Iterative construction of the fundamental solution of a 2nd order elliptic operator.

Companion software for the paper Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients.

We implement in C++ the construction given in the paper.

Compilation

g++ -O2 main.cpp -o main

Usage

You shall configure the dimension (N) at the top of the function_representations.hpp file.

Then, in the main() you shall describe the operator you want to consider. The example currently implemented is for the operator (in dimension N=3, with the three variables x_0, x_1, x_2)

Then, executing ./main will print the first few homogenous terms of the expansion of the fundamental solution for L.

Documentation

An operator is a function<AE(AE)> taking an AsymptoticExpansion=AE and giving back an AsymptoticExpansion=AE. It must be elliptic and its principal symbol must be the same of the Laplacian. See the example in main.cpp. The field of coefficient is denoted by F.

The coefficients defining the operator shall be instances of the class Polynomial. A polynomial is a vector of pairs {exponent, coefficient}. The exponent is a vector of length N (where N is the dimension), and the i-th entry represent the exponent of x_i. For example, the polynomial is constructed by {{{0, 0, 0}, F(1)}, {{2, 0, 0}, F(4)}}.