## $$H_{\alpha}$$-stability of modified Runge-Kutta methods for nonlinear neutral pantograph equations.(English)Zbl 1125.65076

The authors investigate the $$H_\alpha$$-stability of algebraically stable Runge-Kutta methods with a variable stepsize for the following system
$\begin{cases} y'(t)=f(t,y(t))+g(t, y(qt))+c(t)y'(qt),& t>0,\\ y(0)=y_0,\end{cases} \tag{1}$
where $$f,g:[0,\infty)\times C^d\to C^d$$ are continuous and $$c(t)$$ is a continuous $$d\times d$$-matrix function in $$[0,\infty)$$. The first section is an introduction concerning the nonlinear neutral pantograph equation:
$\begin{cases} y'(t)=F(t,(t),y(qt),y'(qt)),& t>0,\\ y(0)=y_0,\end{cases}\tag{2}$
where $$0<q<1$$, $$y_0\in C^d$$, $$F:\mathbb R^+\times C^d \times C^d\times C^d\to C^d$$ continuous, and the stability properties of the appropriate numerical methods used for the numerical resolution of this kind of equations. The second section focuses on the modified Runge-Kutta method $$(A,b, c)$$ with the form:
\begin{aligned} y_{n+1} & =y_n+h_n \sum^s_{i=1}b_if(t^i_n,y_i^{n+1}), \\ y_i^{n+1} & =y_n+\overline {h_n}\sum^s_{j=1}a_{ij}f(t^j_n,y_j^{n+1}),\quad i=1,2, \dots,s\end{aligned}
where $$y_0\in C^d$$, $$f:[0,\infty)\times C^d\to C^d$$ is a continuous function, $$\Delta=\{0=t_0<t_1<\cdots<t_n=T>0\}$$ represents a mesh, $$h_{n+1}=t_{n+1}-t_n$$ the stepsize, $$t^i_n=t_n+ c_ih_n$$ and $$\overline{h_n}=(1+ \alpha_n(h_n))h_n$$ with $$\alpha_n (\eta)$$ such that
$\begin{cases} \alpha_n(\eta)=O (\eta^p) & \text{ as }\eta \to 0,\\ \alpha_n(\eta)>0, & \text{ for all }\eta,\end{cases}$
Asymptotical stability conditions of the analytic solutions of (1) are presented. The third section is devoted to the stability analysis of the modified Runge-Kutta method, giving the conditions for the $$H_\alpha$$-stability. It points out the $$H_\alpha$$-stability for the Radau IA, Radau IIA and Lobato IIIC methods, the odd-stage Gauss-Legendre methods and the one-leg $$\theta$$ methods with $$\tfrac 12\leq\theta\leq 1$$. In the last section one gives two numerical experiments are presented: the linear neutral pantograph equation and the nonlinear neutral pantograph equation.

### MSC:

 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K40 Neutral functional-differential equations
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### References:

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