In this paper, the one-dimensional nonlinear Sine–Gordon equation is solved using the “Exponential modified cubic B-spline differential quadrature method with the leave-one-out cross-validation (LOOCV) approach”. By employing the LOOCV approach to determine the optimal value of the parameter \(\epsilon \) involved in the basis function, the accuracy and effectiveness of the results are improved. The combination of this approach with the exponential modified cubic B-spline differential quadrature method, which is novel in the literature, is likely to attract researchers' interest. Additionally, the procedure is implemented on six examples of the Sine–Gordon equation. The results are presented in the form of tables and figures. It is demonstrated that this approach is straightforward and yields superior outcomes compared to the existing literature. This paper also presents an insightful discussion on the significant application of the Sine–Gordon equation in Josephson junctions and its crucial role in new technologies.
