Euler integration formula

If the model of the system to be simulated can be reduced to the form of equations (2.22), then a timemarching, numerical solution becomes possible by repeated application of, for instance, the first-order Euler integration formula ... 

The mathematical equations used for each numerical int ration method [7, 8] are summarized in this section, where y stands for a nonlinear ordinary differential equation and h for the step size. Modified Euler integration formula ... 

Euler s formula has been improved manyfold over the last century or two. To better account for the turning , or for the concavity of the solution curve y(x), improved integration formulas involve several evaluations of F in the interval [x, x +1] and then average. For example, the classical Runge11-Kutta12 integration formula uses four... 

For the radial integration, we have adopted the Euler-Maclaurin formula proposed by Murray et al. [47], in which the radial points and weights are given by... 

In the case of an Euler integration scheme, approximate formulae were derived to estimate an appropriate and stable time interval. 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

This relation can be easily verified by employing the Euler formula c = cos 6 i sin 0 for the pure imaginary part of the exponential function and by observing the definite integrals ... 

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

In this formula, I is the nuclear quantum number, r(Bj) the first derivative lineshape function, B the resonance position and P the transition probability. 0 and i are the Euler angles expressing the orientation of the magnetic field vector B with respect to the principal axes of the tensors. Integration is needed since in powder samples, the crystallites take all possible orientations with respect to the magnetic field. Since the principal tensor axes and the crystal axes are assumed to be coincident, integration can be restricted to one octant of the unit sphere. 

Here r = hk Jhk, where hj = xJ+1 —Xj is the integration step size. Two variants are presented, as well as a simple Euler formula suitable for starting an integration series. 

The Euler-Maclaurin numerical quadrature formula for such an integral (using n -F 1 equally spaced values of y yi = i/n, for i = 0, n) is ... 

Intercepts and Common Tangents to Agmixing ts. Composition in Binary Mixtures. Euler s integral theorem and the Gibbs-Duhem equation provide the tools to obtain expressions for Agmixing and (9 Agmixing/9y2)r,/) in binary mixtures. This information allows one to evaluate the tangent at any mixture composition via the point-slope formula. For example, if i i = and p,2 = M2 when the mole fraction of component 2 is y, then equations (29-73) and (29-76) yield ... 

This formula is simply the explicit Euler method. The local truncation error is of order of Oih ), that is, the error is proportional to if all the previous values, y , y i,... are exact. However, in the integration from time t = 0, the error of the method at time is accurate up to Oih) because the number of integration step is inversely proportional to h. 

Formulae for the solutions are derived by successive integration of a certain system of linear inhomogeneous equations in which the Euler equations are reduced after the choice of the basis from the root vectors. A similar assertion was later announced in the paper [15]. FVom Theorem 4.2.8 there follows a useful corollary. 

In order to determine the spectrum transmittance H(ja)), it is necessary to calculate the integrals on the right-hand side of Eq. (2.85). According to the Euler formula ... 

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