Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind

Abstract

We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎