Second-order linear partial differential

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). 

This is a second-order linear partial differential equation. Note that the transport terms (Eq. 22-4) are linear per se, while the reaction term (Eq. 22-5) has been intentionally restricted to a linear expression. For simplicity, nonlinear reaction kinetics (see Section 21.2) will not be discussed here. For the same reason we will not deal with the time-dependent solution of Eq. 22-6 the interested reader is referred to the standard textbooks (e.g. Carslaw and Jaeger, 1959 Crank, 1975). 

This equation is a second-order linear partial-differential equation with a rich mathematical literature [1]. For a large class of initial and boundary conditions, the solution has theorems of uniqueness and existence as well as theorems for its maximum and minimum values.1... 

This last equation is a well-known second-order linear partial differential equation. The precise solution is determined by the boundary conditions that T rojt) = To (a constant), or equivalently, < (l,r) = 0 and the solution can be written as... 

If the thermal power W is linearly dependent or independent of the temperature d, the heat conduction equation, (2.9), is a second order linear, partial differential equation of parabolic type. The mathematical theory of this class of equations was discussed and extensively researched in the 19th and 20th centuries. Therefore tried and tested solution methods are available for use, these will be discussed in 2.3.1. A large number of closed mathematical solutions are known. These can be found in the mathematically orientated standard work by H.S. Carslaw and J.C. Jaeger [2.1],... 

Methods for solving mass and heat transfer problems. The convective diffusion equation (3.1.1) is a second-order linear partial differential equation with variable coefficients (in the general case, the fluid velocity depends on the coordinates and time). Exact closed-form solutions of the corresponding problems can be found only in exceptional cases with simple geometry [79,197, 270, 370, 516]. This is especially true of the nonlinear equation (3.1.17). Exact solutions are important for adequate understanding of the physical background of various phenomena and processes. They can serve as test solutions to verify whether the problem is well-posed or to estimate the accuracy of the corresponding numerical, asymptotic, and approximate methods. 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

The parts of Equation 1.54 are linear second-order hyperbolic partial differential equations called wave equations. The general solutions for the wave equations were given by D Alembert in 1747 [16] as... 

Equation 10.100 has therefore been converted from a partial differential equation in C in an ordinary second order linear differential equation in C. ... 

Linear first order parabolic partial differential equations in finite domains are solved using the Laplace transform technique in this section. Parabolic PDEs are first order in the time variable and second order in the spatial variable. The method involves applying the Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes a boundary value problem (BVP) in the spatial direction with s, the Laplace variable as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 3.1 for solving linear boundary value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). Certain simple problems can be inverted to the time domain using Maple. This is best illustrated with the following examples. 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

In establishing the relationship between time-domain and Laplace-domain, we use only first and second order differential equations. That s because we are working strictly with linearized systems. As we have seen in partial fraction expansion, any function can be "broken up" into first order terms. Terms of complex roots can be combined together to form a second order term. 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

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... 

When D is constant, Eq. 4.2 takes the relatively simple form of the linear second-order partial differential equation... 

The three-dimensional, second-order, nonlinear, elliptic partial differential equation may be simplified in the limit of weak electrolyte solutions, where the hyperbolic sine of is well approximated by 4). This yields the linearized Poisson—Boltzmann equation... 

The error analysis of this calculation procedure can be done using the equations in the previous section. It shows that the error made in using this scheme is of the order of 0(h + t). Thus, the scheme introduces an error term equivalent to a second-order partial differential term, which would add up to the RHS of Eq. 10.61, t.e., would decrease the apparent column efficiency. This procedure should not be used, unless very small values of the time increment t are selected. This, in turn, would make the computation time very long. In order to overcome this type of problem. Lax and Wendroff have suggested the addition to the axial dispersion term of an extra term, equivalent to the numerical dispersion term but of opposite sign [51]. This term compensates the first-order error contribution. In linear chromatography, the new finite difference equation, or Lax-Wendroff scheme, can be written as follows ... 

The solid and liquid film linear driving force models can be written under the same general form of a second order Langmuir kinetic model [1]. We can insert the Langmuir isotherm equation q = qsbC)/ l bC)) in the partial differential equation of tire solid fihn linear driving force model (Eq. 14.3)... 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

The elementary theory concerning the character of partial differential equations has developed mainly from the study of the simplified two dimensional, quasi-linear second order equation defined by [55, 174] ... 

If these relations are used, the time differentiation can be replaced by the multiplication by the factor -icot. Then, for a diffusion to a flat surface the diffusion equations in partial derivatives (5.210) - (5.212), (5.223), (5.224), (5.226), (5.228) are reduced to ordinary second order differential equations. Besides, because of the condition c = const, which follows from Eq. (5.211) and the boundary condition (5.239) for the slow relaxation process, the micellar concentration is uniform and one has to solve only the boundary value problem for Eqs. (5.210), (5.212). Therefore, for any step of micellisation and for different relaxation mechanisms we always obtain a system of two ordinary linear differential equations. For the slow process of micellisation (5.148) the characteristic equation has the form... 

Equation (8-62) generates three coupled linear second-order partial differential equations (PDEs). Eor complicated two-dimensional flow problems, this force balance and the equation of continuity yield three coupled linear PDEs for two nonzero velocity components and dynamic pressure. In some situations, this complexity is circumvented by taking the curl of the equation of motion ... 

The wave equation is a linear, second-order partial differential equation. While the solution to this type of differential equation is not particularly difficult, it might present a challenge for the average student taking inorganic chemistry. Therefore, a general solution is presented and it is left as an exercise to demonstrate the validity of this result. One possible solution to the one-dimensional wave equation is the sine wave given by Equation (3.4), where A is the maximum amplitude of the wave. 

Chemical engineers working in the area of transport phenomena must frequently solve problems that involve linear second-order differential equatiOTis. These may occur as boundary value problems in diffusional systems or initial value problems in process control or reacting systems, but most frequently, they are the result of reduction of partial differential equations. 

The third chapter addresses linear second-order ordinary differential equations. A brief discourse, it reviews elementary differential equations, and the chapter serves as an important basis to the solution techniques of partial differential equations discussed in Chapter 6. An applications section is also included with ten worked-out examples covering heat transfer, fluid flow, and simultaneous diffusion and chemical reaction. In addition, the residue theorem as an alternative method for Laplace transform inversion is introduced. 

This is also called the linear-diffusion equation, which in its most elementary form is a linear second-order partial differential equation (PDE). The assmnption of a concentration-independent diffusion coefficient is generally true for diffusion in gases, hquids, and solutions. Polymers above the glass-transition temperature and, especially, rubbers such as PDMS can be expected to behave like liquids for small molecular diffusants. Analytical solutions to the Unear-diffusion equation, eq 2, for various... 

This chapter provides a review of the planform and profile processes related to beach nourishment. The governing equations are introduced to the degree that they are known. The combination of the linearized longshore sediment transport equation and the continuity equation results in a second-order partial differential equation... 

This is a non-linear, second order, partial differential equation. The different characteristic surfaces are shown in Fig. 5.1 together with the corresponding sign of H. The arrow in the figure indicates the direction of the pressure difference, p. 

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