Total differential equation

These total differential equations can be combined with the initial condition (7.210b) and solved. The result is... 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

Transform methods are used to solve two-variable linear differential equations essentially by means of the transformation of a partial differential equation into a total differential equation of one independent variable (in general, the number of variables is reduced by one) [1], The major inconvenience of these methods to find analytical solutions is that the inverse transformation is frequently very difficult or cannot be done at all even for not too complex electrochemical processes. In these cases, the solutions have an integral non-explicit form, from which it is not possible to deduce limit behaviors and the influence of the different variables cannot be inferred for a glance. In Electrochemistry, this method has been extensively used to solve the diffusion equation, which is a two-variable partial differential equation. 

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. 

A further thermodynamic expression for l is possible.4 Since the electrocapillary equation (Eq. 8) is a total differential equation, the second cross-partial-differential coefficients of y are equal ... 

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. 

Partial differential equations, such as Pick s second law (in which the concentration is a function of both time and space), are generally more difficult to solve than total differential equations, in which the dependent variable is a function of only one independent variable. An example of a total differential equation (of second order ) is... 

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. 

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 three characteristics, or conditions, as they are called, of a particular diffusion process cannot be rediscovered by mathematical argument applied to the differential equation. To get at the three conditions, one has to resort to a physical understanding of the diffusion process. Only then can one proceed with the solution of the (now) total differential equation (4.42) and get the precise functional relationship between concentration, distance, and time. 

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... 

Each equation is now a total differential equation in one variable, Q, This is the linear harmonic oscillator equation in terms of the normal coordinate Q. The solution is then expandable as the product of harmonic oscillator functions, one for each normal mode, and the total energy corresponds to the sum of the energies of the 3A atom 6 oscillators. 

This is a partial differential equation in three independent variables x, y, and z. In order to solve such an equation it is usually necessary to obtain three total differential equations, one in each of the three variables, using the method of separation of variables which we have already employed to solve the Schrodinger time equation (Sec. 9b). We first investigate the possibility that a solution may be written in the form... 

This is now a total differential equation, which can be solved by familiar methods. As may be verified by insertion in the equation, a solution is... 

It is evident that this equation has been separated into terms each of which depends upon one variable only each term is therefore equal to a constant, by the argument used in Section 13. We obtain in this way three total differential equations similar to the following one ... 

Problem 17-3. The equation for the isotropic harmonic oscillator is separable also in spherical polar coordinates. Set up the equation in these coordinates and carry out the separation of variables, obtaining the three total differential equations. 

This procedure is useful only for total differential equations in one independent variable, but there are many problems involving several independent variables which can be separated into total... 

Assume that a project causes a small change in the individual s survival prohahility tt. We want to find the individual s monetary valuation of the change in tt. Totally differentiating equation (5.1), adjustingincome so as to keep the individual s level of utility unchanged, yields after straightforward calculations ... 

A Pfaffian form is also known as a total differential equation [10, pp. 326-330], This type of differential equation plays an important role in thermodynamics. Consider the vector function f(x) of the vector argument x. The scalar product f(x) dx is... 

A small mechanical stress or elongation e thus can cause a big change of resistance I iR/R, for which Eq. [15.2] can be deduced by total differential equation of Eq. [15.1] ... 

Because x and y are related through the equation g x,y) = constant, they are not independent variables. To satisfy both Equations (5.14) and (5.15) simultaneously, put the constraint Equation (5.15) into differential form and combine it with the total differential Equation (5.14). Since g = constant, the total differential dg is... 

To obtain a constant distribution of orientations on the giso-curve, the total differential equation for the curve s in polar coordinates is given by... 

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