Ordinary differential equations coefficients

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

A dynamic ordinary differential equation was written for the number concentration of particles in the reactor. In the development of EPM, we have assumed that the size dependence of the coagulation rate coefficients can be ignored above a certain maximum size, which should be chosen sufficiently large so as not to affect the final result. If the particle size distribution is desired, the particle number balance would have to be a partial differential equation in volume and time as shown by other investigators ( ). 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

The sensitivity analysis of a system of chemical reactions consist of the problem of determining the effect of uncertainties in parameters and initial conditions on the solution of a set of ordinary differential equations [22, 23], Sensitivity analysis procedures may be classified as deterministic or stochastic in nature. The interpretation of system sensitivities in terms of first-order elementary sensitivity coefficients is called a local sensitivity analysis and typifies the deterministic approach to sensitivity analysis. Here, the first-order elementary sensitivity coefficient is defined as the gradient... 

Therefore the last term in Eq. (6.37) is equal to zero. We end up with a linear ordinary differential equation with constant coefficients in terms of perturbation variables. 

Stratonovich SDEs, unlike Ito SDEs, may thus be manipulated using the familiar calculus of differentiable functions, rather than the Ito calculus. This property of a Stratonovich SDE may be shown to follow from the Ito transformation rule for the equivalent Ito SDE. It also follows immediately from the definition of the Stratonovich SDE as the white-noise limit of an ordinary differential equation, since the coefficients in the underlying ODE may be legitimately manipulated by the usual rules of calculus. 

Thanks to Assertion 9, the problem of symmetry reduction of Yang-Mills equations by subalgebras of the algebrap(l, 3) reduces to routine substitution of the corresponding expressions for , > into (61). We give below the final forms of the coefficients (60) of the reduced system of ordinary differential equations (59) for each subalgebras of the algebra p(l, 3) ... 

Systems (87) and (91) contain 12 nonlinear second-order ordinary differential equations with variable coefficients. That is why there is little hope for constructing their general solutions. Nevertheless, it is possible to obtain particular solutions of system (87), whose coefficients are given by formulas 2-4 from (91). 

Consider, as an example, system of ordinary differential equations (87) with coefficients given by the formula 2 from (93). We look for its solutions of the form... 

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... 

For a fixed value of x, this is a third-order, nonlinear, ordinary differential equation. Recall that the metric coefficient h x,y) depends on both coordinates, but that the cone angle 9 is a fixed constant. 

This is a functional equation for the boundary position X and the unknown constant parameter n. Upon substituting Eq. (256) into Eq. (251) an ordinary differential equation is obtained for X(t, n), and a family of curves in the phase plane (X, X) can be obtained. For n sufficiently close to unity two functions in the phase plane can be determined which serve as upper and lower bounds for the trajectories. The choice is guided by reference to the exact solution for the limiting case of constant surface temperature. It is shown that the upper and lower bounds are quite close to the one-parameter phase plane solution, although no comparison is made with a direct numerical solution. The one-parameter solution also agrees well with experiments on the solidification of aluminum under conditions of low surface heat transfer coefficient (hi = 0.02 cm.-1). 

The diffusion in a velocity field with a wide velocity spectrum supposedly describing turbulence is considered in the spirit of cascade-renormalization ideas. For the latter case of isotropic turbulence, we construct an ordinary differential equation for the turbulent diffusion coefficient. 

In order to complete the solution of the problem of gas motion under the action of a short impulse, we must not only find the exponents and dimensionless functions, which is accomplished by integrating the ordinary differential equations. We must also determine the numerical coefficients A and B in the formulas. 

St is the total sorbed concentration (M/M), a, is the first-order mass-transfer rate coefficient for compartment i (1/T), / is the mass fraction of the solute sorbed in each site at equilibrium (assumed to be equal for all compartments), Kp is the distribution coefficient (L3 /M), C is the aqueous solute concentration (M/L3), St is the mass sorbed in compartment i with respect to the total mass of the sorbent (M/M), 0 is the volumetric flow rate through the reactor (L3/T), C, is the influent concentration of solute (M/L3), M, is the mass of sorbent in the reactor (M), and V is the aqueous reactor volume (L3). Using the T-PDF, discrete values for the mass-transfer rate coefficients were generated for the NK compartments. The median value of the mass-transfer rate coefficient within each compartment was chosen as the representative value. The resulting system of ordinary differential equations was solved numerically using a 4th-order Runge-Kutta integration technique. 

So, the seasonal model of the global carbon cycle developed in the works by Nefedova and Tarko (1993) and Nefedova (1994) can be simulated by a system of 56 ordinary differential equations with periodic coefficients (i = 1,..., 14) ... 

The calculating procedure is based on sub-division of the Arctic Basin into grids (Eijk. This is realized by means of a quasi-linearization method (Nitu et al., 2000a). All differential equations of the SSMAE are substituted in each box E by easily integrable ordinary differential equations with constant coefficients. Water motion and turbulent mixing are realized in conformity with current velocity fields which are defined on the same coordinate grid as the E (Krapivin et al., 1998). 

In this Section so far, ADM is used to solve theoretical generalized models in the forms of ordinary differential equations). For diffusion-convection problems, the distributions along the axial direction of the packed bed electrode were neglected in certain cases, and mass transfer in the three dimensional electrodes were characterized by an average coefficient kh... 

The governing equations - that is, mainly the component and the total mass balances in the anode channels - are provided here in dimensionless form. The five ordinary differential equations (ODE) with respect to the spatial coordinate describe the development of the five unknowns in one single anode channel, namely the mole fractions, with i = CH4, H2O, H2, CO2, as well as the molar flow density inside the anode channel, y. Here, the Damkohler numbers, Da/, are the dimensionless reaction rate constant of the reforming and the oxidation reaction, respectively, the rj are the corresponding dimensionless reaction rates, and the v, j are the stoichiometric coefficients ... 

The problems of parametric estimation and model identification are among the most frequently encountered in experimental sciences and, thus, in chemical kinetics. Considerations about the statistical analysis of experimental results may be found in books on chemical kinetics and chemical reaction engineering [1—31], numerical methods [129—131, 133, 138], and pure and applied statistics [32, 33, 90, 91, 195—202]. The books by Kendall and Stuart [197] constitute a comprehensive treatise. A series of papers by Anderson [203] is of interest as an introductory survey to statistical methods in chemical engineering. Himmelblau et al. [204] have reviewed the methods for estimating the coefficients of ordinary differential equations which are linear in the... 

Substituting Eq. (14.31) into Eq. (14.30), and equating coefficients of to zero (for a 0), it is found that the functions n("n) constitute the solution of the following ordinary differential equation. 

The photodissociation rate coefficients are included as source and sink terms in a system of time-dependent continuity equations for the atmosphere. Modem values for vertical (eddy) diffusion and solar photon flux are utilized. The system of 2nd-order ordinary differential equations is solved by integration, and yields chemical species abundances as a function of time and altitude. The isotope atmospheric chemistry includes only SO2 isotopologue photodissociation reactions and production of SO isotopologues. Additional isotopic reactions such as SO2 oxidation by OH, SO photolysis, SO disproportionation during self-reaction, and SO dimmer formation, have been neglected. My objective here is to focus only on SO2 photolysis as a S-MIF mechanism. 

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