Second-order differential equations

M. Hochbruck and Ch. Lubich. A Gautschi-type method for oscillatory second-order differential equations. Tech. Rep., Universitat Tubingen, 1998. 

Note that the mathematical symbol V stands for the second derivative of a function (in this case with respect to the Cartesian coordinates d fdx + d jdy + d jdz y, therefore the relationship stated in Eq. (41) is a second-order differential equation. Only for a constant dielectric Eq.(41) can be reduced to Coulomb s law. In the more interesting case where the dielectric is not constant within the volume considered, the Poisson equation is modified according to Eq. (42). 

Equations (11.111) - (11.113) define a boundary value problem for a pair of simultaneous second order differential equations in and x, subject... 

The charge density is simply the distribution of charge throughout the system and has 1 units of Cm . The Poisson equation is thus a second-order differential equation (V the usual abbreviation for (d /dr ) + (f /dx/) + (d /dz )). For a set of point charges in constant dielectric the Poisson equation reduces to Coulomb s law. However, if the dielectr... 

There are now four constants rather than eight. We expect four constants from two second-order differential equations. Dropping the unnecessary subscript 1 and replacing the cumbersome prime notation . 

Much of the language used for empirical rate laws can also be appHed to the differential equations associated with each step of a mechanism. Equation 23b is first order in each of I and C and second order overall. Equation 23a implies that one must consider both the forward reaction and the reverse reaction. The forward reaction is second order overall the reverse reaction is first order in [I. Additional language is used for mechanisms that should never be apphed to empirical rate laws. The second equation is said to describe a bimolecular mechanism. A bimolecular mechanism implies a second-order differential equation however, a second-order empirical rate law does not guarantee a bimolecular mechanism. A mechanism may be bimolecular in one component, for example 2A I. 

Equation (7) is a second-order differential equation. A more general formulation of Newton s equation of motion is given in terms of the system s Hamiltonian, FI [Eq. (1)]. Put in these terms, the classical equation of motion is written as a pair of coupled first-order differential equations ... 

By substimting the definition of H [Eq. (1)] into Eq. (8), we regain Eq. (6). The first first-order differential equation in Eq. (8) becomes the standard definition of momentum, i.e.. Pi = miFi = niiVi, while the second turns into Eq. (6). A set of two first-order differential equations is often easier to solve than a single second-order differential equation. 

Newton s equation of motion is a second-order differential equation that requires two initial values for each degree of freedom in order to initiate the integration. These two initial values are typically a set of initial coordinates r(0) and a set of initial velocities v(0). ... 

Equations (B.15) are exactly the same as those derived by Holstein [1978], and the following discussion draws on that paper. The pair of equations (B.15) may be represented as a single second-order differential equation... 

The general solution of this second order differential equation is... 

If the system dynamics produced a second-order differential equation, it would be represented by... 

Solutions for this second-order differential equation are known for a number of initial and boundary conditions [4]. 

To solve the Poisson equation, we must express as a p function of the coordinates of the system and solve the resulting second-order differential equation to obtain ip(x,y, z), from which trci and hence, AG, p - p°, and 7 can be calculated. 

Equation (7.25) can be substituted into equation (7.20) to give a second order differential equation in ijj. In theory, the resulting equation can be solved to give ip as a function of r. However, it has an exponential term in -ip, that makes it impossible to solve analytically. In the Debye-Hiickel approximation, the exponential is expanded in a power series to give... 

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... 

If the temperature everywhere is constant initially, 0l=o is a constant and the equation may be integrated as a normal second-order differential equation since p is not a function of x. 

Substitution of (A4.2) into (A4.1) leads to a second-order differential equation... 

The latter difference equation clarifies that (6) is an analog of a second-order differential equation. 

The above second-order differential equation can be solved by integration. At the liquid surface, where Z=0, the bulk gas concentration, Cso. is known, but the concentration gradient dCs/dZ is unknown. Conversely at the full liquid depth, the concentration Cso is not known, but the concentration gradient is known and is equal to zero. Since there can be no diffusion of component S from the bottom surface of the liquid, i.e., js at Z=L is 0 and hence from Pick s Law dCs/dZ at Z=L must also be zero. 

At steady state, aCA/9t can be set to zero, and the equation becomes an ordinary second-order differential equation, which can be solved using ISIM. 

Figure 4.17. Flow diagram for solving the second-order differential equation from the axial disperion model.

A second-order differential equation has two solutions of the form of equation (G.2), each with a different set of values for the constant s and the coefficients a. ... 

Thus, the two solutions of the second-order differential equation (G.IO) are... 

We now derive the time-domain solutions of first and second order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, reduced order model, and the effect of zeros will be discussed. 

In establishing the relationship between time-domain and Laplace-domain, we use only first and second order differential equations. That s because we are working strictly with linearized systems. As we have seen in partial fraction expansion, any function can be "broken up" into first order terms. Terms of complex roots can be combined together to form a second order term. 

Example 4.1 Derive the state space representation of a second order differential equation of a form similar to Eq. (3-16) on page 3-5 ... 

The Hermite polynomials introduced above represent an example of special functions which arise as solutions to various second-order differential equations. After a summary of some of the properties of these polynomials, a brief description of a few others will be presented. The choice is based on their importance in certain problems in physics and chemistry. 

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