Friday, July 13, 2018 at 11:00am
Physical Sciences Building, 416
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Institute for Theoretical Physics
BLUES Function Method in Physics
A computational method in physics is proposed that goes "beyond linear use of equation superposition" (BLUES). A BLUES function is defined as a solution of a nonlinear differential equation (DE) with a delta source that is at the same time a Green's function for a related linear DE. For an arbitrary source, the BLUES function can be used to construct an exact solution to the nonlinear DE with a different, but related source. Alternatively, the BLUES function can be used to construct an approximate piecewise analytical solution to the nonlinear DE with an arbitrary source.
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 also predict growth, diffusion and extinction of biological populations, and fit observations in the 21st century of dramatic cosmic events anticipated by Einstein’s general theory of relativity. They are also used to model nonlinear optical properties of metamaterials, and 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 under a bilateral agreement between KU Leuven and Stellenbosch University.
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