Second-order linear equations

Radial coordinate in a (solid) catalyst particle or in a reactor tube Generation rate Vector of generation rates Generation rate of key component k Vector of generation rates of key component k Dimensionless generation rate. Equation 5.101 Roots of characteristic equation (second-order linear differential equations)... 

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

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

Some problems with diffusion or dispersion give rise to the second order linear equation with constant coefficients,... 

The Ilypergeomctric Equation. In certain problems it is possible to reduce the solution to that of solving the second order linear differential equation... 

Bessel s Differential Equation. We showed previously 2fi above) that, if n is an integer, J (ie) is a solution of Bessel s equation (2G. )). We shall now examine the solutions of that equation when the parameter n is not necessarily an integer. To emphasise that this parameter is, in general, non-integral, wc shall replace it by the symbol v, so that wc now consider the solutions of the second order linear differential equation... 

Improved mathematical models. First or second order linear equations adequately fit much calibration data. If neither model is appropriate, the following semi-empirical multiple curve procedure may be used. 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

A solution to equation (E2.1.2) may be achieved by (1) separating variables and integrating or (2) solving the equation as a second-order, linear ordinary differential equation. We will use the latter because the solution technique is more general. 

Solve the following second order linear differential equation subject to the specified "boundary conditions" ... 

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). 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

In order to make the comparison between Ep and Ep/2 measurements summarized in Table 9, the two quantities were measured in separate experiments. A recent study by Eliason and Parker has shown that this is not necessary [57]. Analysis of theoretical LSV waves by second-order linear regression showed that data in the region of Ep are very nearly parabolic. The data in Fig. 9 are for the LSV wave for Nernstian charge transfer. The circles are theoretical data and the solid line is that described by a second-order polynomial equation. It was concluded that no detectable error will be invoked in the measurement of LSV Ep and Ip by the assumption that the data fit the equation for a parabola as long as the data is restricted to about 10 mV on either side of the maximum. This was verified by experimental measurements on both a Nernstian and a kinetic system. 

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... 

Most of the problems involving second-order differential equations which we encounter in chemistry are second-order linear differential equations, which take the general form ... 

From the theory of linear differential equations, the solution of the second-order linear differential equation (5.27) has the general form... 

The motion of ions in a quadrupolar field is described mathematically by a second-order linear differential equation (Eq. (2)), the solutions to which were... 

The model (12.43) and (12.44) consists of two second-order linear differential equations whose solution has the following form ... 

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... 

This is a rather nasty problem to solve numerically, because boundary conditions over the whole range of x are assigned at t = 0 and t = 1. A perturbation expansion around l/P = 0 yields, as expected, the CSTR at the zero-order level at all higher orders, one has a nested series of second-order linear nonhomogeneous differential equations that can be solved analytically if the lower order solution is available. The whole problem thus reduces to the solution of Eq. (133), which has been discussed before. This is, of course, the high-diffusivity limit that corresponds to a small Thiele modulus in the porous catalyst problem. 

In ref 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

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... 

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],... 

Convert the governing equation to finite difference form by using central difference expression accurate to the order h for the first and second derivatives in the spatial variable, x (equation (6.11)). This gives raise to N second order linear ODEs in This system of second order equations is converted to 2N first order linear ODEs in as described in equation (6.12). The variable ui(Q, i = 0..N-I-1 corresponds to the dependent variable, ui at node point i. 

The cost, however, is that some of our terminology i s nontraditional. For example, the forced harmonic oscillator would traditionally be regarded as a second-order linear equation, whereas we will regard it as a third-order nonlinear system, since (3) is nonlinear, thanks to the cosine term. As we ll see later in the book, forced oscillators have many of the properties associated with nonlinear systems, and so there are genuine conceptual advantages to our choice of language. 

Naturally, one can consider Eq. (6) as a second-order linear inhomogeneous equation to find U (R). Its solution is given in quadrature, and the general solution of the homogeneous part of Eq. (6) has the form... 

The mass balance for a first-order reaction in a tubular reactor with a flow velocity of v and the concentration of reactant in feed, Cp undergoing a reaction with the rate constant of k is governed by the following second-order linear equation with constant coefficients ... 

A perturbation analysis about this singular point yields a second order linear ordinary differential equation whose characteristic equation has the roots Ai and A2 where... 

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. 

The Schrodinger wave equation that describes the motion of an electron in an isolated hydrogen atom is a second-order linear differential equation that may be solved after specification of suitable boundary conditions, based on physical considerations. The solution to the equation, known as a wave function provides an exhaustive description of the dynamic variables associated with electronic motion in the central Coulomb field of the proton. 

Substituting the coefficients from (126) into (127), we obtain the system of two second-order linear differential equations... 

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