Partial differential equation linear

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations... 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

Most wave equations e.g. (11) are valid only for sufficiently small amplitudes. At larger amplitudes deviations from linearity may occur as described by a non-linear partial differential equation of the type... 

The solution of this linear partial differential equation could be started by taking... 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

The superposition principle can be used to combine solutions for linear partial differential equations, like the diffusion equation. It is stated as follows ... 

If any two relations solve a given linear partial differential equation, the sum of those two is also a solution to the linear partial differential equation. 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

Summarizing we conclude that the problem of constructing conformally invariant ansatzes reduces to finding the fundamental solution of the system of linear partial differential equations (33) and particular solutions of first-order systems of nonlinear partial differential equations (39). 

Integration of systems of nonlinear partial differential equations (54),(86) has been performed [33,49]. Here we indicate the principal steps of the integration procedure. While integrating (54),(86), we essentially apply the fact that the general solution of system of equations 1,2 from (86) is known [62]. With already known (x) in hand, we proceed to integrating linear partial differential equations 3,4 from (86). Next, we insert the results obtained into the remaining equations and get the final forms of the functions (x), ). 

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. 

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). 

A related but more sophisticated concept is the random field, which occurs in radiation theory. Let u(r, t) be a field governed by some linear partial differential equation independent of time, e.g.,... 

This linear partial differential equation for F can be solved by the standard method of characteristics. The characteristic curves in the (z, t)-plane are determined by... 

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)]. 

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... 

Fluent is a commercially available CFD code which utilises the finite volume formulation to carry out coupled or segregated calculations (with reference to the conservation of mass, momentum and energy equations). It is ideally suited for incompressible to mildly compressible flows. The conservation of mass, momentum and energy in fluid flows are expressed in terms of non-linear partial differential equations which defy solution by analytical means. The solution of these equations has been made possible by the advent of powerful workstations, opening avenues towards the calculation of complicated flow fields with relative ease. 

A widespread concept of "mathematical physics equations includes primarily linear and quasi-linear partial differential equations. But what are "mathematical chemistry equations and "mathematical chemistry in general ... 

If the thermal conductivity k and the product pCp are temperature independent, Eq. 5.3-1 reduces for homogeneous and isotropic solids to a linear partial differential equation, greatly simplifying the mathematics of solving the class of heat transfer problems it describes.1... 

The integral equation approach is a general purpose numerical method for solving mathematical problems involving linear partial differential equations with piecewise constant coefficients. It is commonly used in various fields of science and engineering, such as acoustics, electromagnetism, solid and fluid mechanics,... 

From a mathematical point of view the PCM models can be unified according to the approach they use to solve the linear partial differential equations determining the electrostatic interactions between solute and solvent. This analysis is presented by Cances who reviews both the mathematical and the numerical aspects of such an integral equation approach when applied to PCM models. 

For a linear partial differential equation with relatively simple initial and boundary conditions, and uniform parameter values, an analytical solution may be obtained. Although simplified, analytical solutions can be put to a number of important uses. Analytical models may be used to [9] ... 

Next, one performs the normal mode analysis, i.e. the flow instabilities are governed by discrete eigenmodes those do not interact with each other and are studied separately. Equations (2.3.8) to (2.3.11) are variable coefficient linear partial differential equations but, they do not admit analytic solutions. However, with the help of normal mode analysis, this can be further simplified. As the coefficients of these equations are functions of the wall normal co-ordinate, it is natural to expand the disturbance quantities in the following manner,... 

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... 

It is known [56] that an arbitrary system of quasi-linear partial differential equations which may be written in the form... 

This is a first-order, quasi-linear partial differential equation, although the presence of the maximization gives it an unconventional form. It must be integrated subject to the boundary condition... 

Equations (8) form an infinite system of coupled non-linear partial differential equations for the fransformed potentials,, . For computation purposes, system (8) is also truncated at the Ntii row and colimm, with N sufficiently large for the required convergence. A few automatic numerical integrators for tiiis class of one-dimensional partial differential systems are now readily available, such as those based on tiie Method of Lines [41, 52]. Once the transformed potentials have been computed from numerical solution of system (8), tiie inversion formula Eq.(7.b) is recalled to reconstruct the original potentials, in explicit form along thejc v -iables. 

Formulae (13.23) form a system of linear partial differential equations of motion of a homogeneous isotropic elastic medium. These equations can be presented in more compact form using vector notation. 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

In the application of the heat conduction equation in its general form (2.8) a series of simplifying assumptions are made, through which a number of special differential equations, tailor made for certain problems, are obtained. A significant simplification is the assumption of constant material properties A and c. The linear partial differential equations which emerge in this case are discussed in the next section. Further simple cases are... 

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],... 

Rhee et al. developed a theory of displacement chromatography based on the mathematical theory of systems of quasi-linear partial differential equations and on the use of the characteristic method to solve these equations [10]. The h- transform is basically an eqmvalent theory, developed from a different point of view and more by definitions [9]. It is derived for the stoichiometric exchemge of ad-sorbable species e.g., ion exchange), but as we have discussed, it can be applied as well to multicomponent systems with competitive Langmuir isotherms by introducing a fictitious species. Since the theory of Rhee et al. [10] is based on the use of the characteristics and the shock theories, its results are comprehensive e.g., the characteristics of the components that are missing locally are supplied directly by this theory, while in the /i-transform they are obtained as trivial roots, given by rules and definitions. 

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