Differential equation, linear, boundary general solution

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). 

The theory for classifying linear, second-order, partial-differential equations is well established. Understanding the classification is quite important to understanding solution algorithms and where boundary conditions must be applied. Partial differential equations are generally classified as one of three forms elliptic, parabolic, or hyperbolic. Model equations for each type are usually stated as... 

Let us, for a moment, consider a single particle in one dimension with a Hamiltonian of the type// = p2/2m + V(x). This is a second-order differential operator, and this means that the general solution to the inhomogeneous Eq. (3.51)—considered as a second-order differential equation—will consist of a linear superposition of two special solutions, where the coefficients will depend on the boundary conditions introduced. As a specific example, one could think of the two solutions to the JWKB problem, their connection formulas, and the Stoke s phenomenon for the coefficients. 

The physical argument presented above is consistent with the mathematical nature of the problem since tlie heat conduction equation is second order (i.e., involves second derivative.s with respect to the space variables) in all directions along which heat conduction is significant, and the general solution of a second-order linear differential equation involves two surbitrary constants for each direction. That is, the number of boundary conditions that needs to be specified in a direction is equal to the order of the differential equation in that direction. 

This is a second-order linear ordinary differential equation, and thus its general solution contains two arbitrary constants. The determination of these constants requires the specification of two boundary conditions, which can be taken to be... 

Beginning with this section, we will now consider partial differential equations. Let us consider a general approach for constructing asymptotic expansions of the solutions of singularly perturbed linear partial differential equations, which was proposed in the well-known fundamental work of Vishik and Lyusternik [27]. We will illustrate the idea of this approach (known in the literature as the method of Vishik-Lyusternik) on a simple example of an elliptic equation in a bounded domain with smooth boundary. 

The second stage is to find a solution to the set of equations. Three basic approaches are possible (a) a complete analytical solution, (b) a partial analytical solution which is completed by a numerical method such as numerical integration, (c) a computer solution either based on a simulation of the experiment or a numerical solution of the set of equations. The first approach is always to be preferred since it leads to an exact equation relating experimental measurables to kinetic parameters. Its range of applicability is, however, limited to relatively simple experiments, and as the experiment becomes more complex it is necessary to deal with time dependent boundary conditions, coupled partial differential equations, and perhaps non-linear equations. Then the computer techniques must be employed, and these generally lead to dimensionless plots. 

If the evolving shape itself must be determined over the course of time as part of the solution, the underlying boundary value problem is no longer linear. Analysis of surface evolution in such situations must rely on approximate methods, in general. The two most common approaches are (i) to determine approximate solutions to the governing partial differential equations by numerical methods and (ii) to express the evolving shape in terms of a small number of modes, each with its own time-dependent amplitude. In the former approach, the solution proceeds incrementally. An elasticity... 

When a second-order differential equation is linear (i.e., does not contain nonlinear terms such as uf du/dxY, etc.), the following important superposition principle applies Any two independent solutions that satisfy the equation and its boundary conditions can be added to obtain the general solution of the differential equation. 

Consider the general case in the stationary state in which speed of reaction and rate of diffusion are of the same order of magnitude the solution of the differential equation with symmetrical boundary conditions can be approximated by MacLaurin series. If in first approximation the non-linear terms of the series are discarded following Kozel [91], Broun et al. [92] and Thomas [44] the relations obtained are cited in Table X. 

In this equation, U is some variable that depends on the independent variable x. While this example is a linear seeond order differential equation with eonstant eo-effieients, the equation eould just as easily be some nonlinear funetion of the solution variable, or first derivative or seeond derivative of the variable. In addition the a,b andc eoeffieients eould be funetions of the independent variable. More general equations will be eonsidered as the ehapter develops. As with any seeond order differential equation, two arbitrary eonstants ean be speeified and these are speeified as the value of the independent variable at two different spatial points -these are the boundary values. The speeified values eould just as well be values of the first derivative or some eombination of dependent value and first derivative at the boundary points. All of these are possibilities. 

This section has concentrated on a special type of two point boundary value problem associated with linear second order differential equations. Such systems only have nontrivial solutions for a discrete set of value of one equation parameter catted the eigenvalue of the equation. This is a very important subset of boundary value differential equations. It has been shown fliat the shooting method of solving boundary value problems can be extended to such eigenfunction problems and accurate numerical solutions obtained. A set of callable code functions that can be used for these problems have been developed and discussed. The next section will return to the general problem of solving two point boundary value problems and attack the problem with a different approach. 

Regarding the physics of this issue, the task is to formulate the appropriate equations, describing convection, diffusion and conduction. The equations of motion are coupled to electric equations, like Poisson s law. The mathematical task is to solve the ensuing set of differential vector equations, some of which are non-linear, with the appropriate boundary conditions. General analytical solutions do not exist, but there are numerical solutions and good approximate equations for a number of limiting situations. Although the full mathematical anailysis is beyond the confines of the present chapter, we shall present their main elements because these are needed to understand the physics of the phenomena. 

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