Delta-shaped Functions Research Articles

Wavelet transformation on a finite interval

Integral transformations on a finite interval with a singular basis wavelet are considered. Using a sequence of such transformations, the problem of nonparametric approximation of a function is solved. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet must be zero). The paper develops the previously proposed method of singular wavelets when the tolerance condition is not met. In this case Delta-shaped functions that participate in Parzen – Rosenblatt and Nadaray – Watson estimations can be used as a basic wavelet. The set of wavelet transformations for a function defined on a numeric axis, defined locally, and on a finite interval were previously investigated. However, the study of the convergence of the decomposition on a finite interval was carried out only in one particular case. It was due to technical difficulties when trying to solve this problem directly. In the paper the idea of evaluating the periodic continuation of a function defined initially on a finite interval is implemented. It allowed to formulate sufficient convergence conditions for the expansion of the function in a series. An example of approximation of a function defined on a finite interval using the sum of discrete wavelet transformations is given.

Isolation of a periodic component by singular wavelet decomposition

In this paper, we propose to use a discrete wavelet transform with a singular wavelet to isolate the periodic component from the signal. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet is zero). For singular wavelets, the validity condition is not met. As a singular wavelet, you can use the Delta-shaped functions, which are involved in the estimates of Parzen-Rosenblatt, Nadaraya-Watson. Using singular value of a wavelet is determined by the discrete wavelet transform. This transformation was studied earlier for the continuous case. Theoretical estimates of the convergence rate of the sum of wavelet transformations were obtained; various variants were proposed and a theoretical justification was given for the use of the singular wavelet method; sufficient conditions for uniform convergence of the sum of wavelet transformations were formulated. It is shown that the wavelet transform can be used to solve the problem of nonparametric approximation of the function. Singular wavelet decomposition is a new method and there are currently no examples of its application to solving applied problems. This paper analyzes the possibilities of the singular wavelet method. It is assumed that in some cases a slow and fast component can be distinguished from the signal, and this hypothesis is confirmed by the numerical solution of the real problem. A similar analysis is performed using a parametric regression equation, which allows you to select the periodic component of the signal. Comparison of the calculation results confirms that nonparametric approximation based on singular wavelets and the application of parametric regression can lead to similar results.

Local transformations with a singular wavelet

The paper considers a local wavelet transform with a singular basis wavelet. The problem of nonparametric approximation of a function is solved by the use of the sequence of local wavelet transforms. Traditionally believed that the wavelet should have an average equal to zero. Earlier, the author considered singular wavelets when the average value is not equal to zero. As an example, the delta-shaped functions, participated in the estimates of Parzen – Rosenblatt and Nadara – Watson, were used as a wavelet. Previously, a sequence of wavelet transforms for the entire numerical axis and finite interval was constructed for singular wavelets. The paper proposes a sequence of local wavelet transforms, a local wavelet transform is defined, the theorems that formulate the properties of a local wavelet transform are proved. To confirm the effectiveness of the algorithm an example of approximating the function by use of the sum of discrete local wavelet transforms is given.

The method of approximate fundamental solutions (MAFS) for elliptic equations of general type with variable coefficients

The paper presents a meshless method for solving elliptic equations of general type with variable coefficients. It is based on the use of the delta-shaped functions and the method of approximate fundamental solutions first suggested for solving equations with constant coefficients. The method assumes that the solution domain is embedded in a square and the initial equation is extended onto the square with the help of the CICE −(Chebyshev interpolation + C-expansion) approximation scheme. As a result the coefficients of the equation are approximated by the truncated Fourier series over some orthogonal system in the square. The approximate fundamental solutions (AFSs) satisfy L[u]=I(x), where I(x) is the delta shaped function in the form of the truncated Fourier series. Thus, the AFSs due to the special form of the operator can be obtained in the similar form of truncated series. The next part of the MAFS follows the general scheme of the MFS. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in solution of two-dimensional elliptic equations of general type with different boundary conditions (Dirichlet, Neumann, mixed) in arbitrary domains.

The method of approximate fundamental solutions (MAFS) for Stefan problems

The paper presents a new meshless numerical technique for solving one and two-dimensional Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) first suggested for solving elliptic problems and heat equations in domains with fixed boundaries. The one-dimensional problems in the plane and cylindrical geometries are considered. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for degenerate and singular problems when a region initially has zero thickness. The same technique was developed for 2D Stefan problems with completely or partially unknown boundary.

A meshless method for one-dimensional Stefan problems

The paper presents a new meshless numerical technique for solving one-dimensional problems with moving boundaries including the Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries. The numerical examples are presented and the results are compared with analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness.

On the contact interaction between a strip-shaped punch and a linearly-deformable base through a covering of variable thickness

The contact problem of the frictionless penetration of a punch with strip-shaped section into the surface of a linearly-deformable base protected by a thin elastic layer (covering) of variable thickness, the stiffness of which is comparable to or smaller than that of the supporting elastic body, is investigated. A Fredholm integral equation of the second kind is obtained for the unknown contact pressure with a coefficient in front of the leading term that is a fairly arbitrary function of the longitudinal coordinate. To solve it the Bubnov-Galerkin projection method is used in which the coordinate elements are chosen to be a system of orthogonal polynomials and delta-shaped functions [1, 2] (variational-difference method), together with an algorithm for the required asymptotic expansions [3] when the above-mentioned coefficient is small. In the special case of an elastic half-space protected by a covering of constant thickness, the results obtained are compared with the corresponding characteristics given in [4].