Differential equations, system

In order to eompute the time response of a dynamie system, it is neeessary to solve the differential equations (system mathematieal model) for given inputs. There are a number of analytieal and numerieal teehniques available to do this, but the one favoured by eontrol engineers is the use of the Laplaee transform. 

The above method is the well-known Gauss-Newton method for differential equation systems and it exhibits quadratic convergence to the optimum. Computational modifications to the above algorithm for the incorporation of prior knowledge about the parameters (Bayessian estimation) are discussed in detail in Chapter 8. 

In this section we first present an efficient step-size policy for differential equation systems and we present two approaches to increase the region of convergence of the Gauss-Newton method. One through the use of the Information Index and the other by using a two-step procedure that involves direct search optimization. 

The proposed step-size policy for differential equation systems is fairly similar to our approach for algebraic equation models. First we start with the bisection rule. We start with g=l and we keep on halving it until an acceptable value, pa, has been found, i.e., we reduce p until... 

The main difference with differential equation systems is that every evaluation of the objective function requires the integration of the state equations, In this section we present an optimal step size policy proposed by Kalogerakis and Luus (1983b) which uses information only at g=0 (i.e., at k ) and at p=pa (i.e., at... 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

I presented a group of subroutines—CORE, CHECKSTEP, STEPPER, SLOPER, GAUSS, and SWAPPER—that can be used to solve diverse theoretical problems in Earth system science. Together these subroutines can solve systems of coupled ordinary differential equations, systems that arise in the mathematical description of the history of environmental properties. The systems to be solved are described by subroutines EQUATIONS and SPECS. The systems need not be linear, as linearization is handled automatically by subroutine SLOPER. Subroutine CHECKSTEP ensures that the time steps are small enough to permit the linear approximation. Subroutine PRINTER simply preserves during the calculation whatever values will be needed for subsequent study. 

Model representations in Laplace transform form are mainly used in control theory. This approach is limited to linear differential equation systems or their... 

On the one hand, a property called cooperativity will be used. This property must hold upon the dynamics of the observation error associated to (19). The cooperative system theory enables to compare several solutions of a differential equation. More particularly, if a considered system = /(C, t) is cooperative, then it is possible to show that given two different initial conditions defined term by term as i(O) < 2(0) then, solutions to this system will be obtained in such a way that i(t) < 2(t), where 1 and 2 are the solutions of the differential equations system with the initial conditions (0) and 2(0), respectively. This is exactly the same result established previously in the case of simple mono-biomass/mono-substrate systems. With regard to this property the following lemma is recalled. 

In this study case, although the heat exchanger is a 2-dimensional system (see equation (1), according to results by Luyben [30] the recycle streams increases the order of the model (which is given by the differential equation system). Hence the dimension of the recycle system should be higher than two. We have chosen the embedding m = 3 in such a way that the space-phase... 

The term on the right side of the equations quantitatively defines mass transfer from the dispersed to the continuous liquid, which is explained more fully in section 9.6. The coupled differential equation system can be analytically or numerically integrated for the appropriate practical boundary conditions. 

The method of weighted residuals comprises several basic techniques, all of which have proved to be quite powerful and have been shown by Finlayson (1972, 1980) to be accurate numerical techniques frequently superior to finite difference schemes for the solution of complex differential equation systems. In the method of weighted residuals, the unknown exact solutions are expanded in a series of specified trial functions that are chosen to satisfy the boundary conditions, with unknown coefficients that are chosen to give the best solution to the differential equations ... 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

Note that this equation still retains the radial coordinate r. Therefore, unlike wedge case, there is not a unique ordinary differential that applies at any radius. Rather, there is an ordinary differential for every r position. Such local similarity behavior certainly represents a simplification compared to the original partial-differential-equation system. Nevertheless, the differential equation is more complex than that for the wedge case. 

Identify the characteristics and orders of the differential-equation systems. Does this vary between the alternative formulations ... 

The plug-flow problem may be formulated with a variable cross-sectional area and heterogeneous chemistry on the channel walls. Although the cross-sectional area varies, we make a quasi-one-dimensional assumption in which the flow can still be represented with only one velocity component u. It is implicitly assumed that the area variation is sufficiently small and smooth that the one-dimensional approximation is valid. Otherwise a two- or three-dimensional analysis is needed. Including the surface chemistry causes the system of equations to change from an ordinary-differential equation system to a differential-algebraic equation system. 

Setting W(t) = (2kBT )l/2W(t), the Langevin relation—describing the motion of the particle in the phase space—can be written as a two-variable stochastic differential equation system of the form of... 

The laminar flow assumption eliminates the non-linear term in the partial differential equations system (3.3), thus significantly reducing the computational cost. In addition, the present formulation often admits an exact solution. For example, in the case of an incompressible 2D laminar flow between two motionless parallel plates (i.e. planar SOFC configuration of Figure 3.1), Equation (3.29) reduces to ... 

Hindmarsh, A. C. "GEAR Ordinary Differential Equation System Solver" Lawrence Livermore Laboratory, Report UCID-30001, Revision 3, December, 1974. 

As previous sections, by using the variable sf (Eq. 2.8), the differential equation system (2.2) and the surface conditions (2.37) and (2.38) become dependent only on s-> variable ... 

By including this new variable in the differential equation system (2.107) and in the initial and limiting conditions, these are transformed into a one-variable problem (.v )MKj of identical form to that given by Eqs. (2.9)-(2.12) for static planar electrodes that is, c0 and cR can be expressed as functions of only one variable, SqME and srme, respectively. Thus, by following the same procedure indicated by Eqs. (2.13)—(2.18), one obtains expressions for the concentration profiles ... 

Under these conditions, the differential equation systems for the diffusion mass transport of species O and R is given by... 

The mass transport of the different species in solution is described by the diffusive differential equation system ... 

By considering the chemical reactions of reaction scheme (3.VI), the differential equation system that describes the mass transport of the (k + 1) species involved is... 

If we consider a planar electrode and assume that the chemical reaction in reaction schemes (3.VII)-(3.IXa, 3.IXb) is of first or pseudo-first order, the differential equation systems that should be solved together with the initial and surface concentrations in the three reaction schemes considered are given in Table 3.2. 

The first applied potential is set at a value E at a stationary spherical electrode during the interval 0 < t < i. The diffusion mass transport of the electroactive species toward or from the electrode surface is described by the following differential equation system ... 

Under the appropriate conditions, the mass transport can be mathematically modeled as a linear diffusion problem to the spatial domains shown in Scheme 5.4, When p successive potential pulses (E1, E2, . ., Ep) of the same length r are applied, the mass transport during the pth potential pulse in the presence of sufficient amounts of supporting electrolyte in both phases is described by the following differential equations system ... 

To solve the differential equation system given by Eqs. (6.185) and (6.187), the following solution is assumed ... 

Considering the new variable, the differential equation system that describes mass transport [given by Eq. (2.131)] changes into... 

Taking into account the dimensionless variables, V and c. the differential equation system (A.28) becomes... 

Note that Eq. (A.43) is identical to Eq. (A.l) therefore, the solutions of this homogeneous differential equation system have the following form (see Eq. (A.20)) ... 

For the solutions of the homogeneous differential equation system (A.43) with conditions (A.45)-(A.49), it is obtained... 

Next, this method will be applied to find the concentration profiles of species O and R when a potential step is applied to a spherical electrode of radius rs by assuming that diffusion coefficients of both species are equal (i.e., Do = Z)R = D). The differential equation system to solve is given by Eq. (2.131). The following variable change will be done ... 

The solutions of this differential equation system (B.24) are the following ... 

The solutions of differential equations system (D.l) with the boundary value problem given by Eqs. (D.4)-(D.5) can be written as the following functional series of the variable % with their coefficients being dependent on the variable, vp (i = O, R) ... 

The solutions for the differential equation system (D.40) take the form ... 

In order to solve the differential equation system [Eq. (F.2)], it is convenient to introduce the following variable change ... 

---