On justification of general polynomial projection method for solving periodic fractional integral equations

© 2015, Pleiades Publishing, Ltd. In this paper we subtantiate a general polynomial projection method for solving periodic integral equations with Holder coefficients and fractional integral Weyl operator in the main part

Generalized interpolating polynomial operator An

© Research India Publications 2015. The article describes the construction of a linear operator which puts into correspondence an arbitrary 2π - periodic function with zero mean trigonometric polynomial. During an operator construction the decomposition in Fourier series, the Weil operator of fractional integration, Lagrange interpolation polynomial, the properties of fractional differentiation and fractional integration are used. An operator type is obtained, the corresponding formula is derived. A formula type is shown taking into account the form of the trigonometric complex numbers. The relationship of the generalized interpolation operator An with Fourier operator Sn is cosidered. The approximation of functions by the means of an obtained polynomial operator and the evaluation of error is verified using a computer algebra system Wolfram Mathematica. The approximation of function by a trigonometric polynom obtained by the derived formulas is conducted for different functions at different values of node numbers. The calculations showed that the difference module between the values of 2π - periodic function with a zero mean value and the values of trigonometric polynomial, constructed with the help of an operator An (φ; t), decrease with the order of α integration (the values 0,5 <α < 1 were considered). The value of the modulus is less if a midportion of the interval is taken, presumably it is related to the fact that the difference φ — Anφ converges on the average. The growth of node number n also make the function approximation better

About the convergence of the general projection polynomial method for a class of periodic fractional-integral equations

© 2014, Pleiades Publishing, Ltd. In this paper we provide justification of the general projection polynomial method for solution of periodical fractional integral equations in two Holder spaces

A variant of the method of quadratures for solving integral equations with fractional integral of Weyl in the main part

© Published under licence by IOP Publishing Ltd.In this paper the method of mechanical quadrature solutions fractional integral equation. Computational scheme quadrature method is based on the quadrature formula of rectangles with equidistant nodes, which is the formula of the highest trigonometric degree of accuracy, using a regularizing parameter. This decision is taken for the approximate trigonometric interpolation polynomial constructed from the values that make up the solution of the quadrature method. The substantiation of the method in Holder spaces

Quadrature formulas for calculating the hadamard integral of a special form

© 2016, International Journal of Pharmacy and Technology. All rights reserved.Currently, the issues related to finding solutions to some economic problems, such as problems pertaining to the queuing systems are of great interest. These problems, in turn, result in, inter alia, the need to calculate the Hadamard integrals of a special form. The study includes the construction of an optimal quadrature formula for the approximate solution to the Hadamard integral of a special form; such formula is selected depending on the singularities of integrals. This is because of the fact that an error of the quadrature formula for a hypersingular integral is a point function of any singularity. There is a movable singularity, when parameters of the quadrature formula do not depend on the singularity’s position, and a fixed singularity, when the quadrature formula nodes are the singularity’s point functions. In this way, choice of the quadrature formula nodes depends on the choice of singularities. Assessment of the constructed quadrature formula’s error is established in the considered density class, and the order of formulas with arbitrary multiplicity nodes is optimized. The main results and conclusions are given; they were obtained for quadrature formulas of hypersingular integrals with the singularities, both movable and fixed, which can be used for solving boundary value problems, especially in simulating the fractional (or fractal) dynamic processes

The simulation of the activity dependent neural network growth

It is currently accepted that cortical maps are dynamic constructions that are altered in response to external input. Experience-dependent structural changes in cortical microcurcuts lead to changes of activity, i.e. to changes in information encoded. Specific patterns of external stimulation can lead to creation of new synaptic connections between neurons. The calcium influxes controlled by neuronal activity regulate the processes of neurotrophic factors released by neurons, growth cones movement and synapse differentiation in developing neural systems. We propose a model for description and investigation of the activity dependent development of neural networks. The dynamics of the network parameters (activity, diffusion of axon guidance chemicals, growth cone position) is described by a closed set of differential equations. The model presented here describes the development of neural networks under the assumption of activity dependent axon guidance molecules. Numerical simulation shows that morpholess neurons compromise the development of cortical connectivity.Comment: 10 pages, 2 figure

On justification of general polynomial projection method for solving periodic fractional integral equations

© 2015, Pleiades Publishing, Ltd. In this paper we subtantiate a general polynomial projection method for solving periodic integral equations with Holder coefficients and fractional integral Weyl operator in the main part

A variant of the method of quadratures for solving integral equations with fractional integral of Weyl in the main part

© Published under licence by IOP Publishing Ltd.In this paper the method of mechanical quadrature solutions fractional integral equation. Computational scheme quadrature method is based on the quadrature formula of rectangles with equidistant nodes, which is the formula of the highest trigonometric degree of accuracy, using a regularizing parameter. This decision is taken for the approximate trigonometric interpolation polynomial constructed from the values that make up the solution of the quadrature method. The substantiation of the method in Holder spaces

On justification of general polynomial projection method for solving periodic fractional integral equations

© 2015, Pleiades Publishing, Ltd. In this paper we subtantiate a general polynomial projection method for solving periodic integral equations with Holder coefficients and fractional integral Weyl operator in the main part

Generalized interpolating polynomial operator An

© Research India Publications 2015. The article describes the construction of a linear operator which puts into correspondence an arbitrary 2π - periodic function with zero mean trigonometric polynomial. During an operator construction the decomposition in Fourier series, the Weil operator of fractional integration, Lagrange interpolation polynomial, the properties of fractional differentiation and fractional integration are used. An operator type is obtained, the corresponding formula is derived. A formula type is shown taking into account the form of the trigonometric complex numbers. The relationship of the generalized interpolation operator An with Fourier operator Sn is cosidered. The approximation of functions by the means of an obtained polynomial operator and the evaluation of error is verified using a computer algebra system Wolfram Mathematica. The approximation of function by a trigonometric polynom obtained by the derived formulas is conducted for different functions at different values of node numbers. The calculations showed that the difference module between the values of 2π - periodic function with a zero mean value and the values of trigonometric polynomial, constructed with the help of an operator An (φ; t), decrease with the order of α integration (the values 0,5 <α < 1 were considered). The value of the modulus is less if a midportion of the interval is taken, presumably it is related to the fact that the difference φ — Anφ converges on the average. The growth of node number n also make the function approximation better