Parameter-dependent initial value problems

Sensitivities of Parameter-Dependent Initial Value Problems... 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Sometimes, instead of an initial value problem, the mathematical model of a chemical process is a boundary value problem in which values of the dependent variables are specified at different values of the independent variable t. The shooting technique consists of solving an initial value problem, but with an initial value vector a considered as a parameter to estimate (by optimization techniques) so that boundary conditions are satisfied. In this way, a boundary value problem is transformed into an initial value problem. 

An aspect in which circumstellar environments differ from interstellar is the net mass advection through the medium. Abundances become time dependent— and hence space dependent— in the envelope, due both to the implicit time dependence of the reactions and to the transport of matter through different radii via stellar wind flow. The atomic abundances are fixed at stellar photosphere, rather than having to be assumed for some mixture of physical parameters of temperature and pressure as they must for molecular clouds. It is then essentially an initial-value problem to compute the abundances which will be a function of radius in the envelope. For a steady-state wind, the abundances become strictly a function of radius. Also, unlike a molecular cloud, the density profile of the envelope is specified from the assumption of steady mass loss at the terminal velocity for the wind, so that p(r) = M/(47Tr Voo), where M is the mass loss rate and Woo is the terminal velocity of the wind. 

ODEs depending on parameters. The question addressed there is about the sensitivity of the model with respect to parameter changes. Consider the ordinary initial value problem (IVP) ... 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

The unknown dependent variables of the problem become p, m, Z, T, and Yfc. Equations (86) and (110) provide two algebraic equations relating these variables the other N -f 2 variables are determined by the iV + 1 differential equations given by equations (95) and (105) and by the integro-differential equation given in equation (92). The initial values Po, Zq, Tq, and Yj, o are controlled by the experimenter. Attention will be restricted to lean mixtures, whence Z -> 1 as x -> oo for chemical equilibrium to exist at the hot boundary. Since all the differential equations are of the first order, the additional boundary condition suggests that solutions will exist only for particular values of a parameter, which physically is expected to be the burning velocity Uq. ... 

In the latter case, the authors derived Landau equations (3.114) for the am-phtudes of the unstable modes and determined the dependence of the coefficients of these equations on the parameters of the FP problem. Figure 9 depicts the real parts of the coefficients / i and fh as functions of jo (which is proportional to the initial concentration of the initiator) for the typical parameter values [81] (Ei — E2)/ RgqMo) = 19.79 and Ei/ RgqMo) = 58.4. The wavenumber s = 0.55, which is close to the value Sm at which the neutral stabihty curve has a minimum. 

In contrast to linear parameters, nonlinear parameters cannot be represented simply by the product between the matrix of independent variables, X, and the parameter vector, b (cf. Eq. (6.13)). If, however, approximate initial values, b, are known for the parameter vector, b, then the function y=f x) can be rewritten such that a linear function emerges in dependence on the difference of the parameter to be estimated and its starting value, that is, Lb = b-bQ. This means that the method can be traced back to the solution of a least squares problem. 

A typical example is maghemite with its strongly overlapping sextets of A and B sites. This spectrum can be simply fitted with 2 sextets, but the resulting hyperfine parameters will depend on the initial values used in the fit. This ill-posed problem can only be solved by performing measurements in an external magnetic field, where the outer sextet from the A sites is well separated from that of the B sites (Fig. 3.49), yielding accurate values for both isomer shifts and hyperfine fields. 

Due to the nrmlinear dependence of the a priori, unknown residual sequence E on the parameter vector 0, the last equation leads to a nonlinear-weighted least squares problem, which has to be tackled by nrmlinear optimization methods. However, nonUnear least squares techniques are sensitive to the initial parameter values and if no acciuate estimates are available, the nuniniization procedure is very likely to converge to a local minimum. In order to avoid potential inaccurate convergence problems associated with arbitrary initial estimates, initial values for the coefficients of projection may be obtained by identifying conventional ARMA models for each of the K data... 

In the above relationship, the coefficients Aj to An depend on the initial conditions of the problem and the exponential values, are determined by the parameters of the system and in fact represent the eigenvalues or roots of the characteristic solution of the system. 

It has been pointed out by Reisfeld (8, 9) that the value of h obtained for a vitreous medium depends upon the particular s ion used for the spectral determination. This variation arises owing to the incomplete separation of h and m and is also observed for transition metal ions. The variation, however, is fairly small, but nevertheless was recognised as an initial difficulty in tackling the problem of basicity in oxide-containing media (70). To overcome the problem, the Lewis basicity of an oxide-containing medium was expressed as a ratio of h parameters where... 

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