In this paper, we have introduced a novel numerical technique to solve integro-differential equations by computing an operational matrix using the Hosoya polynomial of the path graph. Integrals and derivatives are fundamental calculus methods that have a wide range of uses in science and engineering. Many scholars are focusing on designing computational schemes for discovering solutions to different problems including derivatives and integrals. Many systems in science and engineering are governed by differential, integral, and integro-differential equations. In the fields of dispersive waves, ocean circulations, and electromagnetic theory, integro-differential equations play a major role. Integro-differential equations also play a crucial role in characterizing physical, biological, and social problems. Polymer rheology, a variety of models of population growth, compartmental systems, mathematical modeling of discrete particle diffusion in a turbulent fluid, aeroelastic phenomena, unsteady

Integro-differential equations can be used in Ecology as well. Indeed, optimal search theory suggests that predators can use long leaps to locate prey that is scattered and dispersed randomly, with Brownian motion being more effective only when prey is abundant. Because of its numerous applications in domains like biochemistry, electrical engineering (Communications networks and coding theory), computer science (algorithms and computations), and operations research, graph theory is quickly becoming a mainstream topic in mathematics (scheduling). The adoption of the Hosoya polynomial method is the most recent method which provides a mathematical formulation in the field of science and engineering. Jalilian

A simple graph is a pair

where

The Hosoya polynomial of a path graph

In particular

The prime objective of this paper is strategy to use the Hosoya polynomial method for the numerical solution of non-linear IDEs with Volterra and Fredholm type equations of the form

where

If

. If we use only first

where the Hosoya polynomial co-efficient vector

We Know that

Similarly,

In general

The derivative of Hosoya polynomial of path graph is

Similarly,

The first derivative of the vector

In general, we may write first derivative of the vector

where

By using relation

Consider the non-linear IDEs (1) subject to the suitable initial conditions. We approximate

Therefore, the residual

The application of the Hosoya polynomial method requires that

Equations (6) with Equation (5) generate a system of linear or nonlinear equations in the unknown expansion coefficients

The analysis of the Hosoya polynomial technique is demonstrated in this part by using the method to solve the non-linear integro-differential problems.

The exact solution is

with

and from initial condition

Solving the above system of equations and equation (9), we get

This yields the numerical solution as

For

The results of the proposed method for Example (1) are exhibited in

The exact solution is

The numerical solution is given as

For

To measure the accuracy of the studied approach, the absolute errors are presented

In

The exact solution is

For

For

The results in

0 2 2 2 0.1 1.9800 1.9798 1.9800 0.2 1.9210 1.9210 1.9210 0.3 1.8253 1.8259 1.8253 0.4 1.6967 1.6967 1.6967 0.5 1.5403 1.5358 1.5403 0.6 1.3623 1.3455 1.3623 0.7 1.1699 1.1282 1.1699 0.8 0.9708 0.8862 0.9707 0.9 0.7727 0.6219 0.7726 1 0.5838 0.3375 0.5831

The exact solution is

The numerical solution for

For

To measure the accuracy of the studied approach, the absolute errors are presented in

The exact solution is

The numerical solution is given as

For

From

Using the Hosoya polynomial of the path graph an operational matrix of the derivative has been derived in this paper. This technique is a novel way to deal with IDEs. Applying the operational matrix, we have obtained the numerical solution of IDEs in both Volterra and Fredhlom sense by considering five examples. The results obtained for the solutions of these examples are presented either in tabular form or with graphical representation. The projected absolute errors reveal that the current method gives higher accuracy even for smaller values of m as compared to the available literature. Solutions obtained in this paper suggest that Hosoya polynomial-based operational matrix method is easy to implement and can be used as an efficient method to solve IDEs. As a future direction of study, we can consider other graph theory polynomials to solve IDEs.