Laplace transform integral

The determination of the density of states for s classical oscillators by the method of Laplace transforms is of limited value because this can be obtained by other methods as well. Of much greater interest is the fact that the product of the quantum oscillators in Eq. (6.48) can be inverted by the Laplace transform method. However, it requires solving the inverse Laplace transform integral (Forst, 1971, 1973 Hoare and Ruijgrok, 1970) ... 

Closer examination reveals that (VII-4) is no longer valid in these circumstances, because the Laplace transform integral (III-l) does not exist. Equation (VII-7), therefore, is restricted to values of e below the critical value defined by the inequality (VII-8) if the electric field is increased beyond this point the plasma breaks down into charged fragments. 

In DM-MP2 calculations, we applied the Chebyshev expansion for the evaluation of matrix exponential, which is implanented in the Expoktt library program [51]. For the numerical quadrature of the Laplace-transformed integrals of Eqs. (9) and (27), we used the r-point Euler-Maclaurin (trapezoidal) quadrature... 

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see Integral Transforms (Operational Methods) ]. The one-sided Laplace transform indicated by L[f t)] is defined by the equation L[f t)] = /(O dt. It has... 

Equations of Convolution Type The equation u x) = f x) + X K(x — t)u(t) dt is a special case of the linear integral equation of the second land of Volterra type. The integral part is the convolution integral discussed under Integral Transforms (Operational Methods) so the solution can be accomplished by Laplace transforms L[u x)] = E[f x)] + XL[u x)]LIK x)] or... 

This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of second-order reactions of Figs. 7-5n and 7-5c required numerical integrations. 

Tustin s Rule Tustin s rule, also called the bilinear transformation, gives a better approximation to integration since it is based on a trapizoidal rather than a rectangular area. Tustin s rule approximates the Laplace transform to... 

The Laplace transform of a derivative dy/dt is found by application of Equation (3-65) and integration by parts ... 

Thus the Laplace transformation constitutes a method of integration, and a table of Laplace transforms plays a role in this process that is analogous to a table of... 

Derivatives 35. Maxima and Minima 37. Differentials 38. Radius of Curvature 39. Indefinite Integrals 40. Definite Integrals 41. Improper and Multiple Integrals 44. Second Fundamental Theorem 45. Differential Equations 45. Laplace Transformation 48. 

The Laplace transformation is based upon the Laplace integral which transforms a differential equation expressed in terms of time to an equation expressed in terms of a complex variable a + jco. The new equation may be manipulated algebraically to solve for the desired quantity as an explicit function of the complex variable. 

F(s) = the Laplace transform of f, expressed in s, resulting from operating on f(t) with the Laplace integral. 

The integral transformation (241) with kernel (250) is seen to be accomplished by taking the Laplace transforms of Eqs. (236) with e ", where s= 1/f, and dividing the transformed quantity by t. Hence, the expected value of is simply given by... 

The above differential equation can be integrated through the use of the Laplace transformation, and, taking due consideration of the limits, this gives... 

These equations can be solved (although not easily) by integration by the method of partial fractions, by matrices, or by Laplace transforms. For the case where [I]o = [P]0 = 0, the concentrations are... 

This equation cannot be integrated directly since the temperature 9 is expressed as a function of two independent variables, distance jc and time t. The method of solution involves transforming the equation so that the Laplace transform of 6 with respect to time is used in place of 9. The equation then involves only the Laplace transform 0 and the distance jc. The Laplace transform of 9 is defined by the relation ... 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

These can be solved by classical methods (i.e., eliminate Sout to obtain a second-order ODE in Cout), by Laplace transformation techniques, or by numerical integration. The initial conditions for the washout experiment are that the entire system is full of tracer at unit concentration, Cout = Sout = L Figure 15.7 shows the result of a numerical simulation. The difference between the model curve and that for a normal CSTR is subtle, and would not normally be detected by a washout experiment. The semilog plot in Figure 15.8 clearly shows the two time constants for the system, but the second one emerges at such low values of W t) that it would be missed using experiments of ordinary accuracy. 

While the simple linear models in the previous sections have been solved by straightforward integration, the present model (and more complicated ones) are more conveniently solved by means of the Laplace transform. 

The Laplace transform of a time-dependent variable X(t) is denoted by Lap X t) or x(5) and is defined by means of the definite integral over the positive time domain ... 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

The Laplace transform of a first-order derivative is defined consistently with eq. (39.48) by means of the integral ... 

Integration of Eq. (33) may be accomplished with Laplace transforms. The result is... 

Example 2.16. Derive the closed-loop transfer function X,/U for the block diagram in Fig. E2.16a. We will see this one again in Chapter 4 on state space models. With the integrator 1/s, X2 is the Laplace transform of the time derivative of x,(t), and X3 is the second order derivative of x,(t). 

Other integral transforms are obtained with the use of the kernels e" or xk among the infinite number of possibilities. The former yields the Laplace transform, which is of particular importance in the analysis of electrical circuits and the solution of certain differential equations. The latter was already introduced in the discussion of the gamma function (Section 5.5.4). 

In practice, the inverse Laplace transformations are obtained by reference to the rather extensive tables that are available. It is sometimes useful to develop the function in question in partial fractions, as employed in Section 3.3.3. The resulting sura of integrals can often be evaluated with the use of the tables. 

Laplace transforms 279-286 mapping of a function 271 integrating factor 56-59, 86, 360 integration 43-61 along a curve 51-52 by parts 46 by substitution 45-46 chemical kinetics 47-48 constant of integration 43,49, 89, 124... 

The inversion of the Laplace transform presents a more difficult problem. From a fundamental point of view the inverse of a given Laplace transform is known as the Bromwich integral. Its evaluation is carried out by application... 

Extensive literature is available on general mathematical treatments of compartmental models [2], The compartmental system based on a set of differential equations may be solved by Laplace transform or integral calculus techniques. By far... 

This equation is linear first-order and may be solved in a variety of fashions. One may use an integrating factor approach, Laplace transforms, or rearrange the equation and obtain the sum of the homogeneous and particular solutions. The solution is... 

The Laplace transform of a function//) is defined by F(s) = L f(t) = I(Te s/(f) dt, where s is a complex variable. Note that the transform is an improper integral and therefore may not exist for all continuous functions and all values of s. We restrict consideration to those values of s and those functions/for which this improper integral converges. The Laplace transform is used in process control (see Sec. 8). 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

The rate equations and their integrals found by Laplace transformation,... 

If we then notice that the bracketed expression in the first integral of (65 ) is nothing else but the Laplace transform of PoKP> t)>we obtain, applying the well-known convolution theorem of Laplace transforms together with the definitions (64) and (65),... 

J. G. McWhirter and E. R. Pike, On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A Math. Gen. 11, 1729-1745 (1978). 

The presence of the h(z, P) factor makes Eq. (7.44) different from a Laplace transform of C(z). If the z dependence of h(z, P) is ignored,(34 36) then calculated concentrations of fluorophore near an interface derived from collected fluorescence are approximations. Also, the P dependence in the tf1,11 causes the integral in Eq. (7.44) to differ from the form of a Laplace transform even after the excitation term is factored out. 

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