Ordinary differential equations separable equation

When solving the problem on controlling the chemical reaction, the spatially homogeneous chemical system under isothermal conditions is described by a set of ordinary differential equations. Unlike equation (3.1), in the following system of kinetic equations the control parameters u(t) are separated... 

In the general case of a piston flow reactor, one must solve a fairly small set of simultaneous, ordinary differential equations. The minimum set (of one) arises for a single, isothermal reaction. In principle, one extra equation must be added for each additional reaction. In practice, numerical solutions are somewhat easier to implement if a separate equation is written for each reactive component. This ensures that the stoichiometry is correct and keeps the physics and chemistry of the problem rather more transparent than when the reaction coordinate method is used to obtain the smallest possible set of differential... 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

Our main motivation to develop the specific transient technique of wavefront analysis, presented in detail in (21, 22, 5), was to make feasible the direct separation and direct measurements of individual relaxation steps. As we will show this objective is feasible, because the elements of this technique correspond to integral (therefore amplified) effects of the initial rate, the initial acceleration and the differential accumulative effect. Unfortunately the implication of the space coordinate makes the general mathematical analysis of the transient responses cumbersome, particularly if one has to take into account the axial dispersion effects. But we will show that the mathematical analysis of the fastest wavefront which only will be considered here, is straight forward, because it is limited to ordinary differential equations dispersion effects are important only for large residence times of wavefronts in the system, i.e. for slow waves. We naturally recognize that this technique requires an additional experimental and theoretical effort, but we believe that it is an effective technique for the study of catalysis under technical operating conditions, where the micro- as well as the macrorelaxations above mentioned are equally important. 

As before, we examine the time evolution of the shape-preserving solution by separating variables. The ordinary differential equation for the time evolution, dw/dt = -aw yields w(t) = (1 + t/x), which is the functional form we use to fit the relaxation... 

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. 

Under conditions of constant shear, dy /dt = 0, Eq. (5.67) becomes an ordinary differential equation, which can be solved by separation of variables and integration using the boundary conditions r = tq at t = 0 and r = r at t = t to give the following relation for the shear stress, r, as a function of time... 

This complex system would be difficult to solve directly. However, the problem is separable by taking advantage of the widely different time scales of conversion and deactivation. For example, typical catalyst contact times for the conversion processes are on the order of seconds, whereas the time on stream for deactivation is on the order of days. [Note Catalyst contact time is defined as the volume of catalyst divided by the total volumetric flow in the reactor at unit conditions, PV/FRT. Catalyst volume here includes the voids and is defined as WJpp — e)]. Therefore, in the scale of catalyst contact time, a is constant and Eq. (1) becomes an ordinary differential equation ... 

Reversible bimolecular reactions such asA + B C + D can be solved exactly by the method of separation of variables and the ordinary differential equations in the variable s are Lame equations. This makes the evaluation of the Fourier-type coefficients very difficult since derivative formulas and orthogonality conditions do not seem to exist or at least are not easily used. In addition to this, even if such formulas did exist, it seems unlikely that numerical results could be easily obtained. It does turn out, however, that these reversible bimolecular processes can be solved exactly and conveniently in the equilibrium limit, and this was done by Darvey, Ninham, and Staff.14... 

There are several ways to solve a third-order ordinary-differential-equation boundary-value problem. One is shooting, which is discussed in Section 6.3.4.1. Here, we choose to separate the equation into a system of two equations—one second-oider and one first-order equation. The two-equation system is formed in the usual way by defining a new variable g = /, which itself serves as one of the equations,... 

Cylindrical Coordinates. The separation-of-variables method also applies when the boundary conditions and initial conditions have cylindrical symmetry (see Eqs. 5.7 and 5.8). If c(r, t) = R r)T t), the resulting ordinary differential equation for R(r) is... 

A different classification scheme for DEs, short for differential equations, separates those DEs with a single independent variable dependence, such as only time or only 1-dimensional position, from those depending on several variables, such as time and spatial position. DEs involving a single independent variable are routinely called ODEs, or ordinary differential equations. DEs involving several independent variables such as space and time are called PDEs, or partial differential equations because they involve partial derivatives. 

To solve the 2nd order ordinary differential equation in Rqs. 4-39, wc define a new variable w as w = dT/dj). This reduces Eq. 4-39u into a first order differential equation than can be solved by separating variables. 

When using an ordinary differential equation (ODE) solver such as POLYMATH or MATLAB, it is usually easier to leave the mole balances, rate laws, and concentrations as separate equations rather than combining them into a single equation as we did to obtain an analytical solution. Writing She equations separately leaves it to the computer to combine them and produce a solution. The formulations for a packed-bed reactor with pressure drop and a semibatch reactor are given below for two elementary reactions. 

Each of the first three terms in Eq (3.40) depends on one variable only, independent of the other two. This is possible only if each term separately equals a constant, say, —a, and — respectively. These constants must be negative in order that > 0. Eq (3.40) is thereby transformed into three ordinary differential equations ... 

The left hand side of (2.164) depends on the (dimensionless) time f+, the right hand side on the position coordinate r+ the variables are separated. The equality demanded by (2.164) is only possible if both sides of (2.164) are equal to a constant —fi2. This constant /( is known as the separation parameter. With this the following ordinary differential equations are produced from (2.164)... 

The alert reader will notice that although the left-hand-side of this equation depends only on x, the right-hand-side depends only on t. So both sides must be equal to the same constant. Now you have two easy ordinary differential equations in one unknown each. Also you have an unidentified flying parameter, namely the constant that both sides of the equation must equal. In the grand tradition of calculus textbooks, let us call this constant C. So now we have two separate equations to deal with, each in only one variable. The first one is ... 

The separation of variables is a common technique used to solve linear PDEs. This technique will be discussed in detail in chapter 7. This technique yields ordinary differential equations for the eigenfunctions. In this section, we will present two numerical techniques for the Graetz problem. 

As a result of our separation of variables strategy, the problem of solving the Schrodinger equation for the quantum corral is reduced to that of solving two ordinary differential equations. In particular, we have... 

The solution to this ordinary differential equation may be written down by separation of variables as... 

In fact, the ordinary differential equations in each variable from such a separation have the standard form of the Lam6 equation [12] ... 

They admit factorizable solutions Eq. (43) and can be separated into ordinary differential equations in each variable, provided the geometric and dynamic parameters are related through Eq. (44), so that... 

This equation differs from Equation 3 in that the flow velocity u is given by u = 2u0 (1 — r2/r02) u0 is as before the average flow velocity, and r0 is the tube radius. As usual we let n(r,x) = X(x)R(r), substitute into Equation A-l, separate variables, and obtain two ordinary differential equations for X(x) and R(r), which are related by a separation constant b2. 

Substituting the expansion (3.4.12) into (3.4.1) and then separating the variables, we obtain the following ordinary differential equation for the functions Hm ... 

---