Quasilinearization approximation

Using cs + cp = constant, this equation relates the reaction velocity to affinity A only. A plot of As/As.max versus affinity shows an inflection point, representing a maximum slope. Therefore, the quasilinear approximation between the reaction velocity and the affinity will be valid with a deviation of less than 15%. 

Because of this, there is a real need for designing the general method, by means of which economical schemes can be created for equations with variable and even discontinuous coefhcients as well as for quasilinear non-stationary equations in complex domains of arbitrary shape and dimension. As a matter of experience, the universal tool in such obstacles is the method of summarized approximation, the framework of which will be explained a little later on the basis of the heat conduction equation in an arbitrary domain G of the dimension p with the boundary F... 

The only way to obtain a well-defined and physically meaningful approximation is by performing again an expansion in powers of a physical parameter. If the lowest order is to be deterministic the parameter has to be such that for small values of it the distribution reduces to a narrow peak. Clearly the parameter 6 = kT is suitable, because the low temperatures have small fluctuations. We shall show that the same method used in X for obtaining the -expansion can be adapted to obtain an expansion of the Fokker-Planck equation in powers of 01/2. We first demonstrate the method for the one-variable quasilinear equation (2.4). 

Of course, the linear model is a special case of the polynomial model. Generally, a model is called quasilinear when / is a linear function of p. This does not exclude the case that/is nonlinear in X, In particular, the polynomial model is quasilinear although the functional dependence on, v may be quadratic, as in Fig. 4 b. Given the data pairs (.i y,), the parameter p yielding the best approximation of / to all these data pairs is found by minimizing the sum of the squares of the deviations between the measured values r, and their modeled counterparts... 

In [157] the authors present an initial-value methodology for the numerical approximation of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations. These problems have a boundary layer at one end (left or right) point. The techniaque which used by the authors is to reduce the original problem to an asymptotically equivalent first order initial-value problem. This is done with the... 

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