Third-order differential equation

If the dynamics produce a third-order differential equation, its representation would be... 

Substitution into the momentum equations, cross differentiating, and subtracting yields the following third-order differential equation,... 

Unseeded Cooling Crystallization. In a similar fashion, a cooling profile can be derived for the unseeded case in which spontaneous nucleation and growth are allowed to occur at constant rates. The actual solutions to the resultant third-order differential equations found in the literature differ due to the different sets of the four initial conditions used by various authors (Karpinski et al. 1980b Nyvlt 1991 Randolph and Larson 1988). Understandably, all of them result in a cooling profile of the general form... 

Duffy, B.R. and Wilson, S.K., A third-order differential equation arising in thin-film flows and relevant to Tanner s law, Appl. Math. Lett., 10, 63, 1997. 

Both of the two resulting equations contain the first derivative of the lengthwise displacement u x). Therefore, the latter can be eliminated in order to solve a single third-order differential equation for the twisting angle 4> x). With the normalized coordinate of Eq. (9.3) and compiled coefficients, the solution to the beam torsion can be finally described by... 

The order of a differential equation is the order of the highest derivative present in that equation. Examples of first-, second-, and third-order differential equations are given below ... 

The third-order differential equation is supplied with the following boundary conditions ... 

A few final points are appropriate in using the supplied odebvstQ code for solving boundary value problems. First for a set of first-order equations, the Runge-Kutta integration routine may also be evoked by using the statement odebiv = odebrk before calling the odebvstQ function. Finally one can mix first-order and second-order differential equations provided one defines all second-order differential equations before defining any first-order differential equations. In all cases the number of boundary conditions must match the total number of derivatives in all the defined differential equations. For example a third-order differential equation with boundary values could be specified as three first-order differential equations or one second-order differential equation followed by one first-order differential equation. In either case an appropriate set of three boundary values equations would need to be specified. 

Also as formulated, the FD method appears to be applicable only to differential equations (or sets of differential equations) of even order (second, fourth, sixth, etc.). If one has a third order differential equation for example, this eould be expressed as one first order equation and one second order equation. However, what does one do with the first order equation, as there will be only one boundary eon-dition A prototype of this problem is the question of whether the FD method developed here ean be used with a single first order differential equation with only one boundary eondition. Such a problem is in fact an initial value problem and can be solved by die techniques of Chapter 10. However, the question remains as to whether the formulism of the FD method can be used for such a problem Several authors have suggested that the way to handle such a problem is to simply take another derivative of the given first order differential equation and convert it into a second order differential equation. For the second boundary value one then uses the original first order differential equation at the second boundary. 

Thus, a fourth-order differential equation such as Equation (D.11) has four boundary conditions which are the second and third of the conditions in Equation (D.8) at each end of the beam. The first boundary condition in Equation (D.8) applies to the axial force equilibrium equation, Equation (D.2), or its equivalent in terms of displacement (u). 

The first term on the right-hand side of this equation is zero, since it is simply the sum of the electrical charge in solution, which must be zero for a neutral electrolyte solution. The third term is also zero for electrolytes with equal numbers of positive and negative ions, such as NaCl and MgSC>4. It would not be zero for asymmetric electrolytes such as CaCE. However, in the Debye-Huckel approach, all terms except the second are ignored for all ionic solutions. Substitution of the resulting expression into equation (7.20) gives the linear second-order differential equation... 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

The difference between these two expressions, neglecting second- and third-order differentials, gives the rate of accumulation of solute in the control volume. Equating this to the rate of solute accumulation expressed as... 

As can be seen by inspection of the set of three first-order differential equations, Eqs. (6) to (8), the model profile depends nonlinearly on the set of 2 unknown parameters, namely the apex radius b and the shape factor p. Moreover, as an additional third parameter, the apex correction error, e, is also taken into account. 

In general, a transform expression may not exactly match any of the entries in Table 3.1. This problem always arises for higher-order differential equations, because the order of the denominator polynomial (characteristic polynomial) of the transform is equal to the order of the original differential equation, and no table entries are higher than third order in the denominator. It is simply not practical to expand the number of entries in the table ad infinitum. Instead, we use a procedure based on elementary transform building blocks. This procedure, called partial fraction expansion, is presented in the next section. 

A third example from classical mechanics is that of a planet or comet orbiting the sun in the x-y plant which can be described by the set of second order differential equations ... 

In keeping with a third order equation, three bovmdary values are speeified with two specified at x = 0 and one at x = oo. Of eourse in any numerieal approaeh the infinite limit will have to be approximated by some large finite value. This problem eould be solved by the shooting method discussed in Section 11.2. The discussion here will be to see if the FD approach is also applicable. The original equation can be converted into three first order differential equations or one first order equation and one second order equation. The latter approach seems most appropriate for the FD solution method. If the first variable in a formulation becomes the function and the second variable becomes the first derivative, the following two equations are obtained ... 

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. 

Aitemativeiy, the beam end couid have compiete rotational restraint and no transverse displacement, i.e., clamped. However, a third boundary condition exists in Rgure D-3 just as in Figure D-2. That is, an axial condition on displacement or force must exist in addition to the conditions usually thought of as comprising a clamped-end condition. Note that the block-like device at the end of the beam prevents rotation and transverse deflection. A similar device will be used later for plates. Whether all of the three boundary conditions can actually be enforced depends on the order of the differential equation set when (necessarily approximate) force-strain and moment-curvature relations are substituted in Equations (D.2), (D.4), and (D.7). 

The differential equation for a cheMcaL reanffon of third order is of the general nn... 

A third order reaction can be the result of the reaction of a single reactant, two reactants or three reactants. If the two or three reactants are involved in the reaction they may have same or different initial concentrations. Depending upon the conditions the differential rate equation may be formulated and integrated to give the rate equation. In some cases, the rate expressions have been given as follows. 

Although it is impossible to estimate accurately the amount of energy required in order to effect a size reduction of a given material, a number of empirical laws have been proposed. The two earliest laws are due to Kick17 and von Rittinger(8), and a third law due to Bond(910) has also been proposed. These three laws may all be derived from the basic differential equation ... 

Rigorous treatment of the self-action problem needs the transformation of Eq.(2.1), (2.5) into a system of integro-differential equations. However, if just some orders of group velocity dispersion and nonlinearity are taken into account, an approximate approach can be used based on differential equations solution. When dealing with the ID-i-T problem of optical pulse propagation in a dielectric waveguide, one comes to the wave equation with up to the third order GVD terms taken into account ... 

---