M. Sowmya and A. S. Vatsala
Methodology to Compute Coupled Lower and Upper Solutions for Reaction Diffusion Equations
Neural Parallel and Scientific Computations 33 (2025) 101-110

https://doi.org/10.46719/NPSC2025.33.06

ABSTRACT.
The applications of reaction-diffusion equations can be seen in various branches of science and engineering. Here we have used the generalized monotone method to construct the solution of a reaction-diffusion equation under initial and boundary conditions, where the forcing function is the sum of increasing and decreasing functions. The monotone sequences obtained by the application of generalized monotone method coupled with coupled lower and upper solutions, converge uniformly and monotonically to coupled minimal and maximal solutions. Not only are the lower and upper solutions relatively easy to compute, they also guarantee the interval of existence. Considering these lower and upper solutions as the initial approximation, we develop a method to compute a sequence of coupled lower and upper solutions on the interval of existence or on any desired interval of existence. The coupled minimal and maximal solutions converge to the unique solution of the reaction diffusion equation if the uniqueness conditions are satisfied. We will also provide some numerical results to support our methodology.

AMS (MOS) Subject Classification: 34A12, 34A34, 34B15, 35K57

Keywords and phrases: Generalized Monotone Method, Reaction Diffusion equation