Singular perturbation solution

The asymptotic solution (6.15) differs only slightly from the base solution, which is the solution when = 0. This is the regular perturbation solution. The asymptotic solution (6.20) deviates substantially from the base solution it is the singular perturbation solution. 

We end this chapter by noting that the application of the singular perturbation method to partial differential equations requires more ingenuity than its application to ODEs. However, as in the case of ODEs, the singular perturbation solutions can be used as a tool to explore parametric dependencies and, as well, as a valuable check on numerical solutions. The book by Cole (1968) provides a complete and formal treatment of partial differential equations by the singular perturbation method. 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

The mathematical theory of dissipative structures is mainly based on approximate methods such as bifurcation theory of singular perturbation theory. Situations like those described in Section VI and that permit an exact solution are rather exceptional. 

The proper singular perturbation treatment has thus to take care of this initial stage. Probably the simplest way to do this is via a matched asymptotic expansion procedure, with the outer solution of the type (5.2.13), (5.2.14), valid for t = 0(1), matched with an initial layer solution that has an internal layer at x = 0. 

Because of the complex nature of the Painleve transcendents and of the resulting difficulties in satisfying the boundary conditions we shall not proceed with the exact analytical solution of b.v.p. (5.3.6) (5.3.8) any further, but rather we turn to an asymptotic and numerical study of this singular perturbation problem. 

We shall now solve the Kramers equation (7.4) approximately for large y by means of a systematic expansion in powers of y-1. Straightforward perturbation theory is not possible because the time derivative occurs among the small terms. This makes it a problem of singular perturbation theory, but the way to handle it can be learned from the solution method invented by Hilbert and by Chapman and Enskog for the Boltzmann equation.To simplify the writing I eliminate the coefficient kT/M by rescaling the variables,... 

Undoubtedly other examples can be found. In the study of the systems cited above, a common property is that the (dimensionless) parameter c is very small compared with the (dimensionless) terms f(x) and g(x) which can be chosen to be of the order of 1. In all of the examples cited c is 2.5 X 10 2 or less. In this paper we summarize a singular perturbation technique for the solution of Equation 1, which is to be solved subject to an initial condition c(x,0), and on the assumption that boundary effects (in x) can be ignored. 

An iterative method for developing higher order approximations to the solution to Equation 1 can be devised by using some of the ideas of singular perturbation theory (15). To display the systematics of the procedure let us rewrite Equation 7 as... 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

A.B. Vasilieva and V.F. Butuzov, Asymptotic Expansion for Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973 (in Russian). 

The condition stated in Definition 2.1 assures that a well-defined n-dimensional reduced model will correspond to each solution (2.12) whenever this condition is violated, the system in Equations (2.7) and (2.8) is said to be in a nonstandard singularly perturbed form. 

Then a singular perturbation method can be used to obtain the electrochemical potential and the fluid velocity by matching the inner and the outer solutions. 

Notice that when r - oo,v - S0/(Km + S0). This is exactly the value of v that we arrived at for r = 0. Thus as r -> oc (on the fast timescale), v approaches the derived initial condition for the slow timescale (r = 0). Hence, the entire transient for Michaelis-Menten kinetics can be represented by combining the short timescale result, Equation (4.29), with the long timescale result, the solution to Equation (4.25). The two results match seamlessly at r = oo and x = 0. This is known as asymptotic matching in singular perturbation analysis [110]. 

Prom Eqn. (2.6.65), it is apparent that this is a singular perturbation problem (as the highest derivative term is multiplied by the small parameter) and then one can use matched asymptotic expansion to obtain (f> by describing the solution in terms of outer and inner solutions. 

An analysis of radial flow, fixed bed reactor (RFBR) is carried out to determine the effects of radial flow maldistribution and flow direction. Analytical criteria for optimum operation is established via a singular perturbation approach. It is shown that at high conversion an ideal flow profile always results in a higher yield irrespective of the reaction mechanism while dependence of conversion on flow direction is second order. The analysis then concentrates on the improvement of radial profile. Asymptotic solutions are obtained for the flow equations. They offer an optimum design method well suited for industrial application. Finally, all asymptotic results are verified by a numerical experience in a more sophisticated heterogeneous, two-dimensional cell model. 

The boundary conditions follow naturally from the conventions Uq = = 0, and similarly for v, which say that there are no microorganisms in the two reservoirs. They are justified by the agreement between the numerically computed rest points of (5.1) and the solutions of (5.2) obtained by using singular perturbation theory (as in [S9]). 

This is a pair of nonlinear equations and no simple solution can be written down, but they can be reduced to a single equation by making a so-called pseudo-steady state hypothesis. This is the assumption that, when Cq is much smaller than Uo, the concentration c is never very large and varies very slowly. We can then set dc/dt = 0. This hypothesis appears to be very pseudo indeed at first sight, but it can in fact be justified by what is known as the singular perturbation theory of differential equations. Setting dcjdt = 0 in Eq. (4.6.7) we can solve for c in terms of a ... 

The main point here is that the solution procedure for this particular problem of a singular (or matched) asymptotic expansion follows a very generic routine. Given that there are two sub-domains in the solution domain, which overlap so that matching is possible (the sub-domains here are the core and the boundary-layer regions), the solution of a singular perturbation problem usually proceeds sequentially back and forth as we add higher order... 

S. Kaplan, Low Reynolds number flow past a circular cylinder, J. Math. Meek 6, 595-603 (1957) S. Kaplan and R A. Lagerstrom, Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers, J. Math. Mech. 6, 585-93 (1957). These and other published and previously unpublished works of Kaplan are reproduced in the following book S. Kaplan, Fluid Mechanics and Singular Perturbations P. A. Lagerstrom, L. N. Howard, and C. S. Lin (eds.). (Academic, New York, 1957). 

The analysis for Sc -> oo closely follows the large-Prandtl-number analysis of Section C. The solution of (11-113) is a singular perturbation with a leading-order outer solution 0 = 0 and an inner solution that satisfies a boundary-layer equation that is obtained by rescaling according to... 

The recent development of tensorial schemes for characterizing the intrinsic hydrodynamic resistance of particles of arbitrary shape, and the application of singular perturbation techniques to obtain asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers constitute significant contributions to oim understanding of slow viscous flow around bodies. It is with these topics that this review is primarily concerned. In presenting this material we have elected to use Gibbs polyadics in preference to conventional tensor notation. For in our view, the former symbolism— dealing as it does with direction as a primitive concept—is more closely related to the physical world in which we live than is the latter notation. 

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