Pure Mathematics for Theoretical Computer Science

This paper develops a two-dimensional discrete mathematical model for a nonlinear advection–diffusion–reaction equation under a divergence-free vortex field. The formulation is treated as a finite-dimensional dynamical system generated by an explicit finite-difference algorithm: first-order upwind operators approximate advection, secondorder centered operators approximate diffusion, and explicit Euler time stepping advances the nonlinear reaction. The resulting update rule is analyzed through Courant–Friedrichs–Lewy restrictions, a diffusion constraint, and a frozen-coefficient von Neumann argument. A Gaussian initial state and homogeneous Neumann boundary conditions define the computational problem. Numerical experiments show stable vortex-driven redistribution, diffusion-induced smoothing, decay of total mass in the presence of the linear loss term, and localized changes caused by the quadratic reaction. Grid-refinement, time-step-sensitivity, and analytical diffusion benchmarks support convergence of the discrete approximation. For an Nx by Ny grid, each time step requires O(NxNy) arithmetic operations and O(NxNy) storage, so the method provides a transparent algorithmic bridge between nonlinear PDE analysis and discrete computation. The study emphasizes the mathematical structure, stability conditions, convergence behavior, and computational complexity of the scheme while retaining pollutant transport as an illustrative application.

The purpose of this work was to benchmark three population-based metaheuristic optimizers—Particle Swarm Optimization, Differential Evolution, and Grey Wolf Optimizer—when used to solve nonlinear ordinary differential equations within the Neural Network Trial Solution methodology. Problems used for testing were the Riccati initial value problem, the nonlinear pendulum IVP, the Bratu boundary value problem, and the Lane-Emden equation with index five. All problems were implemented such that their boundary/initial conditions were satisfied exactly through analytical construction while their residuals at collocation points were minimized through unconstrained optimization. Thirty Monte Carlo runs of each algorithm were performed with same underlying settings to facilitate statistical comparisons between algorithms. Metrics used for comparisons were mean absolute error (MAE), root mean square error (RMSE), maximum error at any point, and rate of convergence. All significant testing was performed with the Wilcoxon signed-rank test. PSO is shown to consistently provide the smallest mean absolute error across three of the four problems, with an MAE as small as 2.78×10−5 on the Bratu BVP, while GWO was shown to stagnate prematurely when solving boundary value problems.