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Abstract: In the present paper, the dynamical differential equations with initial conditions is converted into the model of linear operator action, in which the linear operator is just the infinitesimal generator of the solver of the differential equations. And the resolvent of the linear operator is the Laplace transform of the solver of original differential equations. So the solver of original differential equations can be obtained by inversing the Laplace transform of its resolvent. An iterative algorithm for nonlinear differential equations with initial condition is easily presented by means of numerical Laplace transform inversion. The numerical examples show that the method of this paper is effective. [Copyright &y& Elsevier]

Abstract: In this paper, precise integration is compounded with Runge–Kutta method and a new effective integration method is presented for solving nonlinear dynamical system. Arbitrary dynamical system can be expressed as nonhomogeneous linear differential equation problem. Precise integration is used to solve its corresponding homogeneous equation and Runge–Kutta methods can be used to calculate the nonhomogeneous, nonlinear items. The precise integration may have large time step and the time step of the Runge–Kutta methods can be adjusted to improve the computational precision. The handling technique in this paper not only avoids the matrix inversion but also improves the stability of the numerical method. Finally, the numerical examples are given to demonstrate the validity and effectiveness of the proposed method. [Copyright &y& Elsevier]

In the present paper, based on the classical Magnus expansion, a simple and efficient fourth-order integrator is given for an arbitrary nonlinear dynamic system, which can preserve the qualitative properties of the exact solution. The proposed method can be considered an averaging technique, and only requires evaluations of exponentials of simple unidimensional integrals. Finally, the numerical examples are given to demonstrate the validity and effectiveness of the method of this paper. [Copyright &y& Elsevier]

Highlights: [•] It can be included that e-beam evaporation can preserve compositional congruency. [•] It is observed that Te concentration decreases with increasing of Ge concentration. [•] High-pressure polymorphism occurs due to the ferroelectric phase structure. [Copyright &y& Elsevier]

Abstract: In this paper we consider Newton iteration methods for solving nonlinear equations on matrix Lie groups. Recently, Owren and Welfert have proposed a method where the original nonlinear equation is transformed into a nonlinear equation on the Lie algebra of the group via the exponential map, thus Newton iteration methods may be applied. Based on this we suggest two improved variants of Newton iteration algorithm. One is that the exponential map would be approximated by Cayley map and give a Cayley version Newton iteration method for solving nonlinear equations on quadratic Lie groups, then we show that, the proposed method converges quadratically; Another is a variant of Newton type method with accelerated convergence and the numerical tests reported seem to support that it converges with cubically. [Copyright &y& Elsevier]

Highlights: [•] Spray dry powder is studied for orange oil stabilization. [•] The rate of formation of limonene oxide species is used as a probe. [•] Literature data were used to develop an absolute oxidation shelf life scale. [•] Orange oil stability is directly linked to the M n of the glassy matrix. [•] Reduction of oxygen penetration involves reduced stability to humidity. [ABSTRACT FROM AUTHOR]