Differential equations, solution examples

The simplest are statistical theories, where the input information is reduced to the distribution of units in different reaction states. The reaction state of a unit is defined by the number and type of bonds issuing from the unit. In a reacting system, the distribution fraction of units in different reaction states is a function of the reaction time (conversion) (cf. e.g. [7, 8, 29, 30] and can be obtained either experimentally (e.g. by NMR) or calculated by solution of a few simple kinetic differential equations. An example of reaction state distribution of an AB2 unit is... 

In these last two examples of equations of motion, the objective is to determine functions of the form h = /(/) or x=g(t), respectively, which satisfy the appropriate differential equation. For example, the solution of the classical harmonic motion equation is an oscillatory function, x=g t), where g(f) = cos a>t, and a> defines the frequency of oscillation. This function is represented schematically in Figure 7.1 (see also Worked Problem 4.4). 

Thoulouze-Pratt, E., 1983, Numerical analysis of the behaviour of an almost periodic solution to a periodic differential equation, an example of successive bifurcations of invariant tori. In Rhythms in Biology and Other Fields of Application, Lect. Notes in Biomath, Vol. 49, pp. 265-271. 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

You will recall from algebra that a number is a solution of an algebraic equation if it satisfies the equation. For example, 2 is a solution of the equation. v -8 = 0 because the subslitutiou of 2 for jr yields identically zero. Likewise, a function is a solution of a differential equation if that function satisfies the differential equation, In other words,.a solution function yields identity when substituted into the differential equation. For example, it can be shown by direct substitution that the function is a solution of / - 4y - 0 O ig. 2-72). 

In many cases, the solution to a matrix differential equation can be obtained as the matrix generalization of the equivalent scalar differential equation. For example, the first-order differential equation... 

A second-order system is one whose output, y(t), is described by the solution of a second-order differential equation. For example, the following equation describes a second-order linear system ... 

The differential equation for Example 2 is derived by applyii Newton s second law to the given mass. Moreover, for this problem, the initial conditions tell us that, at time r = 0, we pulled the mass upward by a distance of and then release it without giving the mass any initial velocity. The solution to Example 2 gives the position of the mass, as denoted by the variable y, with respect to time t. It shows the mass vtill oscillate according to the given cosinusoidal function. 

The perturbation method can be applied to algebraic equations as well as differential equations. For example, we may wish to find an approximate solution for the algebraic equation containing a small parameter... 

Section 2.1.3 shows how the process of diffusion in one dimension can be represented by a chain of resistors and capacitors, and Section 2.1.6 shows how porous electrodes can be represented by a similar network. While this approach is valid even for a distributed process with no boundary (like diffusion into infinite space), discretization is even more important for the case where a distributed process is limited in space. In this case, a finite number of discrete elements can describe the system to arbitrary precision, and can be used for numerical calculations, as treated in next chapter, even if no analytical solution is possible. Another convenience of discretizing a distributed process is the resulting ability to add additional subprocesses directly to the equivalent circuit rather than starting the derivation by formulating a new differential equation. For example the equivalent network representing electric response of a pore is given in Figure 4.5.3. 

The absorption of CO2 in the MEA solution is accompanied by a very fast reaction. For the determination of the absorption flux, the approximate expression (6.3.2-5) [Hikita and Asai, 1966] was used. The example was recalculated, determining the absorption flux by numerical integration of the set of second-order differential equations of Example 14.3.l.A, (14.3.1.A-k). As shown in Table 14.3.1.B-3, the differences are negligible, while the amount of CPU time, using the rigorous model, has significantly increased. 

As with the two-element example, the solution of a Generalized Maxwell Model for a given strain input, e(t), can be found either by superposition of n first order differential equation solutions or by solution of the single n order differential equation. The n first order equations are all of the form of Eqs. 5.15,... 

Many phenomena in chemical engineering depend, in complex ways, on space and time, and often a mathematical model requires more than one independent variable to characterize the state of a system, i.e. the systems need to be described using partial differential equations, PDEs. Examples of such phenomena include chemical reactions, heat transfer, fluid flow, and population dynamics. For practical engineering applications, analytic solutions do not exist, and numerical methods need to be applied. This section is not intended to give a complete discussion of PDEs nor of solution methods. Instead, the aim is to introduce the terminology and some issues involved in solving PDEs. The discussion will be limited to linear PDEs that have two independent variables, e.g. space and time, or two space variables for a steady-state problem, with the form... 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

As an illustrative example we consider the Galerkin finite element solution of the following differential equation in domain Q, as shown in Figure 2.20. 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

The example verifies that the solution to this differential equation is a = = Oo... 

In this case, Oq is the maximum amplitude of the stress. The solution to this differential equation will give a functional description of the strain in this dynamic experiment. In the following example, we examine the general solution to this differential equation. 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

Example Consider the differential equation for reaction and diffusion in a catalyst the reaction is second order c" — ac, c Qi) = 0, c(l) = 1. The solution is expanded in the following Taylor series in a. 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

The restrictions on engineering constants can also be used in the solution of practical engineering analysis problems. For example, consider a differential equation that has several solutions depending on the relative values of the coefficients in the differential equation. Those coefficients in a physical problem of deformation of a body involve the elastic constants. The restrictions on elastic constants can then be used to determine which solution to the differential equation is applicable. 

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