Non-Standard Analysis Solution to Differential Equations

Question

The non-standard analytical solution to the derivative of simple functions such as $x^2$ is well-known...

Is there a similar solution for differential equations such as the heat equation or a simple ODE?

Answer 1

For the heat equation, I suggest reading the paper "Applications of nonstandard analysis to partial differential equations—I. the diffusion equation", available for instance at the page http://www.sciencedirect.com/science/article/pii/0270025586900692