Jingjun Zhao

Collocation methods for fractional integro-differential equations with weakly singular kernels

In this paper, the piecewise polynomial collocation methods are used for solving the fractional integro-differential equations with weakly singular kernels. We present that a suitable transformation can convert fractional integro-differential equations to one type of second kind Volterra integral equations (VIEs) with weakly singular kernels. Then we solve the VIEs by standard piecewise polynomial collocation methods. It is shown that such kinds of methods are able to yield optimal convergence rate. Finally, some numerical experiments are given to show that the numerical results are consistent with the theoretical results.

Stability of a class of Runge-Kutta methods for a family of pantograph equations of neutral type

This paper deals with the stability of Runge-Kutta methods for a class of neutral infinite delay-differential equations with different proportional delays. Under suitable conditions, the asymptotic stability of some Runge-Kutta methods with variable stepsize are considered by the stability function at infinity. It is proved that the even-stage Gauss-Legendre methods are not asymptotically stable, but the Radau IA methods, Radau IIA methods and Lobatto IIIC methods are all asymptotically stable. Furthermore, some numerical experiments are given to demonstrate the main conclusions.

Delay-dependent stability analysis of symmetric boundary value methods for linear delay integro-differential equations

The paper is concerned with the numerical stability of linear delay integro-differential equations (DIDEs) with real coefficients. Four families of symmetric boundary value method (BVM) schemes, namely the Extended Trapezoidal Rules of first kind (ETRs) and second kind (ETR

Stability of the Rosenbrock methods for the neutral delay differential-algebraic equations

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Delay-dependent stability of symmetric boundary value methods for second order delay differential equations with three parameters

s), the Top Order Methods (TOMs) and the B-spline linear multistep methods (BS methods) are considered in this paper. We analyze the delay-dependent stability region of symmetric BVMs by using the boundary locus technique. Furthermore, we prove that under suitable conditions the symmetric schemes preserve the delay-dependent stability of the test equation. Numerical experiments are given to confirm the theoretical results.

Stability of Symmetric Runge--Kutta Methods for Neutral Delay Integro-Differential Equations

This paper develops the Rosenbrock methods for the neutral delay differential-algebraic equations (NDDAEs) and proves that the Rosenbrock methods equipped with suitable interpolation are GP-stable under proper assumption for the linear neutral delay differential-algebraic equations with constant coefficients. Furthermore, the GP-stability of the Runge-Kutta methods is also considered. The discussions are supported by the numerical experiments.

Asymptotic stability of linear non-autonomous difference equations with fixed delay

This paper aims at the delay-dependent stability analysis of symmetric boundary value methods, which include the Extended Trapezoidal Rules of the first kind and the second kind, the Top Order Methods and the B-spline linear multistep methods, for second order delay differential equations with three parameters. Theoretical analysis and numerical results are presented to show that the symmetric boundary value methods preserve the asymptotic stability of the true solutions of the test equation.

D-convergence and conditional GDN-stability of exponential Runge–Kutta methods for semilinear delay differential equations

The aim of this paper is to analyze the delay-dependent stability of symmetric Runge--Kutta methods, including the Gauss methods and the Lobatto IIIA, IIIB, and IIIS methods, for the linear neutral delay integro-differential equations. By means of the root locus technique, the structure of the root locus curve is given and the numerical stability region of symmetric Runge--Kutta methods is obtained. It is proved that, under some conditions, the analytical stability region is contained in the numerical stability region. Finally, some numerical examples are presented to illustrate the theoretical results.

Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system

Abstract This paper studies the asymptotic stability of the linear non-autonomous difference equations with fixed delay. It is shown that the difference system is asymptotically stable if and only if the norm of the limit for the coefficient matrix is less than 1 under some conditions. Furthermore, the discretization of the pantograph equation by Runge–Kutta methods with variable stepsize is analyzed. Finally, some numerical experiments are made to demonstrate the main conclusions.

The analysis of operator splitting methods for the Camassa–Holm equation

Abstract This paper is concerned with exponential Runge–Kutta methods with Lagrangian interpolation (ERKLMs) for semilinear delay differential equations (DDEs). Concepts of exponential algebraic stability and conditional GDN-stability are introduced. D-convergence and conditional GDN-stability of ERKLMs for semilinear DDEs are investigated. It is shown that exponentially algebraically stable and diagonally stable ERKLMs with stage order p, together with a Lagrangian interpolation of order q (q ≥ p), are D-convergent of order p. It is also shown that exponentially algebraically stable and diagonally stable ERKLMs are conditionally GDN-stable. Some examples of exponentially algebraically stable and diagonally stable ERKLMs of stage order one and two are given, and numerical experiments are presented to illustrate the theoretical results.