On Euler’s Discretization of Sliding Mode Control Systems with Relative Degree Restriction
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In this paper a new Newton-Raphson (NR) method and a new implicit Euler method are developed for numerically solving the HNC and related equations for simple fluids. The new methods are efficient, stable and easy to implement. The implicit Euler method is derived through a reformulation of the usual HNC problem into the form of an initial value problem (IVP). The IVP formulation allows application of specialized numerical techniques developed for the solution of stiff systems of differential equations. The implicit Euler method is shown to have excellent stability and rate of convergence. The relationship between the implicit Euler method and the NR method is elucidated, and it is shown that in the limit of infinite step-size, the implicit Euler method reduces to the NR method.
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The split-step Euler scheme for SDEs is first brought forward,under Lipschitz and linear growth condition it is proved that the split-step Euler scheme is convergence with strong order1/2.At the same time,the meansquare stability of the theory of split-step Euler-approximation is proved.
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Many filters, power converters, AC electric motors and their shaft mechanics are characterized by even order differential equation systems that are oscillatory in their nature. The low damping causes problems with the accuracy of both the forward and the backward Euler algorithms. However, the symmetric Euler algorithm, which is even simpler than the forward Euler, gives an accuracy that is comparable to the accuracy obtained by the trapezoidal method without requiring matrix inversion. The symmetric Euler algorithm is often confused with the forward Euler and is thus quite unknown. A formula for the numerical stability of the symmetric Euler algorithm is presented. Accuracies of the forward Euler, backward Euler, modified Euler, trapezoidal, Runge-Kutta and symmetric Euler algorithms are compared. Some applications are presented in detail.
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In non linear Finite Element Method, Euler Newton method, Euler Cauchy method and Euler Once iteration method are always used to solve non linear equations.For a typical geometrically non linear problem, the processes of the three methods are presented in detail.Comparing the efficiency and accuracy of the three methods, the following conclusions can be drawn.Euler Cauchy method has the highest efficiency and lowest accuracy in the three methods.Although the efficiency of Euler Once iteration method is lower than that of Euler Cauchy method,the accuracy is higher remarkable.It should be noted that Euler Once iteraton method is unstable, when large load increment step is used to analyze a strong non linear problem.Euler Newton method is more stable and reliable than others, and it is robust for many kinds of non linear problems.But ,its efficiency is low.
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Richardson extrapolation
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Many natural and engineering phenomena can be attributed to the ordinary differential equation problems, and Euler methods are the most commonly used for solving ordinary differential equations. In this study, the theoretical analysis method is used to clarify the principle of Euler’s method to solve the ordinary differential equations firstly. Secondly, one typical example is set to compare the calculation accuracy and efficiency of the forward Euler formula, backward Euler formula, trapezoidal formula and modified Euler formula by self-programming. The results show that the calculation accuracy of Euler methods generally depend on the calculation step. As the step decreases, the calculation accuracy is gradually improved. In addition, the calculation efficiency and accuracy of the modified Euler method are very high, and the calculation accuracy of the trapezoidal formula and modified Euler method are quite close. However, the calculation accuracy of the forward Euler formula and the backward Euler formula can’t compare with those of the trapezoidal formula and the modified Euler method. Therefore, it is recommended to use the modified Euler’s method in engineering calculations, and the trapezoidal formula is secondly recommended.
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Euler introduced the famous Euler method in 1728. A s the simplest and the most analyzed numerical integration, it has become the stepping-s tone of numerical methods for solving Initial value Problems in Ordinary Differential Equations. There has been considerable efforts to improve on Euler method by increasing its order of accuracy. Recently, in [1], Abraham proposed a new improvement on Euler Method called M odified Improved Modified Euler Method. In this work, we investigate the basic prop erties of this new method vis-a-vis the older ones. Our analysis show that the method is converge nt to order 2 and stable when applied to autonomous Initial Value Problem. AMS MSC 2010 Classification: 65L05, 65L06.
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