Linear Higher-Order Differential Equations

We now move onto a few so-called higher order or complex processes. We should remind ourselves that all linearized higher order systems can be broken down into simple first and second order units. Other so-called "complex" processes like two interacting tanks are just another math problem in coupled differential equations these problems are still linear. The following sections serve to underscore these points. 

For linear systems, the differential equation for the jth cumulant function is linear and it involves terms up to the jth cumulant. The same procedure will be followed subsequently with other models to obtain analogous differential equations, which will be solved numerically if analytical solutions are not tractable. Historically, numerical methods were used to construct solutions to the master equations, but these solutions have pitfalls that include the need to approximate higher-order moments as a product of lower moments, and convergence issues [383]. What was needed was a general method that would solve this sort of problem, and that came with the stochastic simulation algorithm. 

This is a more general model than the classical one that assnmes linear system constitutive properties instead of operators, allowing one to find solntions for all variables as the model becomes a classical second-order differential equation. However, a higher degree of generality can be kept by merely assuming commutativity between the inductive damping operator and the temporal derivation. 

In this chapter, a few methods were presented for obtaining a solution to the linear second-order (or higher) ordinary differential equations. To the inexperienced practitioners, these many options could present a dilemma that is, given a problem, which method should one use ... 

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. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

Linear Differential Equations Right-Hand Member t(x) 0 Again the specific remarks for y" + ay + by =flx) apply to differential equations of similar type but higher order. We shall discuss two general methods. 

The First-Order Linear Inhomogeneous Differential Equation (FOLIDE) First-Order Reaction Including Back Reaction Reaction of Higher Order Catalyzed Reactions... 

Incorporation of the superposition approximation leads inevitably to a closed set of several non-linear integro-differential equations. Their nonlinearity excludes the use of analytical methods, except for several cases of asymptotical automodel-like solutions at long reaction time. The kinetic equations derived are solved mainly by means of computers and this imposes limits on the approximations used. For instance, we could derive the kinetic equations for the A + B — C reaction employing the higher-order superposition approximation with mo = 3,4,... rather than mo = 2 for the Kirkwood one. (How to realize this for the simple reaction A + B —> B will be shown in Chapter 6.) However, even computer calculations involve great practical difficulties due to numerous coordinate variables entering these non-linear partial equations. 

Since the coefficients of each member in the set of differential equations are spatially dependent and the equations themselves inhomogeneous, higher order terms are obtained with increasing difficulty. This is in contrast to the case with solving the linear PB equation with the same undulating surface. In this case the set member differential equations are simply homogeneous and have constant coefficients,... 

In general there is no exact solution to any sequence of consecutive higher-order equations. The reason for this is that the differential equations are no longer linear equations (as they were in the case of first-order reactions), and nonlinear equations do not have exact solutions except in very particular cases. However, two exact methods are available for studying some aspects of these systems, and there is one more commonly used... 

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. 

Higher Order Linear Ordinary Differential Equations... 

The boundary conditions are applied in the finite element method in a different way than in the finite difference method, and then the linear algebra problem is solved to give the approximation of the solution. The solution is known at the grid points, which are the points between elements, and a form of the solution is known in between, either linear or quadratic in position as described here. (FEMLAB has available even higher order approximations.) The result is still an approximation to the solution of the differential equation, and the mesh must be refined and the procedure repeated until no further changes are noted in the approximation. 

As a general rule it is more difficult to solve differential equations of higher orders than the first. Of these, the linear equation is the most important. A linear equation of the nth order is one in which the dependent variable and its n derivatives are all of the first degree and are not multiplied together. If a higher power appears the equation is not linear, and its solution is, in general, more difficult to find. The typical form is... 

---