Equation quasilinear

We call this equation quasilinear to express the fact that the coefficient of L(t) is still a constant. Although its solution cannot be given explicitly it can still be argued that it is equivalent to the quasilinear Fokker-Planck equation... 

Quasilinear first-order differential equations are like... 

The equations for flow and adsorption in a packed bed or chromatography column give a quasilinear equation. 

Some analytical solutions to the quasilinear heat conduction equation. ... 

As can readily be observed, they are not monotone, thus causing some ripple . This obstacle can be avoided by refining some suitable grids in time. When solving equations of the form (13) with a weak quasilinearity for the coefficients k = k x,t), f = f u) and c = c x,t), common practice involves predictor-corrector schemes of accuracy 0(r" -f /r). Such a scheme for the choice c = k = 1, f = f u) is available now ... 

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... 

A quasilinear equation of parabolic type. The complete posing of problem (15) includes... 

If we compare Equations 6.79 and 6.11 we notice that the only difference between the quasilinearization method and the Gauss-Newton method is the nature of the equation that yields the parameter estimate vector k(l+l). If one substitutes Equation 6.81 into Equation 6.79 obtains the following equation... 

By taking the last term on the right hand side of Equation 6.83 to the left hand side one obtains Equation 6.11 that is used for the Gauss-Newton method. Hence, when the output vector is linearly related to the state vector (Equation 6.2) then the simplified quasilinearization method is computationally identical to the Gauss-Newton method. 

The above equation represents a set of p nonlinear equations which can be solved to obtain koutput vector around the trajectory xw(t). Kalogerakis and Luus (1983b) showed that when linearization of the output vector is used, the quasilinearization computational algorithm and the Gauss-Newton method yield the same results. 

Donnely, J.K. and D. Quon, "Identification of Parameters in Systems of Ordinary Differential Equations Using Quasilinearization and Data Pertrubation", Can. J. Chem. Eng, 48, 114 (1970). 

Oleinik, O. A., 1959, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation, Usp. Mat. Nauk 14 165. 

The behavior of a reactive wave depen ds on the flow of its reacting and product-gases. The conservation laws lead to systems of partial differential equations of the first order which are quasilinear, ie, equations in which partial derivatives appear linearly. In practical cases special symmetry of boundary and initial conditions is often invoked to reduce the number of independent variables. 

The stationary system, e.g., (5.3.1) (5.3.5) is replaced by its time-dependent counterpart. In this counterpart, the Poisson equation is replaced by the total current continuity equation (1.5), obtained as a linear combination of the original equations. The resulting system is then solved by quasilinearization [9] with a simultaneous solution of quasilinearized equations and subsequent Newton s iterations at each time step. Integration is continued in time until the steady state is reached. This numerical procedure is a modification of that suggested by Mock in [10]. 

Other ideas are connected with two types of purely implicit difference schemes (the forward ones with heat conduction equation... 

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