Numerical Solution of Non-linear Integro-differential Equations using Operational Matrix based on the Hosoya Polynomial of a Path Graph

Abstract

Objectives: Introduction to new numerical techniques to solve differential, difference, and integro-differential equations (IDEs) are always remaining the thrust area of research for many scientists over the centuries. The prime objective of this work is to contribute a new numerical technique to solve IDEs. Method: To address non-linear integro-differential equations, we computed an operational matrix of derivatives based on the Hosoya polynomial of the path graph in this work. Findings: Using the derived operational matrix, we have solved both Volterra and Fredholm integrodifferential equations. Taking suitable examples accuracy of the projected method is demonstrated in this paper in terms of a graphical representation of the absolute error. The results of the examples reveal that the projected method is a suitable method to solve IDEs. Novelty: The application of the Hosoya polynomial of path graph to solve integro-differential equations is a novel approach in the field of numerical analysis.

A new graph-theoretic approach for the study of the surface of a thin sheet of a viscous liquid model

“…The Hosoya polynomial provides valuable information on network invariants based on the distance between vertices in the chemical structure. A reason for considering Hosoya polynomials is that they can be differentiated and integrated several times and are recently shown to have some significance in the study of highly nonlinear functional and fractional differential equations such as Fredholm integral equation [14], Voltera-Fredholm integro differential equations [18], Two-point boundary value problems [19], integrodifferential equations [24], etc. As an extension of this application, we implemented the Hosoya polynomial approach on the second-order nonlinear Buckmaster equation.…”