monomials. We explain below an algorithm, based on Gr¨obner basis, that computes exponential solutions for scalar PDEs. Here we consider only exponential solutions.

Gr¨obner basis of equations

Input: A set of PDEs f1(∂)w = 0, f2(∂)w = 0, ..., fd(∂)w = 0.

Computation:

                       • Fix a term ordering ≺ in A.

                       • Compute a Gr¨obner basis G of the ideal a := f1, f2, ..., fd.

                       • Construct the set of standard monomials Γ := {ν ∈ Zn 0 | ∂ν ∈ in≺(a)}.

Output: Standard monomial set Γ.