Time-convolution integral

The function Q cannot be an operator that introduces multi-point terms such as gradients or space/time-convolution integrals. 

That is, the polarization process represented by P,(t) will be established throughout a time evolution. To mathematically express this process, the following time convolution integral (normally named the Duhamel s integral) [32] is used ... 

Skinner and Wolynes came back to the problem of Eq. (1.8) and solved it with a projection method in Laplace space. An interesting aspect of their work is the development of the contracted distribution function a(a t) (see also Section II) inside the time-convolution integral. They pointed out that this provides perturbation terms neglected erroneously by Brinkman. This interesting feature of their approach is included in the AEP illustrated in this chapter. They explicitly evaluated correction terms up to order The projection technique has also been used by Chaturvedi and Shibata, who used a memoiyless equation as the starting point of their treatment. 

When you come across the term convolution integral later in Eq. (4-10) and wonder how it may come about, take a look at the form of Eq. (2-3) again and think about it. If you wonder about where (2-3) comes from, review your old ODE text on integrating factors. We skip this detail since we will not be using the time domain solution in Eq. (2-3). 

The values of C(t) can be evaluated from the current-time history using the convolution integral... 

The convolution integral and the Exponential Piston Flow Model (EPM) were used to relate measured tracer concentrations to historical tracer input. The tritium input function is based on tritium concentrations measured monthly since the 1960s near Wellington, New Zealand. CFC and SF6 input functions are based on measured and reconstructed data from southern hemisphere sites. The EPM was applied consistently in this study because statistical justification for selection of some other response function requires a substantial record of time-series tracer data which is not yet available for the majority of NGMP sites, and for those NGMP sites with the required time-series data, the EPM and other response functions yield similar results for groundwater age. 

The general case of non-uniform initial conditions and time-dependent boundary conditions can also be expressed analytically, and leads to convolution integrals in space and time. Nevertheless, for most practical applications, the assumptions of uniform initial conditions and constant boundary conditions are adequate. 

The convolution integral appearing in this equation can be easily understood by considering the excitation function as successive Dirac functions at various times t. 

The convolution integral in (1.19) and (1.20) can be solved by the method of numerical integration proposed by Nicholson and Olmstead [47], The time t is divided into m time increments t = md. It is assumed that within each time increment the function 1 can be replaced by the average value Ij ... 

There are good reasons for choosing to carry out certain operations in the Laplace domain rather than performing equivalent operations in the time domain. In particular, integration of a function with respect to time is equivalent in the Laplace domain to division by the Laplace variable s. Conversely, differentiation corresponds to multiplication by s. This latter property enables differential equations to be Laplace transformed and then solved by algebraic means. These Laplace domain operations are all more simple than their time domain counterparts. In addition convolution in the time domain is equivcdent to multiplication in the Laplace domain. Formally, this may be represented by eqn. (24), the left-hand side of which is termed the convolution integral. 

When g(t,T) is considered time-invariant, then Equation 1 reduces to a convolution integral. 

The actual deconvolution of a data set is formally straightforward. Let dik)(x) be the kth iterative estimate of the actual spectrum o(x), where x is nominally time viewed as a sequence-ordering variable. Further, let i(x) be the actual observed spectrum that has been instrumentally convolved with the observing system response function s(x). The observed data set i(x) is assumed to be related to o(x) by the convolution integral equation... 

When the quasi-elastic method is used, the viscoelastic resultant moment can be approximated by substituting the time-dependent stiffnesses for elastic stiffnesses in Equation 8.30 and making use of the convolution integral. The resulting moments are... 

It is important that differentiation and integration in the time domain give multiplication and division, respectively, by the variable v in the frequency domain. The role of convolution integrals will be further discussed in Chapter 5. 

Another non - parametric approach is deconvolution by discrete Fourier transformation with built - in windowing. The samples obtained in pharmacokinetic applications are, however, usually short with non - equidistant sample time points. Therefore, a variety of parametric deconvolution methods have been proposed (refs. 20, 21, 26, 28). In these methods an input of known form depending on unknown parameters is assumed, and the model response predicted by the convolution integral (5.66) is fitted to the data. 

Deconvolution Estimation of the time course of drug input (usually in vivo absorption or dissolution) using a mathematical model based on the convolution integral. For example, the absorption rate time course (i abs) that resulted in the plasma concentrations (c(t)) may be estimated by solving the following convolution integral equation for r abs... 

In a linear response regime the solvent polarization at a given time due to a TD electric field can be expressed as a convolution integral on previous times as [43] ... 

It is possible to suggest that relativistic effects are operating within each wave-particle conceived as a four-dimensional space-time continuum, but that the equations of relativity should be inserted within those equations, descriptive of the properties of holographic matrices convolutional integrals and Fourier transformations. 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

Upon evaluating the convolution integral from the experimental current-potential (time) curve and its limiting values (Eq. 77), kinetic analysis can be performed with the help of Eq. (76). Conversely, Eq. (76) or similar equations can be used to calculate the theoretical current-potential curve, e.g., for the linear potential sweep voltammogram, provided that the values of all the parameters are known. Some illustrative examples were provided by Girault and coworkers [183]. 

The extra-column effects (due to sampling loop, connecting elements, detector, etc.) were determined experimentally and taken into account by the convolution integral during the time domain fitting of experimental column responses. 

Equations (16.43a) and (16.43b) are the relevant constitutive equations to be solved. However, they are not of the convolution type, and for this reason we cannot apply the Laplace transform to them. Nevertheless, note that is a monotonically increasing function of t, and by inverting the functional relationship given by Eq. (16.41) the convolution integrals in terms of the new time scale are obtained. The pertinent equations are... 

Next, the convolution integral is carried out to obtain the final time domain solution as ... 

Now that we have an understanding of the meaning of the RTD curve from pulse input, we will fomiulate a more general relationship between time-varying tracer injection and the corresponding concentration in the efflt ent. We shall state without development that the output concentration from vessel is related to the input concentration by the convolution integral ... 

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