Partial differential equations, Laplace

The Laplace transform technique also allows the reduction of the partial differential equation in two variables to one of a single variable In the present case. 

This partial differential equation is most conveniently solved by the use of the Laplace transform of temperature with respect to time. As an illustration of the method of solution, the problem of the unidirectional flow of heat in a continuous medium will be considered. The basic differential equation for the X-direction is ... 

As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c — 0) while the initial concentration of the solution is different (for example c = c°), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation... 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

Equation 10.19 can be solved in various ways including using the Laplace transformation, as employed in the first edition of this book. For the benefit of readers unfamiliar with Laplace transforms, an alternative method will be used here. For a few simple problems, it is possible to find a combination of the independent variables which can be treated as a single variable, thus transforming the partial differential equation into an... 

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... 

The Origin of Special Functions. The special functions of mathematical physics arise in the solution of partial differential equations governing the behaviour of certain physical quantities. Probably the most frequently occurring equation of this type in all physics is Laplace s equation... 

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

Faced with a partial differential equation (i.e., an equation involving derivatives with respect to more than one independent variable), one can often construct some solutions by looking for solutions that are the product of functions of one variable. We will apply this technique, called separation of variables, to find harmonic functions of three variables. Recall from Section 1,5 that a function is harmonic if and only if it satisfies Laplace s equation, which we write in spherical coordinates (see Exercise 1.12) ... 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

Only one other general solution exists. Two methods may be used to solve a partial differential equation such as the diffusion equation, or wave equation separation of variables or Laplace transformation (Carslaw and Jaeger [26] Crank [27]). The Laplace transformation route is often easier, especially if the inversion of the Laplace transform can be found in standard tables [28]. The Laplace transform of a function of time, (t), is defined as... 

The subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

The Laplace transform method is a powerful technique for solving a variety of partial-differential equations, particularly time-dependent boundary condition problems and problems on the semi-infinite domain. After a Laplace transform is performed on the original boundary-value problem, the transformed equation is often easily solved. The transformed solution is then back-transformed to obtain the desired solution. 

Thus, the Laplace transform removes the f-dependence and turns the partial-differential equation into an ordinary-differential equation.9 The solution to Eq. 5.58 is... 

This partial differential equation is made ordinary by taking its Laplace transform. It is then solved for the transform of u and the transform of the total reaction rate for the particle calculated. There is no need to invert this transform, for what we want is the average reaction rate in the bed and this is the integral of the product of the reaction rate after a time a and the probability of the particle having age a. Because the ages are exponentially distributed, this is none other than the Laplace transform with the dimensionless transform variable set equal to 1. The end result is that the Damkohler number is... 

Equation (9.17) is solved by a Laplace transformation. In chemical kinetics and diffusion, the problems may often be formulated in terms of partial differential equations... 

The solution of Fick s second law gives the variation of flux, and thence diffusion-limited current, with time, it being important to specify the conditions necessary to define the behaviour of the system (boundary conditions). Since the second law is a partial differential equation it has to be transformed into a total differential equation, solved, and the transform inverted1. The Laplace transform permits this (Appendix 1). 

The Laplace transform is essential in order to transform a partial differential equation into a total differential equation. After solving the equation the transform is inverted in order to obtain the solution to the mathematical problem in real time and space. 

The occurrence of partial differential equations in electrochemistry is due to the variation of concentration with distance and with time, which are two independent variables, and are expressed in Fick s second law or in the convective-diffusion equation, possibly with the addition of kinetic terms. As in the resolution of any differential equation, it is necessary to specify the conditions for its solution, otherwise there are many possible solutions. Examples of these boundary conditions and the utilization of the Laplace transform in resolving mass transport problems may be found in Chapter 5. 

In electrochemistry, the most frequent use of Laplace transformation is in solving problems involving -> diffusion to an electrode. -> Fick s second law, a partial differential equation, becomes an ordinary differential equation on Laplace transformation and may thereby be solved more easily. 

The most common approach to solution of partial differential equations of the type represented by (7) involves the use of Laplace transformation (Crank, 1957). The method involves transforming the partial differential equation into a total differential equation in a single independent variable. After solving the total differential equation inverse transformation of the solution can be carried out in order to reintroduce the second independent variable. Standard Laplace transforms are collected in tables. 

A simple first order reaction following reversible charge transfer is one of the few cases for which an analytical solution to the diffusion-kinetic differential equations can be obtained. For reactions (1) and (2) under diffusion-controlled charge-transfer conditions after a potential step, the partial differential equations which must be solved are (18) and (19). After Laplace transforma-... 

One may conclude therefore that the solution of Pick s second law (a partial differential equation) would proceed smoothly if some mathematical device could be utilized to convert it into the form of a total differential equation. The Laplace transformation method is often used as such a device. 

Since the method is based on the operation of the Laplace transformation, a digression on the nature of this operation is given before using it to solve the partial differential equation involved in nonsteady-state electrochemical diffusion problems, namely. Pick s second law. 

Laplace Transformation Converts the Partial Differential Equation into a Total Differential Equation... 

It will now be shown that by using the operation of Laplace transformation, Pick s second law—a partial differential equation—is converted into a total differential equation that can be readily solved. Since whatever operation is carried out on the left-hand side of an equation must be repeated on the right-hand side, both sides of Pick s second law will be subject to the operation ofLaplace transformation (c/. Pq. (4.33)]... 

The solution of Pick s second law is facilitated by the use of Laplace transforms, which convert the partial differential equation into an easily integrable total differential equation. By utilizing Laplace transforms, the concentration of diffusing species as a function of time and distance from the diffusion sink when a constant normalized current, or flux, is switched on at f = 0 was shown to be... 

The following section contains the particularization of the integral Laplace transformation for the case of the stochastic model given by the assembly of relations (4.146)-(4.147). This particularization illustrates how the Laplace transformation is used to solve partial differential equations. We start by applying the integral Laplace operator to all the terms of relation (4.146) the result is in ... 

The analysis of outlet peaks is based on the model of processes in the column. Today the Kubi n - Kucera model [14,15], which accounts for all the above-mentioned processes, as long as they can be described by linear (differential) equations, is used nearly exclusively. Several possibilities exist for obtaining rate parameters of intracolumn processes (axial dispersion coefficient, external mass transfer coefficient, effective diffusion coefficient, adsorption/desorption rate or equilibrium constants) from the column response peaks. The moment approach in which moments of the outlet peaks are matched to theoretical expressions developed for the system of model (partial) differential equations is widespread because of its simplicity [16]. The today s availability of computers makes matching of column response peaks to model equations the preferred analysis method. Such matching can be performed in the Laplace- [17] or Fourier-domain [18], or, preferably in the time-domain [19,20]. 

Partial differential equations are generally solved by finding a transformation tiiat allows the partial differential equation to be converted into two ordinary differential equations. A number of techniques are available, including separation of variables, Laplace transforms, and the method of characteristics. 

The Laplace transformation has proved an effective tool for the solution of the linear heat conduction equation (2.110) with linear boundary conditions. It follows a prescribed solution path and makes it possible to obtain special solutions, for example for small times or at a certain position in the thermally conductive body, without having to determine the complete time and spatial dependence of its temperature field. An introductory illustration of the Laplace transformation and its application to heat conduction problems has been given by H.D. Baehr [2.25]. An extensive representation is offered in the book by H. Tautz [2.26]. The Laplace transformation has a special importance for one-dimensional heat flow, as in this case the solution of the partial differential equation leads back to the solution of a linear ordinary differential equation. In the following introduction we will limit ourselves to this case. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

Parabolic partial differential equations are solved using the Laplace transform technique in this section. Diffusion like partial differential equations are first order... 

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