Series Euler

The following relation exists between the harmonic series of the above equation and the Euler constant (y= 0.5772) for large values of n... 

The simplest method for integrating eq. (14.23) is the Euler method. A series of steps are taken in the direction opposite to the normalized gradient. 

We obtain quite simple power series expressions for the Betti numbers of all the SM in terms of the Betti numbers of S. Similar results hold for the KAn-. The formulas specialize to particularly simple expressions for the Euler numbers of... 

A procedure based on the truncated Taylor series is named after Euler. With increment h = Ax, successive values of y are yi+i = Yi + h f (Xj, yj)... 

Bertaut, E. E (1988) Eulers indicatrix and crystallographic transitive symmetry operations in the hyperspaces E ri). Comptes Rendus de VAcademic des Sciences, Serie II (Mecanique, Physique, Chimie, Sciences de I Univers, Sciences de la Terre). 307(10) 1141-46. (The author uses elementary number theory in this article on crystallographic symmetry operations.)... 

To compare the explicit and implicit Euler methods we exploited that the solution (5.3) of (5.2) is known. We can, however, estimate the truncation error without such artificial information. Considering the truncated Taylor series of the solution, for the explicit Euler method (5.7) we have... 

The first three terms represent the forward Euler algorithm operating on the exact solution, with the last term [in square brackets] providing a measure of the local truncation error. The local truncation error can be identified through a Taylor series expansion of the solution about the time tn ... 

The first three terms represent the implicit Euler algorithm and the remaining [bracketed] term represents the local truncation error. A Taylor series expansion about tn+ (in the negative t direction) yields an expression for y(t )... 

By expanding the Helmholtz free energy F at constant T in an arithmetic series in terms of ujk, we see that the linear terms vanish in view of the equilibrium condition (Euler relation for homogeneous functions of second order, F is given as... 

Since the exponential function may be defined everywhere in the complex plane, we may expand exp(i0) and, using the series expansions for the trigonometric functions, obtain Euler s formula... 

The Taylor series expansion in Chapter 2 makes it possible to derive a remarkable relationship between exponentials and trigonometric functions, first found by Euler ... 

The integral over the Euler angles in Eq. (32) is found analytically using the Clebsch-Gordan series [40, 64, 68]... 

Recall from Section 2.2.1 that the coordinate axes of any individual grain, or crystal, can be transformed to the sample reference coordinate system through a series of three Euler angle rotations. With the ODF, tme three dimensional representations of intensity... 

The simplest method for integrating eq. (14.23) is the Euler method. A series of steps e taken in the direction opposite to the normalized gradient.------------------------------... 

For systems involving recycle streams or intermediate feed locations, the method of successive substitution can be used [Mohan and Govind, 1988a]. Moreover, multiple reactions including side reactions and series, parallel or series-parallel reactions result in strongly coupled differential equations. They have been solved numerically using an implicit Euler method [Bernstein and Lund, 1993]. 

Prime numbers do not occur in a predictable way. There are sequences of primes which can be partially described in a formula, but sooner or later the formula breaks down. One formula, invented by Marin Mersenne (1588-1648) is 2P - 1, where p is a prime number. Although this formula generates many primes, it also misses many primes. Another formula, invented by Leonhard Euler (1707-1783), generates prime numbers regularly for the series of consecutive numbers from 0 to 15 and then stops. The formula is + x + 17, in which x is any number from 0 to 15. 

The advantage of this method, known as Euler s method, is that it can be easily expanded to handle systems of any complexity. Euler s method is not particularly useful, however, since the error introduced by the approximation d[A]/dt = A[A]/Af is compounded with each additional calculation. Compare the Euler s method result in column B of Figure 9-20 with the analytical expression for the concentration, [A]t = [A]oe, in column C. At the end of approximately one half-life (seven cycles of calculation in this example), the error has already increased to 3.6%. Accuracy can be increased by decreasing the size of At, but only at the expense of increased computation. A much more efficient way of increasing the accuracy is by means of a series expansion. The Rxmge-Kutta methods, which we describe next, comprise the most commonly used approach. 

Euler s methods can be derived from a more general Taylor s algorithm approach to numerical integration. Assuming a first-order differential equation with an initial value such as [dy/dx] = / = function of x, and y = f(x,y) with y(xo) = yo. if the f(x,y) can be differentiated with respect to x and y, then the value of y at X = (xo + h) can be found from the Taylor series expansion about the point x = xq with the help ofEq. (16) ... 

To obtain these expressions, take the Taylor series expansions of the elements fIji in Eq. (125) and neglect higher than the first-order terms. If we can calculate the gradients of the Euler angles as... 

Error estimate for Euler method) In this question you ll use Taylor series expansions to estimate the error in taking one step by the Euler method. The exact... 

Error estimate for the improved Euler method) Use the Taylor series arguments of Exercise 2.8.7 to show that the local error for the improved Euler method is G(Ar ). 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

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