Integral transforms Laplace transform

Laplace transformation — One of a family of mathematical operations called integral transforms , Laplace transformation converts a function / (t), usually of time, into another function f(s) of a dummy variable s. The... 

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... 

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... 

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). 

This equation is a partial differential equation whose order depends on the exact form of/ and F. Its solution is usually not straightforward and integral transform methods (Laplace or Fourier) are necessary. The method of separation of variables rarely works. Nevertheless, useful information of practical geological importance is apparent in the form taken by this equation. The only density distributions that are time independent must obey... 

There are numerous useful integral transforms, each of which is specified by a two-variable function Kit, p) called the Kernel function or nucleus of the transform. The Laplace transform or the Laplace integral of a function fit), defined for... 

The most common technique for the derivation of fundamental solutions is to use integral transforms, such as, Fourier, Laplace or Hankel transforms [29, 39]. For simple operators, such as the Laplacian, direct integration and the use of the properties of the Dirac delta are typically used to construct the fundamental solution. For the case of a two-dimensional Laplace equation we can use a two-dimensional Fourier transform, F, to get the fundamental solution as follows,... 

GENERALIZED INTEGRAL TRANSFORMATIONS, A.H. Zemanian. Graduate-level study of recent generalizations of the Laplace, Mellin, Hankel, K. Weierstrass, convolution and other simple transformations. Bibliography. 320pp. 5H x 8H. 65375-7 Pa. 7.95... 

The Laplace integral transformation, used in Section 4.3.1, allows the indentifica-tion of its kernel as K(z,p) = K(z,s) = e . It corresponds to the case when we produce a transformation with time. So, for this case, we particularize the relation (4.151) as ... 

Let the integral transformation of Laplace-Carson be applied to system (50). The functions of time x t) can then be substituted by the x (q) transformation functions, and the time derivatives by the qx (g) — f/xlO) values, where q represents reciprocal time and the x(0) are the initial concentrations of the intermediates. The Laplace-Carson procedure transforms the system of differential equations (50) into a system of algebric equations with respect to X (q) thus ... 

The Laplace transform is an integral transform in which a function of time /(t) is transformed into a new function of a parameter s called frequency, J[s) oi F s), according to... 

Integral transforms can be used to solve ordinary differential equations by converting them to algebraic equations. In what follows, the convolution properties of the different transforms have been listed, followed by the methods of integral transform used to solve (a) one-dimensional diffusion equations in the infinite and semi-infinite domains and (b) Laplace equations in the cylindrical geometries. 

The system equation along with boundary and initial conditions were solved analytically using a Laplace integral transform (1.) a computer program by Cleary (14) was modified for use in this study. 

An integral transform is similar to a functional series, except that it contains an integration instead of a summation, which corresponds to an integration variable instead of a summation index. The integrand contains two factors, as does a term of a functional series. The first factor is the transform, which plays the same role as the coefficients of a power series. The second factor is the basis function, which plays the same role as the set of basis functions in a functional series. We discuss two types of transforms, Fourier transforms and Laplace transforms. 

Section 6.5 Integral Transforms TABLE 6.1 Laplace Transforms... 

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

We make now, similarly as is common with the different integral transforms, a correspondence table between the stochastic variable and the associated characteristic function. Note, there are several integral transforms. The most well-known integral transformation might be the Fourier transform. Further, we emphasize the Laplace transform, the Mellin transform, and the Hilbert transform. These transformations are useful for the solution of various differential equations, in communications technology, all ranges of the frequency analysis, also for optical problems and much other more. We designate the stochastic variable with X. The associated characteristic function should be... 

The goal of the electrochemical modelhng in this chapter is to solve the mathematical model developed in the previous chapter in order to obtain the form of the algebraic (containing no derivatives) function C X,T), i.e., to determine how the concentration of the chemical species varies in space and in time. From this, other information, such as the current passed at the electrode, can be inferred. A munber of analytical techniques exist that may be used for solving partial differential equations (PDEs) of the type encountered in electrochemical problems, including integral transform methods such as the Laplace transform, and the method of separation of variables. Unfortunately these techniques are not applicable in all cases and so it is often necessary to resort to the use of numerical methods to find a solution. 

I2. To solve the heat conduction problem for a slab geometry in Example 11.4 by the method of finite integral transform (or alternately by the Laplace transform or the separation of variables method), it was necessary to find eigenvalues for the transcendental equation given by Eq. ll.SSfc, rewritten here for completeness... 

This set of equations (Eqs. 11.184) can be solved readily by the Laplace transform method, as was done in Problem 10.17. Here we are going to apply the method of generalized integral transforms to solve this set of equations. 

This linear partial differential equation was solved analytically in Chapters 10 and 11 using Laplace transform, separation of variables, and finite integral transform techniques, respectively. A solution was given in the form of infinite series... 

This equation describes many transient heat and mass transfer processes, such as the diffusion of a solute through a slab membrane with constant physical properties. The exact solution, obtained by either the Laplace transform, separation of variables (Chapter 10) or the finite integral transform (Chapter 11), is given as... 

Since the enzyme concentration in the reaction solution does not change significantly during the steady state of the hydrolysis process, Q can be treat as a constant. Equation (6) and Equation (7) can be integrated with Laplace Transform Method (21). The boundary conditions are C, = 0 and Q = 0 at < = 0. Thus, the Laplace Transforms of Equation (6) and Equation (7), respectively, are ... 

This is an integral transform that maps the function of time, /(0> into function called F(s) or/ (s) of the parameter s, called the frequency, because if t is in s, then s must be in s. Of course, integration over the parameter t between 0 and oo assures that t will not appear after integration. During the transformation, no information about f(f) is lost and the transform contains the same amount of information, only displayed in the frequency domain. Complex equations are usually much simpler in the Laplace domain. In general, the parameter s can be complex (see Sect. 2.6),... 

From the definitions given in Section 4.4.2, it is apparent that the interfacial impedance can be calculated from the perturbation and response in the time domain, in which the excitation can be any arbitrary function of time. In principle, any one of several linear integral transforms can be used (Macdonald and McKubre [1981]) to convert from the time domain into the frequency domain, but the two most commonly used are the Laplace and Fourier transforms ... 

Another Unear-scaUng MP2 algorithm was proposed in [129] that is based on the atomic-orbitals Laplace-transform (LT) MP2 method [137]. In this method, the energy denominators (5.32) in (5.30) are ehminated by Laplace-transformation, which paves the way to express the MP2 energy directly in the basis. The price to pay is the additional Laplace integration, which is carried out by quadrature over a few (8-10) points. For each of the quadrature points an integral transformation has to... 

In addition to the Fourier transform, there are the Laplace transform and the Melhn transform. They are all called integral transforms. The general formula is... 

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