Pantograph Differential Equations Research Articles

Convergence rate of Euler–Maruyama scheme for stochastic pantograph differential equations

This paper investigates the convergence rate of Euler–Maruyama scheme for a class of stochastic pantograph differential equations, which may not satisfy the general linear growth condition. We reveal that the convergence order for the equations driven by Brownian motion is 12. We also demonstrate that, along with the value of the exponent p⩾2 increasing gradually, the convergence rate for the equations with Markovian jump is decreasing. This conclusion is entirely different from the case without Markovian jump.

Approximate Solutions of Hybrid Stochastic Pantograph Equations with Levy Jumps

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in <svg style="vertical-align:-0.0pt;width:16.1625px;" id="M1" height="15.8375" version="1.1" viewBox="0 0 16.1625 15.8375" width="16.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> </svg> sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milo&#x161;evi&#x107; and Jovanovi&#x107; (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory.

Oscillation of a pantograph differential equation with impulsive perturbations

Some sufficient conditions are obtained for the oscillation of all solutions of a pantograph differential equation with impulsive perturbations of the formx′(t)=P(t)x(t)-Q(t)x(αt),t⩾t0,t≠tk,(∗)x(tk+)=bkx(tk),k=1,2,…(∗∗)Our results reveal the fact that the oscillatory properties of all solutions of impulsive differential equations (∗) and (∗∗) may be caused by the impulsive perturbations (∗∗), though the corresponding differential equations without impulses admit a nonoscillatory solution. Some examples are also given to illustrate the applicability of the results obtained.

Direct operatorial tau method for pantograph-type equations

The Lanczos tau method is applied to find Chebyshev polynomial approximations for the solutions of pantograph differential equations. The results are accompanied by an error analysis. Numerical examples, calculated by our Matlab package Chebpack confirm the theory and prove the importance for practice of this approach.

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.

On collocation methods for delay differential and Volterra integral equations with proportional delay

To compute long term integrations for the pantograph differential equation with proportional delay qt, 0 < q ⩽ 1: y′(t) = ay(t) + by(qt) + f(t), y(0) = y0, we offer two kinds of numerical methods using special mesh distributions, that is, a rational approximant with ‘quasi-uniform meshes’ (see E. Ishiwata and Y. Muroya [Appl. Math. Comput., 2007, 187: 741-747]) and a Gauss collocation method with ‘quasi-constrained meshes’. If we apply these meshes to rational approximant and Gauss collocation method, respectively, then we obtain useful numerical methods of order p* = 2m for computing long term integrations. Numerical investigations for these methods are also presented.

Stability of the Discretized Pantograph Differential Equation

Stability of the Discretized Pantograph Differential Equation

Stability of the discretized pantograph differential equation

In this paper we study discretizations of the general pantograph equation \[ y ′ ( t ) = a y ( t ) + b y ( θ ( t ) ) + c y ′ ( ϕ ( t ) ) , t ≥ 0 , y ( 0 ) = y 0 , y’(t) = ay(t) + by(\theta (t)) + cy’(\phi (t)),\quad t \geq 0,\quad y(0) = {y_0}, \] , where a, b, c, and y 0 {y_0} are complex numbers and where θ \theta and ϕ \phi are strictly increasing functions on the nonnegative reals with θ ( 0 ) = ϕ ( 0 ) = 0 \theta (0) = \phi (0) = 0 and θ ( t ) &gt; t , ϕ ( t ) &gt; t \theta (t) &gt; t, \phi (t) &gt; t for positive t. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a, b, c and the stepsize which imply that the solution sequence { y n } n = 0 ∞ \{ {y_n}\} _{n=0}^\infty is bounded or that it tends to zero algebraically, as a negative power of n.