A Comparative Study of PSO and LOOCV for the Numerical Approximation of Sine–Gordon Equation with Exponential Modified Cubic B-Spline DQM

This research aims to compare the results of numerical simulations obtained using two different optimization algorithms: particle swarm optimization (PSO) and leave-one-out cross-validation (LOOCV), with the exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM). In this paper, the sine–Gordon (SG) equation has been solved numerically by using the “exponential modified cubic B-spline differential quadrature method with PSO” and the “exponential cubic B-spline differential quadrature method with LOOCV”. This paper captures the researcher’s attention by comparing the numerical results of these two methodologies but introducing an innovative combination of Expo-MCB-DQM with PSO and LOOCV, which optimizes parameter value involved in the basis functions. The exponential cubic B-spline basis functions involve a parameter \((\upepsilon )\) $(\mathrm{ϵ})$ that needs to be assigned to a value in order to find numerical solutions. Until now, the value of a parameter had been determined by the hit-and-trial approach, leading to unstable results. The comparison of numerical results obtained by both of these methodologies is presented in the form of tables and figures. Furthermore, to validate the authenticity and effectiveness of both methodologies, we apply them to six problems of the SG equation. The numerical solutions obtained through these methodologies are close to exact solutions and also comparable to the other numerical methods established in the literature.