Physical Models for Sustainability using Fredholm Integro-Differential Equations: Applicability and Analysis of Chebyshev Polynomial Method

Numerical analysis is concerned with the mathematical derivation, explanation and evaluation/analysis of algorithms, models and methods used to obtain numerical solutions for mathematical problems. This paper explores the reliability of the Chebyshev Polynomial Method (CPM) for solving a specific class of equations known as the second-order Fredholm Integro-Differential Equations (FIDEs). A series expansion of the Chebyshev polynomial is derived, used in solving these integral equations, and later on examined in terms of accuracy and convergence of solutions. The evaluation process involves a hybrid approach, combining manual methods and mathematical programs like MAPLE and MATLAB. In addition, three numerical examples were solved in which two truncation points are considered per each example. Furthermore, the performance of the CPM is reported in terms of accuracy, convergence, suitability, reliability and effectiveness in the context of the exact solution.


Introduction
Numerical analysis is an interdisciplinary branch of science involving both mathematics and computer science [1].Some cases in which numerical techniques were applied include weather prediction [2], heat transfer [3], modelling in fluid [4] and so on.When the above-mentioned cases are being modelled (converted to mathematical language), equations are used.These could be differential equations, integral equations, or some combination of both.Integro-Differential equations are equations containing both integral and derivatives of a function [5].This makes obtaining their analytical solutions very challenging.It is therefore no surprise that there are numerous numerical approaches to the solution of these equations.One of these approaches is the Chebyshev polynomial method which utilises Chebyshev polynomial series.These polynomials have some unique properties such as Orthogonality which allows for efficient approximation of functions during expansions, minimax, recursive formulation and fast convergence, to name a few [6].The aforementioned properties of Chebyshev polynomials suggest it to be a top-shelf approach when conducting numerical analysis.Also, integro-IOP Publishing doi:10.1088/1755-1315/1342/1/012004 2 differential equations are complex equations having applications in epidemiology [5], signal processing and neural networking [7], biomedical implants [8], and so on.Hence, obtaining such solutions will be important.These birthed this study which is tasked with testing and exploring the reliability of the Chebyshev polynomial method giving attention to a very special class of equations known as second-order Fredholm integro-differential equations (FIDEs).Kang et al. [9] presented a new highly accurate numerical approach for Fredholm integral equations of the second using a Gauss-type Clenshaw-Curtis quadrature.They also applied the scheme to integro-differential Schrodinger equations with nonlocal potentials.Also, Chang and Kang [10] employed a novel method derived from the Clenshaw-Curtis quadrature in the numerical solution of the integro-differential Schrödinger equation.Mohd [11] applied the Adomian Decomposition method to solve linear second-order Fredholm integro-differential equations numerically.They employed a Maple package to obtain approximate solutions to second-order FIDEs.In this paper, the numerical solution of second-order Fredholm integrodifferential equations via the Chebyshev polynomial method will be obtained.The rest of the paper is as follows: section 2 presents the methodology of the study; section 3 consists of three numerical examples and results and section 4 presents the discussion of results and also concludes the paper.

Chebyshev Polynomials of the First Kind
As earlier stated, this study gives focus on Chebyshev Polynomials of the first kind.Chebyshev polynomials of the first kind denoted by   (), are originally defined on the interval −1 ≤  ≤ 1 and as: When n = 0: When n = 1: Equations ( 2) and (3) above are the first and second terms of the sequence for Chebyshev Polynomials of the first kind in the interval−1 ≤  ≤ 1.

Formulation of Recursive Relation
In this section, a recursive relation for the Chebyshev polynomials of the first kind defined on the interval [-1, 1] is formulated.Due to the definition Chebyshev Polynomial of the first kind has, obtaining results for  = 2, 3 and higher terms, will be difficult, hence obtaining a recursive relation is essential.

Analysis of Chebyshev Polynomial Method
The analysis of Chebyshev polynomial method for the solution of second-order Fredholm integro-differential equations is presented as follows.
i. Define the Chebyshev polynomial series   (): In this study, two truncation points (N=3 and N =5) are considered.

This section presents three numerical examples and results
. The results generated via Chebyshev polynomial method for N =3 and N = 5 in the context of the exact solution are presented in Table 1.The graph comparing the solutions and errors of the approximate solutions using Table 1 are displayed in Figures 1 and 2.  The comparative result of the Chebyshev polynomial method and exact solution is shown in Table 2.The graph comparing the solutions and errors of the approximate solutions using Table 2 are displayed in Figures 3 and 4.   The comparative result of the Chebyshev polynomial method and exact solution is shown in Table 3.The graph comparing the solutions and errors of the approximate solutions using Table 3 are displayed in Figures 5 and 6.   3 Figure 6: The graph comparing the errors of the approximate solutions using Table 3 4

. Discussion and Conclusion
Firstly, it is noteworthy to say that this method transforms FIDEs and the conditions into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients.The selected FIDEs encompass diverse practical scenarios hailing from various scientific disciplines.These equations undergo a rigorous numerical analysis to evaluate both the convergence and accuracy of the obtained numerical solutions.Moreover, it is seen from Tables 1, 2 and 3 that the Chebyshev polynomial method agreed with the exact solution for both N = 3 and N = 5.The columns for Exact, N=3 and N =5 contained very similar results.Afterwards, the above results for each example were graphically represented.In Figures 1, 3 and 5, the graphs comparing the solutions: Exact, CPM(N=3) and CPM(N=5) are presented, and it was observed that the curve of the Chebyshev polynomial method followed that of the exact solution elegantly for N = 5 than N = 3, except for Example 3 in which both solutions are similar.This implies that the Chebyshev polynomial method converges faster to the exact solution for N = 5.In other words, the higher the value of N, the more accurate is the Chebyshev polynomial method.Also, from Figures 2, 4 and 6, it is seen that the errors generated via the Chebyshev polynomial method for N = 5 are smaller when compared to those generated for N = 3, except for Example 3 in which both errors were identical.Lastly, the computational effort required in arriving at the earlier presented results was found to be minimal.The results obtained show that the Chebyshev polynomial method yields high-accuracy solutions suitable for the solution of such FIDEs, with good convergence and requires little computations, hence confirming its reliability as a good approach for the numerical solution of second order FIDEs.
The methodology can further be extended or applied to the higher order fractional order FIDEs.

Figure 4 :Example 3 :
Figure 4: The graph comparing the errors of the approximate solutions (Example 2)