An Analysis of the Theta-Method for Pantograph-Type Delay Differential Equations

&e pantograph equation arises in electrodynamics as a delay differential equation (DDE). In this article, we provide the θ-method for numerical solutions of pantograph equations. We investigate the stability conditions for the numerical schemes. &e theoretical results are verified by numerical simulations. &e theoretical results and numerical simulations show that implicit or partially implicit θ-methods, with θ> (1/2), are effective in resolving stiff pantograph problems.


Introduction
Pantograph equations are a special kind of functional differential equations with proportional delays [1]. e name pantograph was derived in 1971 from the work of Ockendon and Tayler [2]. Over the past few years, pantograph equations have gained increasing importance in the investigation of various scientific models. ese arise in industrial applications and in a variety of fields of pure and applied mathematics, such as electrodynamics, control systems, number theory, probability, and quantum mechanics. Since most of these types of differential equations cannot be solved analytically, researchers have developed numerical methods to solve them.
ere are a lot of research articles in the solution methods of pantograph equations, such as the exponential approximation [3], Taylor method [4], collocation method using Hermite polynomials [5], and improved Morgan-Voyce collocation method [6]. Legendre wavelet solutions are discussed in [7], and homotopy perturbation method is examined in Ref. [8]. e variational iteration method is discussed in Ref. [9], and the homotopy analysis method is discussed in Ref. [10]. e authors in Ref. [11] discuss the modified Chebyshev collocation method for pantograph equations. A pseudospectral method based on the Legendre principle is examined in Ref. [12], and a modified procedure based on the residual power series method is explored in [13]. Other spectral methods and recent research about numerical treatments of pantograph equations exist in Refs. [14][15][16][17].
Despite the fact that there are several polynomial approximation methods, such as spectral and pseudospectral methods, a comprehensive study of step methods for solving pantograph delay differential equations is lacking. In this article, we present some numerical schemes using the ϑ-method to solve pantograph equations numerically and stability conditions for such schemes. Although the maximum order of ϑ-methods is 2, the numerical schemes are very suitable and effective for pantograph equations in a long-run time interval [18][19][20]. e general delay differential equations (DDEs) take the following form: where F: [t 0 , +∞) × C n × C n ⟶ C n and y(t): [t 0 , +∞) × R + ⟶ C n . Time delay can be a constant, time-dependent, or state-dependent. A classic case that has been discussed in many papers is when c(t) � t − τ, and τ is positive. e pantograph equations are functional differential equations whose delays are proportional, where 0 ≤ c(t) ≤ t, c(0) � 0. Differential equations with variable or proportional delays are more difficult to solve numerically than equations with constant delays. e purpose of this article is to consider a nonlinear pantograph type of DDE: (2) e lag function c(t) � λt, and ψ ∈ [λt 0 , t 0 ] is a continuous initial function. We remark that this kind of equation plays an interesting role in modeling many phenomena including biological and nonlinear dynamical systems. Considering the numerical aspect, it is important to identify how the numerical methods preserve the qualitative behavior of the analytic solutions. Herein, we investigate the stability properties of ϑ-methods when applied to pantograph delay differential equations.
is study is organized as follows. Analytical stability conditions are discussed in Section 2. In Section 3, we provide continuous ϑ-methods for the pantograph equation. Numerical stability analysis and conditions of the asymptotic stability are discussed in Section 4. Some numerical simulations showing the effectiveness of the theoretical results are provided in Section 5, and concluding remarks are given in Section 6.

Nonlinear Stability
One of the important characteristics of differential equation (2) is the sensitivity of a particular solution to small changes in the "parameters," which occur in the equation, or in the "initial conditions," that gives rise to a definition of stability. If we consider another system, defined by the same function F(t, u(t), u(λt)) of (2) but with another initial condition, then y 0 ≠ u 0 . Assume that 〈u, v〉 is the inner product of vectors u, v ∈ C N , such that ‖u‖ � 〈u, u〉 1/2 .

depends on the initial condition), and ξ is the unique positive solution of
where Proof. Assume for the inner product 〈., .〉 on C n such that (6) holds. For every t ≥ t 0 , we have (see [22]) It follows from the definitions of ρ 1 (t) and ρ 2 (t), and from the Schwartz inequality that Define Note that Y(t) > 0 for every t > t 0 because we assume that the function f is such that (2) has unique solution y(t) for every initial condition y(t 0 ) � y 0 . en, 2 Complexity so we have and hence, Applying eorem 1 to the above inequality yields the following: Here, G depends only on the initial conditions en, the solution of y ′ (t) � αy(t) + βy(λt), with ρ 1 (t) � Re(α) and ρ 2 (t) � |β|, is algebraically asymptotically stable (see [23]).

Continuous ϑ-Methods for Pantograph Equations
Here, we study a discretization of pantograph equation (2). Let 0 < λ < 1 be a positive real and T ≡ t 0 , t 1 , . . . , t n , . . . be the assigned mesh points, with the property that t n ⟶ ∞ as n ⟶ ∞. Let h n � t n+1 − t n , where n � 0, 1, . . . . be the nonconstant step-size. We divide the whole interval if the solution [0, T) into bounded subintervals We then partition any of them into a fixed number m of subintervals of the same size using the grid points, such that Let y n denote an approximate to the exact solution y(t) of (2) at t n . en, applying the linear ϑ-method to (2) yields y n+1 � y n + h n ϑF t n+1 , y n+1 , y λt n+1 where y(t) extends the values of the numerical solution to nonmesh points. Since, for λ ∈ (0, 1), it is usually the case that λt n ∉ T, we shall then require a densely defined approximation via linear interpolation: To find the solution at nonmesh points, assuming a uniform constant step-size h n � h, we have dense outputs for y(λt): y λt n � (⌊λn⌋ + 1 − λn)y ⌊λn⌋ +(λn − ⌊λn⌋λn)y ⌊λn⌋+1 n � 0, 1. . . . , (20) where ⌊.⌋ is denoting the integer part, so that t ⌊λn⌋ < λt n < t ⌊λn⌋+1 . Furthermore, we define the global mesh by partitioning every primary interval into a fixed number m of subintervals of the same size, so that the mesh points are defined by the following recursion formula: erefore, when ϑ � 1, it corresponds to the so-called backward (or implicit) Euler method. However, when ϑ � 0, it corresponds to the forward method. Both of them are of 1− stage RK methods. Normally, the order of ϑ methods is 1, but for ϑ � 1/2, which corresponds to the trapezium rule method, the order is 2.

Numerical Stability
In this section, we can use the discrete Halanay inequality [21] to examine the stability conditions of the discrete schemes for linear pantograph (22). Theorem 3. Given α + |β| < 0 and ξ h is the solution of the characteristic equations (see [21]):

Complexity
where m ∈ N is a positive integer number, and 0 < αh < 1. Assume that p n ∞ − m is a sequence, satisfying numbers satisfies the recurrence relation: are the values that occur in (25) and G h max ℓ∈T pℓ.

Complexity
For r 3 � 1, We have following theorem and remarks. e ϑ-method, as applied to equation (2) is asymptotically stable if and only ϑ > 1/2.

Complexity 5
It is important to mention that explicit methods are easy to implement and are recommended for solving nonstiff problems. For stiff problems, however, the standard explicit methods with poor stability properties suffer a lot from stepsize reduction and turn out to be inefficient in terms of overall computational costs [27][28][29].
Remark 2. In order to solve stiff problems numerically, we should use implicit or partially implicit ϑ-method's schemes.

Conclusion
Time delays significantly increase the complexity and stability of differential equations. It is possible for a delay to stabilize an otherwise unstable system or unstabilize a stable one. We developed and discussed an efficient numerical scheme for solving pantograph-type delay differential equations using the ϑ-method. Some stability conditions of analytical and numerical solutions of the problem have been investigated. From the theoretical results, numerical simulations can conclude that the ϑ-method, applied to pantograph equations, is a reliable method and asymptotically stable if and only if ϑ > 1/2, regardless of the step-size of the scheme, especially with respect to the long-time behavior of the solutions. For ϑ < 1/2, the stability of the numerical solution is almost unstable. Most of pantograph equations are stiff problems, and it is recommended to use an implicit or partially implicit sachems.
We can extend the stability analysis, in the coming research, to other classes of higher-order Runge-Kutta methods with pantograph equations. e schemes can also be extended to stochastic pantograph delay differential equations [30,31].
Data Availability e authors confirm that the data supporting the findings of this study are available within the article.

Conflicts of Interest
e authors declare no conflicts of interest.