Utilization of Chebyshev collocation approach for differential, differential-difference and integro-differential equations

Abstract Numerical solutions to differential and integral equations have been a very active field of research. The necessary tools to solving differential equations can add much fuel for acceleration of scientific development. Therefore, it is quite challenging for scientists, especially mathematicians, to work on the solution methodologies for differential equations. In this study, we have discussed the use of Chebyshev collocation method for solving a large variety of problems including various types of integral equations, integro-differential equations, and differential-difference equations. The method under consideration relies on a trial solution which is a linear combination of basic functions derived from Chebyshev polynomials. The trial solution is then used to obtain a residue function whose vanishing weighted residual can be used to obtain a system of algebraic equations which gives the values of constants used in the trial solution. Plugging the constants back into the trial solution, we obtain an approximate solution to the problem at hand. We have provided the comparison between exact and approximate solutions for different problems and found an exceptionally good agreement in results. The comparison has been provided through plots and tables to portray the accurateness of Chebyshev collocation method. The behaviour of the solutions is also displayed with the help of the graphs.


Introduction
Nowadays, very new ideas are taking place to modify existing mathematical techniques to get solutions to physical problems. Many physical problems are modelled with the help of differential equations and to deal with such problems, there is a dire need of very extensive repository of modified algorithms. Several techniques are available in the literature to solve such problems, but few of them have some inbuilt deficiencies like calculation of Adomian's Polynomials, small parameters assumption and calculation of Lagrange multiplier etc.
"Chebyshev polynomials" was named as the great Pafnuty Lvovich Chebyshev . In 1854, he first time proposed "Chebyshev polynomials" defined on the interval [À1, 1], which was reported in detail in (Chan, 2018;Kim, 2012;Siyi, 2015). Some Polynomials were known before his publication, but later, he provided a general idea about Orthogonal Polynomials. The first and second kinds of classical Chebyshev polynomials (CP) have been known for a long time through De Moivre's formula in the eighteenth century by Chebyshev (Kafash, Delavarkhalafi, & Karbassi, 2012;Li & Wenpeng, 2017;Lv & Shen, 2017;Montijano, Montijano, & Sagues, 2013;Sinwell, 2004). CP have a wide range of applications in the fields of number theory, sequence of polynomials, polynomials approximation and Fourier series. In addition to numerical analysis, CP have a significant importance in polynomial approximation, integral equations and differential equations.
The problem related to the tangent of the curve has been discussed by many mathematicians since Archimedes explored these problems in antiquity. The first attempt to determine the tangent to the curve that resembled the modern method of calculus came from Gilles Personne de Roberval during the 1630s and 1640s. Nearly at the same time, when Roberval was devising his method, Pierre de Fermat used the notion of maxima and the infinitesimal to find the tangent to a curve. The credit of discovering "Differential" goes to Fermat. The differential equation was introduced by English Physicist Issac Newton and German mathematician Gottfried Leibniz (in 1676), where Newton first time solved a differential equation by using infinite series and Leibniz introduced the term "differential equations" (Asmussen, 2007). Afterwards, Swiss mathematicians (Bernoulli brothers), Jacob Bernoulli and Johann Bernoulli did not accept Newton's theory of fluxion. From their point of views, it was originally Leibniz theory and Newton (Elsgolts & Norkin, 1973;Lindstr€ om, 2008;Shafia, 2018).
With the more generalization, systematic development of the theory of differential-difference equations (DDE) was not begun until Schmidt published an important paper about 50 years ago. A DDE is a two variables equation consisting of a coupled ordinary differential equation and recurrence equation (Bellman & Cooke, 1963). The idea of an integral equation begins with the need for finding the area under the curve. In 1635, an Italian mathematician, Cavalieri was the first scientist who presented the idea of an integral (Cavalieri, 1635). Cavalieri's work was centred on the observation that a curve can be considered to be sketched by a moving point and an area to be sketched by a moving line. Fractional calculus (FC) goes back to the initial theory of differential calculus. Its applications have emerged over the past two decades, especially in the field of dynamical systems theory and still, the concept of FC is at an early stage. Consequently, fractional differential equations and fractional integro-differential equations arise as more general form of differential equations and integro-differential equations. Actually, FC was introduced, when two mathematicians L'Hospital and Leibnitz shared and brought this idea into the discussion. It was L'Hospital who first put the question about the Leibnitz notation for nth derivative that is D n Dx n by replacing the power n with n ¼ 1 2 : In response to L'Hospital, Leibnitz encouraged the idea by saying that it could solve many other problems that are related to daily life and so on. Several mathematicians namely Fourier, Abel, Reimann, Mittag-Leffler and many others collaborated to expand and enhance the concept (Ahmadi Darani & Nasiri, 2013). He, Ji, and Mohammad-Sedighi (2020) used the variational principle to compute the solitary pattern solution of generalized KdV equations. Exact solution is addressed by them via exp-function method. The computed and discussed the solitary wave solution and discontinuous solution. In their report, authors have shown the effectiveness and applications of exp-function method for other large scale problems. Wazwaz (2001) used an analytical scheme proposed by Adomian to handle the Lane-Emden type equations. He has discussed several cases of considered problem and reported the rapid convergence of used approach. Nasab, Kılı çman, Babolian, and Atabakan (2013) presented the wavelet analysis for the solution of several singular problems. They found the accuracy of the used method and efficiency by quoting the several test problems. Singh and Singh (2018) used CP approach for the fractional modelling of Bloch equation arising in mathematical physics. They presented the error analysis by considering several definitions of fractional derivatives. Several modelled problems have been tested for the validity of the solution and efficiency of used algorithm. Aziz and Sarler (2013) used the wavelets procedure to discuss the numerical solution of elliptic models. They used the wavelet approach on several different types of problems with different boundary conditions. They presented the convergence order of used procedure. Also, they listed the error bounds for used algorithms. Modelling of fractional vibration phenomenon was studied by Singh, Srivastava, and Kumar (2017) numerically. They have several test problems and discussed the efficiency by discussing the stability and convergence analysis. Anjum and He (2020a) used homotopy perturbation method (HPM) for the solution of non-linear oscillator. They showed the highly effectiveness of used scheme on such problems. In another article, Anjum and He (2020b) discussed the solution of some other physical systems by coupling Laplace transform with homotopy perturbation scheme. Non-linear duffering oscillator's solution was computed via perturbation scheme by El-Dib and Matoog (2020). They found the periodic nature of the solution. In another exploration, El-Dib and Moatimid (2019) used homotopy perturbation with Laplace transform for the solution of circular problem. They recorded the periodic nature of the solution and presented the stability analysis by discussing the role of emerging parameters.
The main objective of the present study is to extend the use of CP for the solution of some linear and non-linear differential equations, DDEs, and integral equations. It is very significant in the submitted scheme that this is very efficient, well-organized, dynamic, having a great level of accuracy, time-saving and also for having the feature of numerical and graphical representations.
2. Analysis of Chebyshev polynomials approach for solving higher-order linear differential/differential-difference equations Linear differential/differential-difference equation of mth-order has variable coefficients shows an only negative shift in the differential term / n, f x nÀ2f , m ¼ 0, 1, 2, :::, Q, f ¼ 0, 1, :::, Also, it follows from Equation (6) that To get the matrix X f ð Þ ðxÞ in terms of the X x ð Þ, the procedure is as under Accordingly, we obtained matrix relations using the matrix form for Equations (7) and (8) into Equation (5), Similarly, we have the matrix relations between X t ð Þ ðx À sÞ and XðxÞ as follows: and from Equations (10) and (11), we have Where and by means of Equations (5), (7) and (9)

Implementation of Chebyshev collocation technique to solve DDEs
We will find the solution of DDEs by using the CPES.
subject to the ICs, Exact solution of Equation (26) is Here in this problem Q ¼ 8 and The values of collocation points are calculated as follows & x Assuming, the problem has CP solution stated as We construct the fundamental matrix equation of the given problem by using Equation (25), 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 , To get the solution of the problem (34), we define the truncated CP series where Q is a given integer, T ðÞ are the CP of degree m, and d are unknown constant coefficients.

Implementation of Chebyshev collocation method to solve various types of integral equations
To find the solution of different problems use CPES to solve the solution of various types of integral equations.

Conclusion
We have introduced a Chebyshev collocation method to solve integral, differential and differentialdifference equations with Cauchy condition. Several important figures have been displayed. Figures 1-8 are prepared to monitor the similarity of exact and approximate solution for different problems. Moreover, tables 1-8 are constructed to notice the compartment of exact and approximate solution. It has been observed that the proposed technique is a useful mathematical tool for finding the numerical solution of the particular problems arising in several physical phenomena. In this scheme, there is a trial solution, which is a linear combination of basic functions derived from CP. The proposed scheme is appropriate for solving the linear and non-linear problems. Solutions are elegantly expressed as polynomial of degree n which makes their representation and re-use very easy. The minimization of residual error which is the basic hallmark of Chebyshev collocation method can be used to achieve highly accurate results. Moreover, the comparison plots and tables have also been provided, which depicts the accurateness of the Chebyshev collocation method.

Disclosure statement
No potential conflict of interest was reported by the authors.