Particle Swarm Optimization Numerical Simulation with Exponential Modified cubic B-Spline DQM

Optimization techniques refer to a collection of mathematical algorithms and methodologies used to discover the best possible solutions for specific problems or situations. One of the most significant optimization techniques is particle swarm optimization (PSO). PSO is a metaheuristic optimization algorithm that finds extensive application across diverse fields such as healthcare, the environment, industry, commerce, smart cities, and other general domains. This paper describes the algorithm, advantages, and disadvantages of PSO along with its applications. By employing PSO technique with the “exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM)” for finding the optimal value of the parameter \(\varepsilon \) $\epsilon$ is determined, which enhances the stability and accuracy of the numerical solutions. This study introduces an innovative combination of Expo-MCB-DQM with PSO, which will attract the researcher’s attention. It has been applied to six numerical problems of the Sine–Gordon equation to check the authenticity and effectiveness of this combined technique. The results obtained from this combined technique have been found to be favorable.