Abstract
This paper focuses on the stability and oscillations of Euler-Maclaurin method for linear differential equations with piecewise constant arguments <svg style="vertical-align:-2.3205pt;width:143.9875px;" id="M1" height="20.6" version="1.1" viewBox="0 0 143.9875 20.6" width="143.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path id="x1D462" d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74
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q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" /></g><g transform="matrix(.017,-0,0,-.017,55.162,17.65)"><use xlink:href="#x1D462"/></g><g transform="matrix(.017,-0,0,-.017,64.308,17.65)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,70.189,17.65)"><use xlink:href="#x1D461"/></g><g transform="matrix(.017,-0,0,-.017,75.986,17.65)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,84.706,17.65)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g><g transform="matrix(.017,-0,0,-.017,97.506,17.65)"><path id="x1D44F" d="M452 333q0 -82 -46 -164t-116 -128q-81 -53 -158 -53q-26 0 -51 13t-37 32q-26 41 -6 134l91 431q7 40 2.5 47t-34.5 7h-33l2 25q36 4 72.5 13t58.5 15.5t28 6.5q10 0 4 -29l-91 -391h2q65 77 130 116.5t105 39.5q36 0 56.5 -32t20.5 -83zM365 316q0 76 -35 76
q-31 0 -93.5 -49t-107.5 -110q-12 -37 -17 -70q-9 -65 12 -96.5t59 -31.5q31 0 56 14q53 29 89.5 105t36.5 162z" /></g><g transform="matrix(.017,-0,0,-.017,105.479,17.65)"><use xlink:href="#x1D462"/></g><g transform="matrix(.017,-0,0,-.017,114.624,17.65)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,120.506,17.65)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,126.37,17.65)"><use xlink:href="#x1D461"/></g><g transform="matrix(.017,-0,0,-.017,132.167,17.65)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g><g transform="matrix(.017,-0,0,-.017,138.031,17.65)"><use xlink:href="#x29"/></g> </svg>. The necessary and sufficient conditions under which the numerical stability region contains the analytical stability region are given. Furthermore, the conditions of oscillation for the Euler-Maclaurin method are obtained. We prove that the Euler-Maclaurin method preserves the oscillations of the analytic solution. Moreover, the relationships between stability and oscillations are discussed for analytic solution and numerical solution, respectively. Finally, some numerical experiments for verifying the theoretical analysis are also provided.
Highlights
In the present paper we will consider the following differential equations with piecewise constant arguments (EPCA): u (t) = au (t) + bu ([t]), t ≥ 0, (1)u (0) = u0, where a, b, u0 ∈ R and [⋅] denotes the greatest integer function
Let the stepsize h = 1/m; we shall use the Euler-Maclaurin method with n = 2, the θ-methods with θ = 0.5, and the 2-Radau IA method to get the numerical solution at t = 10
The analytic solution and the numerical solution of (39) are both oscillatory and asymptotically stable according to Theorems 28 and 29, which
Summary
In the present paper we will consider the following differential equations with piecewise constant arguments (EPCA): u (t) = au (t) + bu ([t]) , t ≥ 0, (1). In [23, 24], the stability of numerical solution in Runge-Kutta methods for (1) and the mixed type EPCA was studied, respectively. Different from [29], the novel idea of our paper is that we will study both stability and oscillations of the numerical solution in the Euler-Maclaurin method for (1), and their relationships are analyzed quantitatively. With respect to the numerical analysis of delay differential equations (DDEs), few results on the Euler-Maclaurin method were obtained except for [33].
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