Swarm Intelligence for Solving Some Nonlinear Differential Equations

The Euler method is a well-known numerical technique that is employed for solving initial value problems of ordinary differential equations (DE's). The solution obtained through Euler's method is subject to significant inaccuracies, which tend to propagate with each successive iteration. The Particle Swarm Optimization (PSO) algorithm is a highly effective method for finding optimal solutions to both linear and nonlinear optimization problems. In this particular investigation, the PSO technique was utilized to solve initial value problems associated with ordinary DE's. The Euler method, on the other hand, employs equidistant grid points to approximate solutions, which can result in significant errors and a substantial deviation from the actual solution. The PSO algorithm is a reliable approach for achieving optimal solutions to linear and nonlinear optimization problems, including initial value problems associated with ordinary differential equations. In contrast, the Euler method uses evenly spaced grid points to estimate solutions, which can lead to significant inaccuracies and deviation from the true solution. To address this issue, a new approach was developed using PSO to determine non-uniform grid points that minimize the approximation errors. Increasing the number of non-uniform grid points enhances accuracy and reduces errors. Numerical calculations were performed to compare this approach with traditional Euler formulae, demonstrating its superiority in overcoming existing limitations and delivering numerous benefits.