Joseph O Indekeu, Kristian K. Müller-Nedebock, “BLUES function method in computational physics”, Journal of Physics A: Mathematical and Theoretical, 51, 165201 (2018) http://dx.doi.org/10.1088/1751-8121/aab345 or https://arxiv.org/abs/1802.10090.
Here is a popular summary prepared by Joseph Indekeu:
BLUES function method
In the 17th century, with the invention of calculus, Newton and Leibniz introduced differential equations. In linear differential equations the unknown function appears to the first power. There are very few methods of solving other, nonlinear, differential equations. These can describe complex behavior and chaos. They can predict growth, diffusion and extinction of biological populations. They can fit observations in the 21st century of dramatic cosmic events anticipated by Einstein’s general theory of relativity. They can model nonlinear optical properties of metamaterials. They can depict interface dynamics in inanimate condensed matter.
Linear differential equations permit superposition of solutions, one of the most powerful tools in computational science, paradigmed by the Fourier transform. Also the Green’s function method exploits the superposition principle of the linear theory, using Dirac’s delta-function as the mathematical atom for building the material system. Now, a new function method that goes beyond linear use of equation superposition and is therefore named BLUES, is jointly proposed by a KU Leuven statistical physicist and a Stellenbosch University biophysicist. Superposition of solutions of a nonlinear problem is normally not permitted. Surprisingly, however, the transgression may be only lightly penalized or even rewarded. The demonstration of a case in point is given in Joseph O. Indekeu and Kristian K. Müller-Nedebock, “BLUES function method in computational physics”, 2018 J. Phys. A: Math. Theor. 51, 165201. The research was carried out in the framework of a bilateral agreement between KU Leuven and Stellenbosch University.