Quasilinearization and invariant imbedding in optimization

Abstract

concentration of this certain component and relatively m a l l weights are used for the other components. Furthermore, instead of the least squares criterion, any other criterion can also be used in the procedure discussed above.An estimation problem with multipoint boundary value (3) in difference equations can also be solved by essentially the same approach. For example, it is frequently necessary to have a certain specified purity of a certain side stream. Since this side stream is located at a plate inside the column and not at the terminals, a multipoint boundary-value problem is formulated. Many other specifications of the conditions of a distillation column are possible ( 5 ) . As we shall see in later papers, the present approach has certain distinct advantages over the conventional procedures for some of the specifications.It should be emphasized that the problem formulated does not always have solution. For example, the specified purity of the end product must be within the capability of the specified column in order for a solution to exist. The estimation approach avoids this problem. This is because of the fact that the least squares criterion only demands a solution that is as near as possible to the desired one. ACKNOWLEDGMENTThe work was partially supported by National Science Foun-

A modified Crank—Nicolson technique with non-equidistant space steps

“…Since the nonlinear source term S(y, z) is part of the right hand side f(y), the above equation has to be solved iteratively. In order to obtain fast convergence of the iterations, the quasilinearization technique can be applied [2]:…”

“…Since the nonlinear source term S(y, z) is part of the right hand side f(y), the above equation has to be solved iteratively. In order to obtain fast convergence of the iterations, the quasilinearization technique can be applied [2]:where p is the iteration index. These nonlinear equations can be linearized by the quasi.linearization technique and the system of tridiagonal linear eqns (12) again results except that the values for DA(L) and AL(L,2) are now given by: …”

“…Among the vast variety of methods which have been proposed, two stand out as having been used extensively and with good success: the classical Crank-Nicolson technique [I] combined with quasilinearization [2], and the method of orthogonal collocation [3,4]. However, in reaction and reactor modelling for highly exothermic reactions in which very steep concentration and temperature gradients are commonly encountered, both of these methods have significant drawbacks.…”

The Initial-Value Adjusting Method for Problems of the Least Squares Type of Ordinary Differential Equations

“…holds, and by Lemma 4 we obtain the assertion that ^"O/, e) has the derivative with respect to e at (*f(e), e) such as Let us consider the following problem originally mentioned in [4], which occurs relating to the tubular flow chemical reactor with axial mixing.…”

On nonlocal three-point boundary value problems of Duffing equation with mixed nonlinear forcing terms

“…To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization. The origin of the quasilinearization lies in the theory of dynamic programming [4,5]. Agarwal [6] discussed quasilinearization and approximate quasilinearization for multipoint boundary value problems.…”