Differential equation theory

Finding analytical solutions to nonlinear differential equations is a persistent challenge in physics and mathematics, as the standard principle of superposition does not apply. The BLUES (Beyond Linear Use of Equation Superposition) function method addresses this by defining a "BLUES function" that acts as both a solution to a nonlinear differential equation with a delta source and a Green’s function for a related linear operator. This duality allows for the construction of an analytic iteration sequence that systematically calculates approximants to the exact solution of complex nonlinear problems.

 

This framework has been successfully applied to a broad spectrum of mathematical problems, including Ordinary and Partial Differential Equations (ODEs and PDEs), as well as coupled systems and fractional differential equations. In the context of physical systems, the method has provided high-precision analytic approximants for phenomena such as interface growth under shear, traveling wavefronts in reaction-diffusion-convection equations, and even epidemiological models like the SIRS model. By providing a "middle ground" between exact solutions and purely numerical schemes, the BLUES method allows for a deeper understanding of the universal principles governing nonlinear dynamics, from heat transfer in semi-infinite rods to telegrapher equations and the evolution of Aeolian dunes. Current work is exploring numerical implementations of the method that serve to accelerate the convergence of the BLUES method.