Enhanced moving least square method for the solution of volterra integro‑differential equation: an interpolating polynomial

This paper presents an enhanced moving least square method for the solution of volterra
integro-differential equation: an interpolating polynomial. It is a numerical scheme
that utilizes a modified shape function of the conventional Moving Least Square (MLS)
method to solve fourth order Integro-differential equations. Smooth orthogonal polynomials
have been constructed and used as the basis functions. A robust and unrestricted
trigonometric weight function, along with the basis function, drives the shape
function and facilitates the convergence of the scheme. The choice of the support size
and some controlling parameters ensures the existence of the moment matrix inverse
and the MLS solution. Valid explanation and illustration were made for the existence
of the inverse linear operator. To overcome problems of near-singularity, the singular
value decomposition rule is used to compute the inverse of the moment matrix. Gauss
quadrature rule is used to compute the integral at the initial test points when the exact
solution is unknown. Some tested problems were solved to show the applicability of
the method. The results obtained compare favourable with the exact solutions. Finally,
a highly significant interpolating polynomial is obtained and used to reproduce the
solutions over the entire problem domain. The negligible magnitude of the error at
each evaluation knot demonstrates the reliability and effectiveness of this scheme.