Euler-Type Methods

EULER-TYPE METHODS 5.2.1 Euler s Method for Single ODE... 

This form is a closed type formula since it does not allow direct steps from x, to x.,, but uses the basic Euler s method to estimate y j, thus... 

The set of ODEs represented by Equations 8.30 can be solved by various means. They are first-order, initial-value problems of the type introduced in Chapter 2 for multiple reactions. We use Euler s method. Appling it to Equations 8.31 and 8.33 gives... 

The numerical methods that are available in Stella are Euler s method, Runge-Kutta second order, or Runge-Kutta fourth order. One of the menu items allows the user to specify the length of simulation time, as well as the time increment dt, and the type of numerical method that is to be employed. 

Equations that arise in modeling the dynamics of homogeneous systems are initial value problems, generally approached with techniques of the Euler type. Initial value problems involve derivatives with respect to time these must be discretized, which can be done using the forward Euler method... 

The nonlinear study of the critical and postcritical behavior of a cantilever bar is performed by means of the linear equivalence method, introduced by one of the authors. A Bernoulli-Euler type model is considered, which yields approximate formulae for the ratio f/ (where Z is the bar length and f Is the maximum deflection) as a function of the supraunitary ratio P/Pcr (P is the axial load and Per is Its critical value). Two other approximate formulae concerning the displacement of the bar end along Its straight axis and the rotation of the corresponding cross section are also deduced. All these formulae give numerical results very close to the exact ones and much better than other postcritical estimations. 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

With Euler s simple method, very small time intervals must be chosen to achieve reasonably accurate profiles. This is the major drawback of this method and there are many better methods available. Among them, algorithms of the Runge-Kutta type [15, 28, 29] are frequently used in chemical kinetics [3], In the following subsection we explain how a fourth-order Runge-Kutta method can be incorporated into a spreadsheet and used to solve nonstiff ODEs. 

We present how to treat the polarization effect on the static and dynamic properties in molten lithium iodide (Lil). Iodide anion has the biggest polarizability among all the halogen anions and lithium cation has the smallest polarizability among all the alkaline metal cations. The mass ratio of I to Li is 18.3 and the ion size ratio is 3.6, so we expect the most drastic characteristic motion of ions is observed. The softness of the iodide ion was examined by modifying the repulsive term in the Born-Mayer-Huggins type potential function in the previous workL In the present work we consider the polarizability of iodide ion with the dipole rod method in which the dipole rod is put at the center of mass and we solve the Euler-Lagrange equation. This method is one type of Car-Parrinello method. 

In section 9.2 we illustrated one explicit method, Euler s forward method. In the present section, we likewise used only one type of implicit method, based on the trapezoidal or midpoint rule. All our examples have used constant increments Af higher computational efficiency can oftenbe obtained by making the step size dependent on the magnitudes of the changes in the dependent variables. Still, these examples illustrate that, upon comparing equivalent implicit and explicit methods, the former usually allow larger step sizes for a given accuracy, or yield more accurate results for the same step size. On the other hand, implicit methods typically require considerably more initial effort to implement. 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

There are two kinds of failure due to buckling. The first, general buckling, involves bending of the axis of the cylinder, resulting in instability. This is the type addressed by Euler and designed for by a slenderness ratio method. 

It turns out that one may present an exact integration of the equations for geodesics of the metrics (pahD on the group SL(m,C). Metrics of the type (pabD appeared for the first time in the course of construction of nonlinear differential equations integrable by the inverse scattering method. FVom the paper [38] it readily follows that the Euler equation X = [X,ipahD ) on a classical Lie algebra of series Afn-i serves as a commutativity equation for a pair of operators. 

The nonlinear study of bifurcations of the elastic equilibrium of a straight bar involves, in a way, a change of the physical point of view, mostly due to the mathematical difficulties related to the direct approach of the Bernoulli-Euler (B.-E.) equation. This aspect gave rise to various models describing the same phenomenon, such as Kirchhoff s pendulum analogy [1], as well as to different methods of calculus, such as Thompson s potential energy method [2], [3]. In this paper, we use the linear equivalence method (LEM) to a B.-E. type model, thus deducing an approach for the critical and postcritical behavior of the cantilever bar. 

Equations 5.38 and 5.39 are two first-order ODE-IVPs that can be solved by any of the methods previously described. Calculations using the Euler method with h = 0.1 and = 0.5 are shown in the following spreadsheet for f up to 2. The formulas for the right-hand side of the equations for y and z are shown in bold type. 

The dynamic model of the manipulator is obtained using the Newton-Euler dynamic modeling formulation. This method is independent of the type of manipulator configuration. When Newton-Euler equations are applied, they yield and equation of motions for the manipulator which can be written as [12] ... 

The ATB model is based on the rigid body dynamics which uses Euler s equations of motion with constraint relations of the type employed in the Lagrange method. The model has been successfully used to study the articulated human body motion under various types of body segment and joint loads. The technology of robotic telepresence will provide remote, closed-loop, human control of mobile robots. 

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