Design of a Novel Second-Order Prediction Differential Model Solved by Using Adams and Explicit Runge–Kutta Numerical Methods

In this study, a novel second-order prediction diﬀerential model is designed, and numerical solutions of this novel model are presented using the integrated strength of the Adams and explicit Runge–Kutta schemes. The idea of the present study comes to the mind to see the importance of delay diﬀerential equations. For veriﬁcation of the novel designed model, four diﬀerent examples of the designed model are numerically solved by applying the Adams and explicit Runge–Kutta schemes. These obtained numerical results have been compared with the exact solutions of each example that indicate the performance and exactness of the designed model. Moreover, the results of the designed model have been presented numerically and graphically.


Introduction
e historical delay differential equations (DDEs) are applied in the pioneer work of Newton and Leibnitz in the last years of the 16 th century. To understand the worth and importance of the DDEs, one can see their extensive and wide-ranging applications in the field of scientific wonders. Few mentioned applications are population dynamics, economical systems, engineering systems, transports, and communication models [1][2][3][4]. Many researchers worked to solve DDEs in different years, e.g., Kondorse studied DDEs in the seventh decade of the 17 th century, but properly used the applications of DDEs in the 19 th century. Kuang [2] and Hale and LaSalle [5] presented the detailed theory, solution schemes, and applications of DDEs. Perko [6] studied linear/ nonlinear differential models for the dynamical system and configuration. Beretta and Kuang [7] worked on the geometric constancy of DDEs with the constraints of delay values. Frazier [8] explained the DDEs of the second kind by applying the wavelet Galerkin scheme. Rangkuti and Noorani [9] established the exact solution of DDEs by applying the iterative method named as a coupled variation scheme with the support of the Taylor series method. Chapra [10] discussed the scheme of Runge-Kutta for solving both types of differential delay and nondelay models. Adel and Sabir [11] presented the numerical solutions of a nonlinear second-order Lane-Emden pantograph delay differential model via the Bernoulli collocation method. Sabir et al. [12,13] solved the nonlinear functional differential models of second and third order. Erdogan et al. [14] applied the finite difference method on a layer-adapted mesh for singularly perturbed DDEs. Some more details of DDEs are provided in references [15][16][17][18][19][20]. e literature form of the second-order DDE is given as follows [21]: where θ(x) is used for the initial condition and τ 1 is the delayed term in the abovementioned equation. It is clear in understanding that the appearance of this term y(x − τ)in any differential equation shows the DDE that means to subtract some values from time. e question arises here when we add some values in time then what happens, i.e., y(x + τ). is clearly indicates the prediction and aim of the present work is related to design a new model based on the prediction differential equation. e general form of the new second-order prediction differential model along with initial conditions is presented as where τ is used as a prediction term in equation (2). is prediction differential model can be used to forecast the weather, transport, engineering, stock markets, technology, biological models, astrophysics, and many more. Moreover, the obtained numerical results from both of the schemes have been compared with the exact solution to verify the correctness and exactness of the designed model presented in equation (2). Some salient features of the designed model are given as follows: (i) e novel prediction model is successfully designed by considering the worth of the delay differential model (ii) For verification of the designed model, the obtained numerical results have been compared with the exact solutions (iii) Easily comprehensible procedures with effortless implementation, conserved accuracy in close locality of the input interval, broader, and extendibility applicability are other considerable advantages.
e remaining parts of the paper are organized as follows. Section 2 shows the designed methodology. Section 3 represents the detailed results. e conclusions and future research directions of the present study are provided in the last section.

Methodology
In the present study, the strength of predictor-corrector Adams technique [22,23] and explicit Runge-Kutta numerical technique [24,25] is exploited to solve the second-order prediction differential model.

Predictor-Corrector Adams Numerical Scheme.
To find the numerical solutions of the novel designed prediction differential model, the predictor-corrector numerical technique is applied, which takes further two steps to complete.
Step 1: the approximate measures of prediction are accomplished Step 2: to find if the numerical solutions of correction are capable with the similar contributions of prediction.
e generalized Adams-Bashforth two-step numerical scheme using the predictor-corrector techniques is given as e Adams-Moulton two-step corrector scheme is shown as follows: e 4-step predictor-corrector scheme is provided as follows: e Adams-Bashforth-Moulton 4-step scheme is written as follows:

Explicit Runge-Kutta Numerical
Scheme. e explicit Runge-Kutta scheme is applied to solve the novel designed second-order prediction model. e general form of the explicit Runge-Kutta scheme is considered as y n+1 � y n + g s j�1 b j I j , I 1 � h x n , y n , I 2 � h x n + c 2 g, y n + g a 21 I 1 , ⋮ I s � h x n + c s g, y n + g a s 1 I 1 + a s 2 I 2 + a s 3 I 3 + . . . + a ss−1 I s−1 .

Simulations and Results
In this section of the study, the prediction differential model presented in equation (2) is solved by using the four numerical examples based on the predictor-corrector Adams technique and explicit Runge-Kutta method. Furthermore, the obtained numerical results using both the schemes have been compared with the exact solutions of each example. Example 1. Consider the second-order prediction differential equation along with the initial conditions given as follows: e exact solution of equation (9) is 1 + sin x.
Example 2. Consider the second-order prediction differential equation along with boundary conditions given as follows: e exact solution of equation (10) is Example 3. Consider the second-order prediction differential equation involving trigonometric functions along with initial conditions given as follows: e exact solution of equation (11) is sin x.
Example 4. Consider the second-order prediction differential equation along with initial conditions given as follows: e exact solution of equation (12) is e x . It is clearly seen that the prediction term is involved in the form of y(x + π), y(x + 1) in Examples 1 and 2, respectively. e prediction terms are involved four times in Example 3, i.e., (dy/dx)(x + 1), y(x + 1), cos(x + 1), and sin(x + 1). Moreover, the prediction terms appeared twice in Example 4, i.e., (dy/dx)(x + 1) and y(x + 1). e graphic illustration based on the numerical results for all four examples is provided in Figure 1

Conclusions
e present study is carried out to design a novel secondorder prediction differential model by manipulating the strength of the Adams numerical scheme and explicit Runge-Kutta scheme. e designed novel prediction differential model will be very useful and can be applied in many applications. Four different variants of the designed model have been solved by using the Adams and Runge-Kutta schemes and compared the obtained numerical results with the exact solutions. e overlapping of the exact and numerical reference solutions show the worth and accuracy of the novel designed prediction differential model. It is clear in understanding that the proposed methods are valuable and suitable for solving the second-order prediction differential model due to accurate results for all the examples of the second-order prediction differential model. For solving all four examples, the proposed Adams and explicit Runge-Kutta schemes are found to be very good in terms of accuracy and convergence. Software used for solving the prediction differential model is MATLAB R 2017(a) package and Mathematica 10.4.

Data Availability
Our manuscript is not data-based.

Conflicts of Interest
e authors declare that they have no conflicts of interest.