Solving Integral Equations

Most applied mathematics has been developed in terms of the so-called “differential equations”. These are relations which involve an unknown function and its derivatives. Much less widely used is another type of relations: the so-called “integral equations”. They involve the operation inverse to the derivative, which is the integral. As is well known, integrals are generally much more difficult to compute than derivatives and arguably it is for this reason that solving integral equations has not become common practice.

Also the software world is still quite devoid of good packages and frameworks for solving integral equations. Mathematica for example can solve symbolically some simple integral equations through the DSolve functionality (the same tool that solves differential equations), but it has no way to solve more complex integral equations (NDSolve does not apply). For these, numerical integral equation algorithms would be required because, by weakening the requirement for a symbolic solution — that is, accepting even just numerically interpolated solutions — the range of solvable integral equations would vastly increase.

Thus, I am happy to announce that I have started to change this scenario by developing the first Wolfram resource function to solve numerically a particular type of integral equations: the Thermodynamic Bethe Ansatz (TBA).

You can read the rest of this computational essay on the Wolfram Community (Staff Picks, September 23, 2025). See also the documentation.