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In this paper, we investigate two-step Runge–Kutta methods to solve Volterra integro-differential equations. Two-step Runge–Kutta methods increase the order of convergence in comparing the classical Runge–Kutta method without extra computational cost. First, the local order conditions and convergence theorem are derived. Then, stability properties of two-step Runge–Kutta methods corresponding to the basic and convolution test equations are analysed. Furthermore, one-stage method with order four and two-stage method with order six are constructed and we plot the stability regions. Numerical examples are presented to confirm the theoretical analyses. [ABSTRACT FROM AUTHOR]

This paper investigates the relationship between business strategy and sustainability of Chinese SMEs and how the environmental uncertainty may affect this relationship. It analyses the impact of active and passive business strategies on firms' sustainability and tests the potential moderating role of economic policy uncertainty in this relationship. The empirical analysis is performed by employing fixed-effect and GMM estimations on data collected from 937 Chinese A-share non-financial companies listed on the Shanghai and Shenzhen stock exchanges from 2010 to 2021. The results show that the high-risk active business strategy of innovation and new market development is associated with greater business sustainability, and this relationship is stronger for SMEs than for non-SMEs in China. Moreover, higher economic policy uncertainty strengthens the positive relationship between active business strategy and firm sustainability, implying that in periods of higher environmental uncertainty firms which pursue active business strategy achieve greater business sustainability. These findings are useful for devising business strategy and can assist in formulating policy initiatives seeking to ensure sustainable development of SMEs in China. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Galerkin–Legendre spectral method for solving the two-dimensional distributed-order time fractional fourth-order partial differential equation. By utilizing the composite Simpson formula to discretize the distributed-order integral, we transform the considered equation into a multi-term time fractional sub-diffusion equation. Then the L 2 - 1 σ formula is used to approximate the multi-term Caputo fractional derivatives and the Legendre spectral method is employed for the spatial discretization. The scheme is proved to be unconditionally stable and convergent in both L 2 - and L ∞ -norms with fourth-order accuracy in distributed order, second-order accuracy in time and spectral accuracy in space. Finally, some numerical tests are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a conservative linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Schrödinger equation. We construct5 a time-discrete system, to which, the mass conservation, semi-discrete error estimates and the suitable regularity of the numerical solution are obtained. With the spatial direction discreted by FEMs, the fully discrete conservative linearized finite element scheme is presented. Moreover, by a new error splitting technique, an unconditional-norm error estimates are derived by the boundedness of the fully-discrete numerical solution in-norm. Finally, some numerical examples are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study Galerkin finite element methods for a class of higher dimensional multi-term fractional diffusion equations. The finite difference approximation of Caputo derivative on non-uniform meshes is used in temporal direction and Galerkin finite element method is used in spatial direction. We prove that semi-discrete and fully discrete finite element schemes are unconditionally stable. Meanwhile,-norm convergence properties of the two schemes are proved rigorously. To confirm our theoretical analysis, we give some numerical examples in both two-dimensional (2D) and three-dimensional (3D) spaces. Finally, a moving local refinement technique in temporal direction is used to improve the accuracy of numerical solution. [ABSTRACT FROM AUTHOR]

In this work, we propose two classes of two-step Milstein-type schemes : the double-implicit Milstein scheme and the split two-step Milstein scheme, to solve stochastic differential equations (SDEs). Our results reveal that the two new schemes are strong convergent with order one. Moreover, with a restriction on stepsize, these two schemes can preserve the exponential mean square stability of the original SDEs, and the decay rate of numerical solution will converge to the decay rate of the exact solution. Numerical experiments are performed to confirm our theoretic findings. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the strong convergence of the split-step theta methods for non-autonomous stochastic differential equations under a linear growth condition on the diffusion coefficient and a one-sided Lipschitz condition on the drift coefficient. Then, if the drift coefficient satisfies a polynomial growth condition, we further get the rate of convergence. Finally, the obtained results are supported by numerical experiments. [ABSTRACT FROM PUBLISHER]