Second-order partial differential

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

There are three basic classes of second-order partial differential equations involving two independent variables ... 

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d f jd d. The basic idea is to approximate f 2 z. polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df jdx and d f jdx -. The polynomial approximation is a local one that applies to some region of space centered about point x. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

This is Fick s second law of diffusion, the equation that forms the basis for most mathematical models of diffusion processes. The simple form of the equation shown above is applicable only to diffusion in one dimension (x) in systems of rectangular geometry. The mathematical form of the equation becomes more complex when diffusion is allowed to occur in more than one dimension or when the relationship is expressed in cylindrical or spherical coordinate geometries. Since the simple form shown above is itself a second-order partial differential equation, the threat of added complexity is an unpleasant proposition at best. 

The wave equation is a second-order partial differential equation in three variables. The usual technique for solving such an equation is to use a procedure known as the separation of variables. However, with r expressed as the square root of the sum of the squares of the three variables, it is impossible... 

The material and energy balances of a tubular vessel are based on the conservation law, Eq 2.42, applied to a differential ring between r and r+dr and z and z+dz. A material balance is derived, for example, in problem P5.08.01, and is quoted in Table 2.6 along with the heat balance. The result is a pair of second order partial differential equations, usually nonlinear, that must be solved numerically. Table 2.6 indicates one possible procedure, but computer software is plentiful. 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

Equation (9.27) defines the so-called axial dispersion coefficient Dax as a model parameter of mixing. Nd is the dispersion flow rate, c the concentration of the tracer mentioned earlier, and S the cross-sectional area of the column. The complete mole flow rate of the tracer consists of an axial convection flow and the axial dispersion flow. The balance of the tracer amount at a cross section of the extractor leads to second-order partial differential equations for both phase flows at steady state. For example, for continuous liquids ... 

The classical Yang-Mills equations of SU(2) gauge theory in the Minkowski spacetime R1,3 form the system of nonlinear second-order partial differential equations of the form... 

More recently Perry and Pigford (P2) made a similar calculation for the absorption of a solute accompanied by a slow second-order reaction, A + B 2C. The problem, which involves the simultaneous solution of three coupled, second-order, partial-differential equations, was worked out by means of an electronic digital computer. 

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 ... 

Second-Order Partial Differential Equations and Green s Functions... 

Inhomogeneous second-order (partial) differential equations... 

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... 

Mathematically, this case is described by the set of second-order partial differential equations, which are usually solved numerically. The general unidirectional (in the x-coordinate) diffusion-reaction equation for any species i, is... 

This treatment leads to a system of stiff, second-order partial differential equations that can be solved numerically to yield both transient and steady-state concentration profiles within the layer (Caras et al., 1985a). Because the concentration profile changes most rapidly near the x = L boundary an ordinary finite-difference method does not yield a stable solution and is not applicable. Instead, it is necessary to transform the distance variable x into a dummy variable y using the relationship... 

The Schrodinger equation is a second-order partial differential equation, involving a relation between the independent variables x, y, z and their second partial derivatives. This kind of equation can be solved only in some very simple cases (for example, a particle in a box). Now, chemical problems are N-body problems the motion of any electron will depend on those of the other N — 1 particles of the system, because all the electrons and all the nuclei are mutually interacting. Even in classical mechanics, these problems must be solved numerically. 

With //., and a independent of E, the solution of this second order partial differential equation is composed of transverse harmonic waves... 

In the general case, eqs 4 and 5 constitute a system of nonlinear coupled second-order partial differential equations. To specify the boundary conditions for this problem, it is necessary to include the external (interphase) heat and mass transfer, as both the concentration and the temperature at the external surface of the catalyst pellet may differ from the corresponding values in the bulk of the surrounding fluid phase. 

All of eighteenth- and nineteenth-century mathematical physics was based on continua, on the solution of second-order partial differential equations, and on microscopic extensions of macroscopic Newtonian ideas of distance-dependent potentials. Quantum mechanics (in its wave-mechanical formulation), classical mechanics, and electrodynamics all have potential energy functions U(r) which are some function of the interparticle distance r. This works well if the particles are much smaller than the distances that typically separate them, as well as when experiments can test the distance dependence of the potentials directly. 

It is also convenient to convert the Dirac equation into a second-order partial differential equation, by multiplying both sides of Eq. (3.6.22) by E/ =i ( /i /() + im0c]. After some travail, using quantities that are more familiar, the result is... 

Since from Eqs. (1) and (2) V(V ) = V( V H) = 0, the following second-order partial differential equations that describe the propagation of -> electromagnetic waves in a vacuum are obtained ... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

When ux = uy = uz = 0, indicating no convective motion of the gas, Eq. 10.15 reverts to the pure diffusion case. The terms ux, uy, and uz are not necessarily equal, nor are they usually constant, since convective velocities decrease as a surface is approached. Equation 10.15 thus represents a second-order partial differential equation with variable coefficients. These types of equations are usually quite difficult to solve. However, often it is sufficient to consider only the steady-state solution, i.e., the case where dc/dt = 0, indicating that the concentration at any point within the system is not changing with time. Then Eq. 10.15 becomes... 

Taking the divergence of Equation (35) and substituting it into Equation (37), a second-order partial differential equation of the Helmholtz type is finally obtained for the PI approximation... 

A recent attempt at a direct stochastic theory by Weinberger [94] using the deterministic flow term as an external (precomputed) constraint should be mentioned here. The intractability of a large coupled system of second-order partial differential equations for the generating function is then reduced to a (nonlinearly coupled) system of ordinary differential equations. The price is the loss of proper population regulation and possible extinction. 

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