Nonclassical Method Research Articles

Applications of nonclassical symmetries

In this paper we discuss new applications of the nonclassical method of symmetry reduction. The equations treated are well known in mathematical physics and are used to describe multisoliton solutions of the Burger equation. They are also useful in the study of Kac-Moody algebras, and in the examination of nonlinear waves in the theory of elasticity. With the help of the nonclassical symmetry method the partial differential equations are reduced to ordinary differential equations which are solvable explicitly.

A stopping criterion for classical iterative methods in inexact affine-invariant Newton techniques

We present a method for the computation of stopping criterion for linear classical iterations in inexact affine-invariant Newton techniques. We show that the same methodology does not hold for nonclassical iterative methods.

Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source

Nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term. Mathematical forms are obtained for the source terms which permit a nonclassical symmetry reduction. In addition to the known source terms obtainable from classical symmetry methods, new source terms are found which also admit symmetry reductions. A number of examples are considered and several new exact solutions are constructed, some of which are illustrated graphically.

Nonclassical Symmetries and Bäcklund Transformations

In this paper we show how to obtain well-known Bäcklund transformations of several equations by applying the nonclassical symmetries method. In particular, we derive the Bäcklund transformation between the modified Korteweg-de Vries and the Korteweg-de Vries equations, the Bäcklund transformation between the Calogero and the third-order linear equations, the auto-Bäcklund transformations of the potential Korteweg-de Vries, the potential modified Korteweg-de Vries equation, and the Burgers equations. As a consequence, the n-solitons solution of the Korteweg-de Vries equation can be obtained by the nonclassical symmetries method.

The direct method and the singular manifold method for the Fitzhugh-Nagumo equation

I explicitly show that a statement made recently by Nucci and Clarkson in this journal on whether the non-classical method is more general than the direct method for the Fitzhugh-Nagumo equation can be completed using the Painlevé analysis. My conjecture is that the direct method combined with the singular manifold method can be equivalent to the non-classical method.

The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh-Nagumo equation

In this paper it is shown that there exist exact solutions of the Fitzhugh-Nagumo equation which can be obtained using the nonclassical method for determining symmetry reductions of partial differential equations developed by Bluman and Cole [J. Math. Mech. 18 (1969) 1025], but which are not obtained using the direct method as developed by Clarkson and Kruskal [J. Math. Phys. 30 (1989) 2201].

A nonclassical stress analysis for a laminated composite section

In this study the nonclassical method of stress analysis, developed by Ardiç, Santare & Chou [ Int. J. Sol. Struct., 26 (1990) 643-53; Int. J. Engng Sci., 27 (1989) 1397-405], is applied to a laminated composite body containing continuous fiber-reinforced layers. For this problem the nonclassical method is performed in two steps. First, each lamina is considered to be homogeneous and anisotropic. The effects of adjacent laminae on the average strains in each lamina are determined. These strains are used as input far-field strains for the laminae, and by using the nonclassical method for three-dimensional, continuous fiber-reinforced materials in Ardiç et al. [ Int. J. Sol. Struct., 26 (1990) 643-53] the effects of heterogeneity on the stresses in the fibers and the matrix of each lamina are found. By summing the stresses calculated from the input far-field strains and the heterogeneity stresses the stress fields in the fibers and matrix are determined separately. Some sample results are plotted for comparison with classical methods.

On the conditional symmetries of Levi and Winternitz

It is pointed out that the 'conditional symmetries' of Levi and Winternitz (1989), and the 'non-classical method' first introduced by Bluman and Cole (1989), have a natural group theoretical setting. This corresponds to classifying solutions to a PDEs according to their full symmetry and not just according to their symmetries which are also symmetries of the PDE.

A reduce package for determining first integrals of autonomous systems of ordinary differential equations

We will discuss three different methods for finding symmetry solutions based on the Fréchet derivative common to each procedure. The methods discussed are Lie's standard procedure of symmetry analysis, the nonclassical method, and the derivation of potential symmetries. A ferromagnet in a strong external field represented by a nonlinear telegraph equation serves as an example describing the application of all three methods. The symmetry methods discussed are realized in a Mathematica package called MathLie performing all of the required calculations.

SOME PROPERTIES OF SOLITON IN NON-CLASSICAL VARIATION FIELD

This paper is concerned with some characteristic properties of soliton by utilizing non-classical variational methods. (1-3) We discussed short range of soliton and the stability of soliton of KdV equation. The main results are given in theorem 1 and theorem 2.

Nonclassical determination of spatial coordinates of ocean-bottom acoustic transponders

Most current methods for acoustic-transponder-array survey are based on the assumption that certain parameters are known and held fixed. Often, however, most of these parameters are very poorly known. Using observed data, the authors outline how these inadmissible assumptions can be eliminated using a nonclassical method for determination of the spatial coordinates of ocean-bottom acoustic transponders and statistical analysis. The problems involved in the elimination process and their remedies, and the precision achievable are discussed. The influence of the number of data points and their spatial distribution is demonstrated, based on statistical tests.

Bearing Estimation Using Data from Many Beams

Let the location of a source of power be given by its polar coordinates. The bearing of the source is the polar angle of its location. Let a set of directional receiving patterns, or beams, be located at the origin of the coordinate system. Each beam is pointed in a different direction. The total power received on each beam is due to the source and noise. Models are presented to describe the dependence of the received powers and their estimates on the source power and bearing. The effect of randomness (noise), such as variation in the medium, is included. Based on these models, we first consider bearing estimation in the absence of randomness but including all sidelobe information. The concept of “virtual source power” is introduced for this purpose. It is then used for bearing estimation when randomness is present giving rise to several nonclassical estimation methods as well as a method using confidence intervals. A variety of squared error techniques are presented, along with comments concerning practical methods of computation.