Approximate solution of linear Volterra-Fredholm integral equations via exponential spline function

This paper presents a novel numerical scheme for solving linear Volterra-Fredholm integral equations (V-FIEs) of the second kind, utilizing exponential spline functions (ESFs) in combination with fractional derivatives. The method simplifies computational implementation by converting the original integral equation into a matrix system. To prove the precision and stability of the suggested approach, a thorough convergence analysis is carried out. Numerical experiments, backed by graphical representations, validate the method's high accuracy and computational efficiency, even with a limited number of subintervals. All simulations and visualizations are implemented using Python. The results indicate that the suggested ESF approach performs noticeably better than traditional methods.