Solution Singular

In the case of some equations still other solutions exist called singular solutions. A singular solution is any solution of the differential equation which is not included in the general solution. 

Unlike the case of the neutral reactants, where analytical solution reveals the auto-model behaviour in coordinated r/ , in our case of charged particles the singular solutions arise on the spatial scale of the order of the recombination radius ro thus preventing us from such a simplified analytical analysis. Therefore, we will compare semi-qualitative arguments for the new law, n(t) oc r5/4, with numerical calculations of our kinetic equations. 

The differential equation (9) has, however, another solution, namely, the singular solution obtained by eliminating the constant between (10) and its derivative with respect to the constant. 

Similar to scalar field problems, in order to obtain an integral representation for the momentum eqns. (10.63) and (10.64) for the flow field (u, p), Green s formulae for the momentum equations (Theorems (10.2.1) and (10.2.2)) are used together with the fundamental singular solution of Stokes equations, i.e.,... 

Let us now consider the same flow domain (represented in Fig. l.a) with the boundary conditions of vanishing velocity on F] and F2 (the fluid is sticking at the wall on Fi and F2). This problem too has been largely studied for a Newtonian fluid. In this case, singular solutions of the homogeneous Stokes problem exist if a is a solution of the following equation (14) 5in(am) 2 sin((0) 2 aof) 0)... 

Let us notice Uq and ao (resp. U i and Oi) velocity and stress tensor at X=0 (resp. derivative according to X of velocity and stress tensor at X.=0 ). The velocity field Uo is the solution of an homogeneous Stokes problem with a "stick-slip singularity ( = jt) and a singular solution (Uo(r,8)= r IJo(e)) exists. As it is easily verified Ui is the solution of the following inhomogeneous Stokes problem ... 

A mathematical model may be constructed representing a chemical reaction. Solutions of the mathematical model must be compatible with the observed behavior of this chemical reaction. Furthermore if some other solutions would indicate possible behaviors so far unobserved, of the reaction, experiments maybe designed to experimentally observe them, thus to reinforce the validity of the mathematical model. Dynamical systems such as reactions are modelled by differential equations. The chemical equilibrium states are the stable singular solutions of the mathematical model consisting of a set of differential equations. Depending on the format of these equations solutions vary in a number of possible ways. In addition to these stable singular solutions periodic solutions also appear. Although there are various kinds of oscillatory behavior observed in reactions, these periodic solutions correspond to only some of these oscillations. 

As shown below in modelling this reaction various solutions were considered. For certain values of parameters it is shown that there are three singular solutions, two stable and one unstable as well as limit cycles (multiple solutions). 

This bifurcation study has been reviewed at length by Ray (1977). The system possesses three singular points and limit cycles as oscillatory solutions. It should be noted that although the applications and the differential equations are significantly different from each other, some of the models of glycolysis studied by Sel kov also possess three singular solutions and limit cycles very similar (topologically) to those obtained by Uppal, et al., see Section III.F. 

This model was slightly altered by Sel kov and Betz (1973). The model still has only one singular solution, and a stable limit cycle corresponding to oscillations. 

Two singular solutions, one saddle (S2), one unstable focus (St) and a stable limit cycle. 

There are in fact two solutions after the singular solution, X = A, Y — BjA bifurcates, the new solution is a stable limit cycle. Lefever and Nicolis (1971) give these solutions for the point with parameter values A = 1,5 = 3, lying in the parameter plane region corresponding to a limit cycle and an unstable singular point, simultaneously. 

Multiple solutions more complex than this are also present in the literature. For example for two-dimensional systems Aris and Amundson (1958) discussed multiple singular solutions as well as multiple limit cycles appearing simultaneously. 

S. Kim, Singularity solutions for ellipsoids in low-Reynolds number flows With applications to the calculation of hydrodynamic interactions in suspensions of ellipsoids, Ini. J. Multiphase Flow 12, 469-91 (1986). 

One major limitation of SAAM II is that it forces the user to assume a normal prior distribution, which for pharmacokinetic problems is not realistic as most pharmacokinetic parameters are log-normal in distribution and their values will always be positive. Adapt II is not limited to normal prior distributions, but also allows for log-normal prior distributions. Hence, Adapt II may be preferred to SAAM II for some Bayesian problems. Nevertheless, using prior information can be useful to solve a problem with a near singular solution or if it is... 

Clairaut s equation introduces a new idea. Hitherto we have assumed that whenever a function of x and y satisfies an equation, that function, plus an arbitrary constant, represents the complete or general solution. We now find that a function of x and y can sometimes be found to satisfy the given equation, which, unlike the particular solution, is not included in the general solution. This function must be considered a solution, because it satisfies the given equation. But the existence of such a solution is quite an accidental property confined to special equations, hence their cognomen, singular solutions. Take the equation dy a a... 

Ordinary differential equations have two classes of solutions —the complete integral and the singular solution. Particular solutions are only varieties of the complete integral. Three... 

This result satisfies equation (4), but, unlike the particular solution, is not included in the complete integral (2). Such a solution of the differential equation is said to be a singular solution. 

Geometrically, the singular solution represents two plane surfaces touched by all the spheres represented by equation (2). The singular.solution is thus the envelope of all the spheres represented... 

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