Taylor first-order

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

Finally, we show how to relate the modified Schrodinger equation evolution X(m) to the usual evolution T (t) [14]. Consider the modified Schrodinger equation, Eq. (12). We approximate f H) in this equation with a first-order Taylor series expansion. 

Equation (E.4) may be obtained by considering the Taylor expansion of the laser field up to first order... 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

The accuracy of any expression for first order errors can be improved as much as desired (so long as the expression is differentiable) by including additional terms of its Taylor expansion. For example, the error magnification factor in the age equation for the 230Th/238u term, (p, can be improved to include the second derivative, so the estimate of the effect of a given error in cp on the age becomes ... 

Laplace transform is only applicable to linear systems. Hence, we have to linearize nonlinear equations before we can go on. The procedure of linearization is based on a first order Taylor series expansion. 

What we do is a freshmen calculus exercise in first order Taylor series expansion about the... 

In case you forgot, the first order Taylor series expansion can be written as... 

As soon as we finish the first-order Taylor series expansion, the equation is linearized. All steps that follow are to clean up the algebra with the understanding that terms of the steady state equation should cancel out, and to change the equation to deviation variables with zero initial condition. 

To handle the time delay, we do not simply expand the exponential function as a Taylor series. We use the so-called Pade approximation, which puts the function as a ratio of two polynomials. The simplest is the first order (1/1) Pade approximation ... 

This is a form that serves many purposes. The term in the denominator introduces a negative pole in the left-hand plane, and thus probable dynamic effects to the characteristic polynomial of a problem. The numerator introduces a positive zero in the right-hand plane, which is needed to make a problem to become unstable. (This point will become clear when we cover Chapter 7.) Finally, the approximation is more accurate than a first order Taylor series expansion.1... 

Since x, and m, are functions of time, we need to linearize (10-26). A first order Taylor expansion of x is... 

This is the result expected if r is effectively a control parameter. Taylor expansion with respect to 1 /k introduces the correction terms as derivatives. To first order in l/k, we obtain... 

The authors describe the use of a Taylor expansion to negate the second and the higher order terms under specific mathematical conditions in order to make any function (i.e., our regression model) first-order (or linear). They introduce the use of the Jacobian matrix for solving nonlinear regression problems and describe the matrix mathematics in some detail (pp. 178-181). 

In Section III.D, we shall investigate when this happens. For the moment, imagine that we are at a point of degeneracy. To find out the topology of the adiabatic PES around this point, the diabatic potential matrix elements can be expressed by a first order Taylor expansion. 

As discussed in Section (8.2), Equations (8.64) and (8.65) is a set of (n + m) nonlinear equations in the n unknowns x and tn unknown multipliers A.. Assume we have some initial guess at a solution (x,A). To solve Equations (8.64)-(8.65) by Newton s method, we replace each equation by its first-order Taylor series approximation about (x,A). The linearization of (8.64) with respect to x and A (the arguments are suppressed)... 

Given an initial guess x0 for x, Newton s method is used to solve Equation (8.84) for x by replacing the left-hand sidex>f (8.84) by its first-order Taylor series approximation at x0 ... 

Galceran, J., Taylor, S. L. and Bartlett, P. N. (1999). Application of Danckwerts expression to first-order EC reactions. Transient currents at inlaid and recessed microdisc electrodes, J. Electroanal. Chem., 466, 15-25. 

Assuming Taylor series expansion using only zero- and first-order terms (dropping second and higher order terms), we arrive at the linear or linearized system described by... 

An often-used method consists of expanding f in Eq. (8.1) as a Taylor series about a certain vector that is close to x(r). In particular, if a first-order expansion is carried out on the current estimate of the state vector, we obtain... 

Expanding the function g in a Taylor series to the first order with respect to m, we get the approximation... 

The last summation in Eq. (4.17) which contains second derivatives is again omitted. This is consistent with the original approximation of setting all second derivatives equal to zero, but implies that even the first-order term in the Taylor expansion of Eq. (4.15a) is not fully taken into account. The resulting m normal equations Sj = 0 ( = 1,m) are... 

The drift velocity and diffusivity for a Stratonovich SDE may be obtained by using Eq. (2.239) to calculate the first and second moments of AX to an accuracy of At). To calculate the drift velocity, we evaluate the average of the RHS of Eq. (2.244) for AX . To obtain the required accuracy of At), we must Taylor expand the midpoint value of that appears in Eq. (2.244) to first order in AX about its value at the initial position X , giving the approximation... 

To evaluate the drift velocity, we take the average of Eq. (2.315) for AX , while Taylor-expanding the midpoint value of D P to first order in AX about its value at X . This yields... 

To evaluate the second term in Eq. (2.402), we must evaluate the average for one timestep. Because is regarded as constant over a timestep, the change in Eq. (2.397) for arises from the change in P, during the step, which causes a change in the constraint force induced by tj, . Taylor-expanding P to first order in AR yields... 

Equations (7) can be viewed as a formal Taylor-series expansion, around the averaged part of the one-particle density matrix, of the HF energy functional E[p] [16, 18], this defining a shell-correction series . In Eqn (13) the first-order term of this expansion is expressed in terms of the single-particle energies e,. 

To improve upon this approximation, we make a Taylor expansion of the left-hand side of Eq. (15) and then make a resummation to infinite order with respect to commutators [f, Hs] of the fast subsystem s and to first order with respect to commutators [f, Hq] of the slow subsystem q. The result is ... 

We then expand K f) in a Taylor series and only retain terms up to the first order... 

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