The present study aims to design a mathematical system based on the Lane–Emden third-order pantograph differential model by using the general forms of the pantograph as well as the Lane–Emden models. The designed model is divided into two types along with the various singularity details at each point. The shape factors and the pantograph points are discussed for each type of the newly designed nonlinear third-order pantograph differential model. The Bernoulli collocation scheme is implemented to find the numerical results of the novel model. To show the reliability of the designed novel nonlinear model, four different variants have been solved. Moreover, the comparison of the obtained results with the exact solutions is presented to check the accuracy of the designed novel model.

1. Introduction

The historical pantograph differential (PD) model is a form of delay differential system that has great importance due to its immense applications in various biological and scientific phenomena. Some examples of these applications are dynamical population models, control problems, communication systems, absorption of light using stellar matter, economical models, engineering, transport, quantum mechanics, electronic systems, infectious diseases, and propagation systems [17]. Due to the huge significance of the PD model, many researchers have investigated various efficient techniques to find some accurate solutions to those types of equations. For example, Yuzbas and Karaçayır [8] proposed a generalized form of the multitype pantograph differential equation, which has been solved using a new Galerkin-based approach. In addition, Yuzbas and Karaçayir [9] investigated the application of the same Galerkin-based technique for simulating the results of the pantograph-type Volterra-Fredholm integrodifferential equations with functional upper limit with mixed boundary conditions. Liu and Li in [10] employed a Dirichlet series scheme for finding an analytical solution to the PD model and performing the existence and uniqueness of the suggested algorithm. A Laplace decomposition method has been used in [11] to solve the system of multipantograph differential equations. Yüzbaşi and Ismailov in [12] employed a collocation matrix approach based on the Taylor polynomials for solving the generalized PD equations through the multiple examples that prove the proposed scheme is efficient in acquiring accurate results. Also, the authors of [13] applied a Bessel-based technique for understanding the behavior of the solution of the multipantograph equation systems with mixed conditions. There are other methods for analyzing the PD model such that including the exponential approximation technique [14], shifted Legendre method [15], backpropagated intelligent network technique [16], Gudermannian neural network [17], Hermite polynomial solutions [18], and Vieta–Lucas series method [19]. These extensive works on the solutions of PD models open the door to exploit the wider applications of similar models of the real-life phenomenon. For more details regarding the applications and techniques for solving the PD models, one may refer to [2023] and the references therein.

The study of singular differential systems has attracted huge attention from the research community for the last few decades. There are many singular models, out of which Lane–Emden (LE) is one of the historical models that have a singularity at the origin and labels a variety of phenomena in the radiators cooling, gas clouds, cluster galaxies, and polytrophic stars. The LE is such a model that has a variety of applications in dusty fluid systems [24], physical science [25], reactions of catalytic diffusion [26], the density state of a gaseous star [27], isothermal spheres of gas [28], electromagnetic theory [29], catalytic diffusion-based reactions [26], classical mechanics [30], mathematical physics [31], oscillating magnetic fields [32], stellar structure systems [33], and anisotropic continuous media [34]. The wide applications of such models drive the scientists to investigate more into the solutions to these types of equation to understand their behavior which shall reflect more understanding of their applications. For example, Wazwaz [35] adapted a modified version of the Adomian decomposition method to solve the Lane–Emden type equations. Also, a multisingular system of the third-order Emden–Fowler equation has been solved using a neuro-evolution computing algorithm in [36]. In addition, Nisar et al. [37] derived an evolutionary reliable algorithm for solving the second kind of the Lane–Emden problems whereas a similar approach was proposed by Sabir et al. in [38] based on a Morlet wavelet neural network for the same problem. Roul [39] adapted a collocation approach based on the use of B-spline as a basis function to find the solution to a strongly nonlinear singular boundary value problem having possible application in the electrohydrodynamic flow of fluid. Sabir et al. [40] employed the Gudermannian neural network technique for solving the three-point boundary value problems resulting in some new results for this type of problem. In addition, Roul [41] investigated the solution of a wide class of singular Lane–Emden equations using a novel algorithm based on the combination of the modified Adomian and B-spline collocation approaches through several examples. Several other techniques have been employed for solving different kinds and forms of the singular problem including a stochastic algorithm for solving the nonlinear singular periodic problem [42], the fourth-order B-spline collocation method [43], optimal homotopy technique [44], and other successful methods that found in [4549]. The mathematical form of the typical singular LE system is presented as [38]

In the above equation, indicates the shape factor, while x = 0 represents the singular point at the origin and A is a constant value. The current work intends to propose an extension to the second-order Lane–Emden pantograph delay differential model presented by Adel and Sabir [50] to the third-order nonlinear pantograph differential Lane–Emden (PD-LE) model.

Over the past few years, Bernoulli collocation (BC) techniques have played an important role to solve different types of application-based problems. Many researchers worked to expand the BC technique to handle those challenges that arise in different areas of science and engineering. For example, Tohidi and Kilicman proposed a BC scheme to solve the generalized form of the PD model [51]. A modified version of the BC method was designed to find an approximate solution to a higher-order complex differential equation in a rectangular domain [52]. The accurate results of the static beam problem have been obtained by using the BC scheme [53]. Toutounian et al. [54] applied the BC scheme for solving the boundary value problems arising in the variation calculus. The study of a few more famous problems is presented by using the BC scheme such those including time-fractional cable equation [55], fourth-order Sturm–Liouville problem [56], stochastic Itô–Volterra integral equations of Abel type [57], and nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type [58].

The main motivation of this paper is to investigate the solution of a general form of the third-order Lane–Emden equation due to its continuous and important application in the real world. In addition, the applications of collocation methods and their ability to provide fast and accurate solutions to different models were one of the main reasons to simulate such an important model to understand more about their dynamics. Also, third-order equations are difficult in finding their solutions, and thus, the addressed method succeeded in achieving highly accurate results. The presented new method based on the Bernoulli collocation approach adheres to several advantages in finding the solution to such models over other existing techniques since it is a fast, straightforward, and simple to apply method. Only a few numbers of bases are considered to achieve high accuracy, and the computational cost for finding these solutions is low. It is worth mentioning that this is the first time that this type of model is examined using this novel technique.

In this work, the numerical investigations of the newly designed hybrid pantograph–LE-based model have been accomplished via the BC approach. Some novel aspects of the designed novel system are described as follows:(i)A novel design of nonlinear pantograph differential Lane–Emden is proposed using the pantograph differential and standard Emden–Fowler equation(ii)The designed model is divided into two categories along with the formulation of a differential system(iii)The values of the shape factors, pantograph terms, and singular points at origin are provided for both categories of the novel-designed model(iv)Two different numerical nonlinear variants of each category have been discussed using the BC scheme(v)The comparison of the proposed results from the BC scheme is presented to check the correctness of the novel designed system

The rest of the paper parts are organized as follows: Section 2 designates the construction of the designed model. Section 3 presents the function approximation, fundamental relations, Bernoulli operational matrix, BC scheme, and a few related theorems. Section 4 describes the results and discussions. Conclusions and future research guidance are listed in the last section.

2. Construction of the Third-Order PD-LE Model

This section describes the third-order PD-LE system, which is designed by using the pantograph differential and typical LE models. The novel third-order PD-LE system is divided into two categories, and the detail of each class is presented along with its mathematical procedures. The shape factors, pantographs, and singular points are also discussed based on the novel model. For the derivation of the novel nonlinear third-order PD-LE equation, the mathematical formulation is expressed aswhere , and are a positive real number. To find the nonlinear PD-LE model, p and q are integer values that must be fixed and written as

The following two possibilities that satisfy the above (3) are written as

Using (3) and (4), the mathematical form of the model (2) is categorized as

Category 1: Using (2) in (4), the updated form of the model becomes asand then, by calculating the derivative, the obtained form is given as

Substituting the expression (7) into the equation (2) gives the third-order nonlinear PD-LE; that is,

The singularity is performed twice at and while the pantographs are noticed in the first, second, and third terms. Moreover, the shape factors are and . It is noticed for k = 1, the third term vanishes, and the value of the shape factor becomes 2.

Category 2: the restructured form of the (2) using the (5) becomes as

By calculating the derivative, the obtained form is given as

Using (10) in (2), the nonlinear third-order PD-LE model is given as

The single singularity appears at zero in (11). k is the shape factor, and pantograph expressions appear twice in (11).

3. Fundamental Relationships/Function Calculation

This section states the basic Bernoulli polynomials (BPs) properties. BPs play a significant role in the fields of mathematics, including finite difference theory and the integral representations of the periodic function. BPs are the type of nonorthogonal polynomials, and their basic definitions are found in references [59]. The classical form of the BPs is given as

These above polynomials designate the various properties, such as,

BPs have many advantages/merits, and the execution of the Bernoulli operational matrix approach is quite easy. One of the main advantages of the BPs is that the operational differential matrix has fewer nonzero elements than other polynomials. The first subdiagonal elements are nonzero in Bernoulli’s operational matrix of its derivative, and the coefficient values of the individual terms in Bernoulli polynomials are much smaller than the coefficients, which appear in the classical orthogonal polynomials to produce fewer computational errors. Due to the wide range of applications as well as the advantages of the BPs, this method is reliable to be implemented as compared to other approaches.

3.1. Differential Form of the Bernoulli Operational Matrix

BPs have various interesting properties and relations, including the derivative to solve different categories of problems. The approximation form of the Bernoulli approximation scheme is expressed aswhere is the unidentified coefficient values, is a positive integer taken as , and is the BP of the first kind, which can be formulated according to (12) in the following form aswhere is the lower triangle matrix having nonzero diagonal elements with . It can be considered as an invertible matrix, and the Bernoulli vector takes the form aswhere and be the , while is matrix which is given as

Additionally, both the coefficient vectors and the polynomial are written as

Consider the solution form (14) is written in the following form:

According to (16), the relationship between and its derivative is written aswhere is an operational matrix based on the BPs differentiation, and the derivative equation can take the following form:

Finally, the solution derivative takes the following form:

The approximate outcomes are signified in the (19) and (22) to operate the approximate results of the given system.

3.2. Bernoulli Matrix Approach

The proposed BPs are used to solve both types represented in the system (8) and (11), respectively. The method begins with the approximate outcomes using the BPs, given as

For the above system, the derivative takes the following form:

Category 1: Our primary focus here is to examine category 1 represented by (8). To overcome the singularity in the system (8), the main equation is transformed into regular ones using the L’Hopital rule [60, 61] based on the expression into the system (8), since asand hence,and for the second expression , the L’Hopital rule is applied, which will transform intoand hence,

Equation (8) will be reduced to the following form:whereand . Thus, the double singularity will be handled here using the previous transformation.

Next, we substitute the solution and its approximation from the (17) and (18) aswhere any function except will be extended by using any appropriate expansion. To solve the equation (31) using the adjustable collocation points is given as

By substituting these collocation points into (31), one can find as

To solve the above equation, it is essential to handle the nonlinear term stated in the theorem.

Theorem 1. The approximation of the function can be performed according to the next relation aswhere

Proof. The proof can be found in [62].
Next, by substituting in equation (32) and approximate values using equations (20) and (23), the following theorem is obtained.

Theorem 2. If the assumed approximate solution to the problem (8) is represented by (19) and (22), then the discrete form of the Bernoulli system can be given in the following form:

Proof. If we replace each term of (8) with its corresponding approximation given by equations (20)–(23) and by substituting collocation points defined previously, (33) can be written in matrix form aswhere andIt is noted that the matrix is used for the delayed terms. The technique applications of the differential form of model (8) are converted into a matrix equation that corresponds to linear algebraic system in unknown value of Bernoulli coefficients. The improved matrix form becomes asThe initial form of system (8) is provided using the matrix formFinally, the corresponding rows are replaced with the initial conditions represented in (36) with the augmented matrix form of (35), which shall result in a new augmented form asThe unknown coefficients can be calculated by applying the next algorithm [50].
Category 2: Here, category (2) is examined for equation (11) using the same previously introduced technique which is applied to equation (11), and thus, the approximation from equations (18) and (19) along with Theorem 1 will reach the following theorem.

(i)Input form of the integer N.
(ii)input (double) tol.
(iii)Input (array) (initial guess, with dimension can be selected to satisfy the boundary conditions)
(iv) is an algebraic linear system and is achieved.
(v)If then , (the program is terminated).

Theorem 3. If the assumed approximate solutions to the problem (11) are equations (17)–(19), then the discrete form of the Bernoulli system can be given in the following form:

Proof. If each term of (11) is replaced with its corresponding approximations given in equations (17)–(19), and substituting , which is the collocation points defined previously, (39) can be written in matrix form aswhere , where , and have been defined previously after (34). The conditions for category 2 are in the same form as (36).Finally, the new augmented matrix for system (37) which can take the following form after replacing the conditions in equation (38) with the defined conditions in equation (39) is written asThen, the new system in (39) shall be solved using the same previously mentioned algorithm to find the unknown coefficients. In the next section, the verification of the residual error function is presented to confirm the validity of the proposed technique.

4. Residual Error Function

The solution accuracy for category I represented by equation (8) and the same for category II based on the residual error function are observed in this section asor (positive) for .

The proposed technique must be tested for the numerical solutions based on the (8) and (11). The max is defined for the truncated limit and is then improved till the variation becomes smaller at each point than the maximum . Calculating the residual error represented by (40) will ensure the effectivity of the proposed algorithm along with calculating the regular absolute error.

In the next section, we shall test the proposed method for several nonlinear examples to verify its effectiveness.

5. Numerical Results

In this section, four different numerical variants are accomplished to check the exactness and perfection based on the nonlinear PD-LE model along with the efficiency and accuracy of the Bernoulli collocation scheme. The absolute error (AE) is calculated from the proposed solutions based on the BC scheme, and the true results are provided at the selected points. All the computations have been performed in MATLAB R2015a. No MATLAB built-in syntax has been used while preparing the coding of the examples. Also, the tolerance for the prosed scheme in all the examples is assigned to be

5.1. PD-LE Model of Category 1

In this category, two different equations of PD-LE third-order model-based examples will be presented. The simplified form of (8) for k = 2 is written as shown in the following problem.

Problem 1. Consider the following nonlinear PD-LE model:The exact solution of (37) is . The numerical results to express the maximum AE are provided in Table 1 for different values of . It can be found from this table that the method provides accurate solutions in terms of the absolute error with an error of , and the CPU time is 5.8 sec. Also, the proposed technique is witnessed to achieve high accuracy results even for a little number of bases. In addition, Figure 1 gives the approximate and exact solution for (31) with N = 6. It can be noticed from the figures that the resulting approximate solution fits well with the exact results proving the method is efficient. Also, to confirm the validity of the proposed technique, the results of the residual error are illustrated in Table 2.

Problem 2. Consider the third-order nonlinear PD-LE model involving exponential values in the forcing factor, that is,The exact form of (42) is . The exact and proposed solutions are presented in Table 3 for different values of along with the maximum AE for at different grid points which shows that the method is accurate with little grid points. It is noticed based on Table 4 that the BC scheme is efficient that provides few error values. In addition, Figure 2 gives the approximate versus the exact solution of the problem.

Problem 3. [63, 64] To test the effectiveness of the proposed scheme, a special case is considered of the third-order PD-LE model with variable coefficients in the form asThe exact form of (49) is . We test the method for system (44), and the results are presented in Table 5 which demonstrated the absolute error along with a comparison with the neuro-swarming-based heuristics technique presented in [63] and the intelligent computing paradigm [64]. It can be seen from the results that our technique provides better results compared to the other method. It is worth mentioning that the time consumed for processing the results is 3.03 sec while there is no consumed time mentioned in the other compared work. In addition, Figure 3 demonstrates the approximate and exact solution for the problem.

5.2. PD-LE Model of Category 2

In this category, two different problems of PD-LE third-order systems will be presented. The efficient form of equation (11) for k = 1 is written as follows.

Problem 4. Consider the third-order nonlinear PD-LE model having multitrigonometric ratios is shown asThe exact form of (44) is . The scheme is applied to the given equation along with the maximum AE provided in Table 6 for different values of together with its maximum error of for . Also, the residual error results for N are provided in Table 7. In addition, Figure 4 gives the approximate and exact solutions for (44) with N = 6.

Problem 5. Consider the third-order nonlinear PD-LE model having exponential values that can be presented asThe exact solution of (45) is . The exact and approximate solution in Table 8 along with the maximum AE is dissipated in Table 9 for different values. In addition, Figure 5 gives the approximate and exact solution for (45) with N = 6.

Problem 6. [64] Finally, a special form based on the category 2 is considered for the third-order PD-LE model to compare the effectiveness of the proposed algorithm in the form asThe exact solution of (52) is . We test our proposed algorithm for acquiring the results and compare these results to the results in [52] in Table 10. As can be seen, from these results, our method gives a better solution compared to the other methods. Also, Figure 6 illustrates the behavior of the solution compared to the exact one, and they are in perfect agreement with each other. In addition, the consumed time for the data is found to be 3.511 sec, but there is no reported CPU time in [52] to compare.

6. Conclusion

In this work, the design of a novel third kind of nonlinear pantograph differential Lane–Emden model is carried out using the pantograph differential model and the typical form of the Lane–Emden system. Two different categories of the novel model together with the shape factors, singular point values, and pantographs of each category are discussed in detail. The singularity at the origin performs twice in category 1, while a single singularity is observed in category 2. Likewise, the single shape factor exists in the typical Lane–Emden system, whereas the shape factor appeared twice in category 1, and the single shape factor is calculated in category 2. For the excellence of the novel system, numerical experimentations have been accomplished by applying the famous Bernoulli collocation approach. A detailed residual error analysis is performed to support the efficiency of the scheme. The operative outcomes have been achieved to solve the novel designed model applying the Bernoulli collocation approach. The absolute error plots and tables have been presented based on the obtained numerical values for . The satisfactory absolute error values for all four problems have been obtained, and one can observe that these values lie in and . One can determine based on these accessible observations that the designed singular model is accurate, and the numerical results are precise.

In the future, the Bernoulli collection scheme can be implemented to solve different nonlinear and higher-order models with real applications in different areas of science and engineering [6569].

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Waleed Adel conceptualized the study, developed the methodology, provided the software, performed data curation, and wrote the original draft. Zulqurnain Sabir wrote the original draft, performed data curation, and reviewed and edited the article. Hadi Rezazadeh performed supervision and reviewed and edited the article. A. Aldurayhim reviewed and edited the article and provided funding.