Narrow Results

This paper presents a novel approach to solve nonlinear second-order Volterra integro-differential equations (VIDEs) using the Iterative Collocation Method. We focus on specific VIDEs with smooth functions, aiming to obtain numerical solutions without the need to solve algebraic equation systems. Our method, based on Lagrange polynomials within a spline space, offers direct implementation, high convergence order, and efficient determination of approximation solution coefficients. Through theoretical analysis and numerical experiments, we illustrate the efficacy and accuracy of our approach, contributing to the improvement of numerical techniques for solving nonlinear VIDEs.

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ_0,2, or to the real Clifford algebra ℝ_0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ_0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ_0,0 ≃ ℝ, ℝ_0,1 ≃ ℂ and the trivial case ℝ_1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝ_p,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.