Multiple Power Series Research Articles

Adaptation of conformable residual series algorithm for solving temporal fractional gas dynamics models

In this paper, we introduced, discussed, and investigated analytical-approximate solutions for nonlinear time fractional gas dynamics equations in terms of conformable differential operator. The proposed algorithm relies upon the conformable power series method and residual error of the generalized Taylor series in terms of the conformable sense. This technique provides analytical solutions in the form of rapid and accurate convergent series in terms of the multiple fractional power series with easily computable components. In this direction, error estimation and convergence analysis for solutions of fractional gas dynamics equations are provided as well. Eventually, several physical examples are tested to justify the theoretical portion and give a clear explanation of dynamic systems for the proposed model for different orders of fractional case The obtained numeric-analytic results indicate that the current algorithm is simple, effective, and profitably dealing with the complexity of many nonlinear fractional dispersion problems.

Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

This article presents an effective and simple approximation scheme to analytically approximate solution behavior of a multi‐dimensional reaction–diffusion Brusselator system. This system has applications in cooperative processes of chemical kinetics and biochemical reaction processes. The proposed residual power series method is based on multiple power series expansion and residual error function formulation. The convergence properties and error bounds of the proposed scheme are theoretically established and numerically verified. Two benchmark problems in two‐ and three‐dimensional space are considered to validate the efficiency and accuracy of the proposed scheme. The obtained results verified reliability, accuracy, and simplicity of the proposed scheme against the available results in literature.

Analytic technique for solving temporal time-fractional gas dynamics equations with Caputo fractional derivative

<abstract><p>Constructing mathematical models of fractional order for real-world problems and developing numeric-analytic solutions are extremely significant subjects in diverse fields of physics, applied mathematics and engineering problems. In this work, a novel analytical treatment technique called the Laplace residual power series (LRPS) technique is performed to produce approximate solutions for a non-linear time-fractional gas dynamics equation (FGDE) in a multiple fractional power series (MFPS) formula. The LRPS technique is a coupling of the RPS approach with the Laplace transform operator. The implementation of the proposed technique to handle time-FGDE models is introduced in detail. The MFPS solution for the target model is produced by solving it in the Laplace space by utilizing the limit concept with fewer computations and more accuracy. The applicability and performance of the technique have been validated via testing three attractive initial value problems for non-linear FGDEs. The impact of the fractional order <italic>β</italic> on the behavior of the MFPS approximate solutions is numerically and graphically described. The <italic>j</italic>th MFPS approximate solutions were found to be in full harmony with the exact solutions. The solutions obtained by the LRPS technique indicate and emphasize that the technique is easy to perform with computational efficiency for different kinds of time-fractional models in physical phenomena.</p></abstract>

Continuability of multiple power series into sectorial domain by means of interpolation of coefficients

Continuability of multiple power series into sectorial domain by means of interpolation of coefficients

Approximation of functions of several variables by multidimensional $S$-fractions with independent variables

The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called "multidimensional $S$-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional $S$-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.

On the convergence of multidimensional S-fractions with independent variables

The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$

Local Limit Theorem for the Multiple Power Series Distributions

We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev.

An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative

Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations (FPDEs) corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the (n − 1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.

Convergence of two-dimensional hypergeometric series for algebraic functions

ABSTRACT Description of convergence domains for multiple power series is a quite difficult problem. In 1889 J.Horn showed that the case of hypergeomteric series is more favourable. He found a parameterization formula for surfaces of conjugative radii of such series. But until recently almost nothing was known about the description of convergence domains in terms of functional inequalities relatively moduli of series variables. In this paper we give a such description for hypergeometric series representing solutions to tetranomial algebraic equations. In our study we use the remarkable observation by M. Kapranov (1991) consisting in the fact that the Horn's formulae give a parameterization of discriminant locus for a corresponding A-discriminant. We prove that usually the considered convergence domains are determined by a signle or two inequalities , where ρ is a reduced discriminant.

Development of an N-Multiple Power Series Into an N-Dimensional Regular C-Fraction

We propose one of possible algorithms for the development of a formal N -multiple power series into a functional N -dimensional regular C -fraction corresponding to this series. In this case, we use the algorithms of development into the corresponding and two-dimensional corresponding fractions for power and double power series.

On the distribution of multiple power series regularly varying at the boundary point

Abstract Let B(x) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1)n and diverges at the point 1 = (1, …, 1). Random vectors (r.v.)ξx such that ξx has distribution of the power series B(x) type is studied. The integral limit theorem for r.v. ξx as x ↑ 1 is proved under the assumption that B(x) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B(x) are one-sided weakly oscillating at infinity.

Continuation of multiple power series in a sectorial domain

AbstractFor multiple power series centered at the origin we consider the problem of its analytic continuability into a sectorial domain. The condition for continuability is formulated in terms of a holomorphic function that interpolates the series coefficients. For series in one variable this problem has been studied in the works of E. Lindelöf, N. Arakelian, and others.

Multidimensional Associated Fractions with Independent Variables and Multiple Power Series

We establish the conditions of existence and uniqueness of a multidimensional associated fraction with independent variables corresponding to a given formal multiple power series and deduce explicit relations for the coefficients of this fraction. The relationship between the multidimensional associated fraction and the multidimensional J-fraction with independent variables is demonstrated. The convergence of the multidimensional associated fraction with independent variables is investigated in some domains of the space ℂN. We construct the expansions of some functions into the corresponding two-dimensional associated fractions with independent variables and demonstrate the efficiency of approximation of the obtained expansions by convergents.

The Approximation Solutions for Higher Dimensional Integro-Differential Equations

This work deals with approximation solutions to a type of integro-differential equations in several complex variables. It concerns the Cauchy formula on higher dimensional domains. In our study, we make use of multiple power series expansions and an iterative computation method to solve a kind of integro-differential equation. We introduce a symmetrized topology product area which is called a bicylinder. We expand functions and derivatives of them to power series. Moreover we obtain unknown functions by comparing coefficients of the series on both sides of equations. We express the approximation solutions by a regular product of matrixes.

Representation of a quotient of solutions of a four-term linear recurrence relation in the form of a branched continued fraction

The quotient of two linearly independent solutions of a four-term linear recurrence relation is represented in the form of a branched continued fraction with two branches of branching by analogous with continued fractions. Formulas of partial numerators and partial denominators of this branched continued fraction are obtained. The solutions of the recurrence relation are canonic numerators and canonic denominators of $\mathcal{B}$-figured approximants. Two types of figured approximants $\mathcal{A}$-figured and $\mathcal{B}$-figured are often used. A $n$th $\mathcal{A}$-figured approximant of the branched continued fraction is obtained by adding a next partial quotient to the $(n-1)$th $\mathcal{A}$-figured approximant. A $n$th $\mathcal{B}$-figured approximant of the branched continued fraction is a branched continued fraction that is a part of it and contains all those elements that have a sum of indexes less than or equal to $n$. $\mathcal{A}$-figured approximants are widely used in proving of formulas of canonical numerators and canonical denominators in a form of a determinant, $\mathcal{B}$-figured approximants are used in solving the problem of corresponding between multiple power series and branched continued fractions. A branched continued fraction of the general form cannot be transformed into a constructed branched continued fraction. For calculating canonical numerators and canonical denominators of a branched continued fraction with $N$ branches of branching, $N>1$, the linear recurrent relations do not hold. $\mathcal{B}$-figured convergence of the constructed fraction in a case when coefficients of the recurrence relation are real positive numbers is investigated.

Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series

AbstractIn the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

On the convergence of multidimensional S-fractions with independent variables

In this paper, we investigate the convergence of multidimensional S-fractions with independent variables, which are a multidimensional generalization of S-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. For establishing the convergence criteria, we use the convergence continuation theorem to extend the convergence, already known for a small region, to a larger region. As a result, we have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional S-fraction with independent variables. And, also, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional S-fraction with independent variables. In addition, we have obtained two new convergence criteria for S-fractions as a consequences from the above mentioned results.

On the convergence of multidimensional regular C-fractions with independent variables

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

Development of the Theory of Branched Continued Fractions in 1996–2016

We perform the analysis of investigations in the theory of branched continued fractions carried out for the last 20 years in the directions developed under the general supervision of Prof. V. Ya. Skorobohat’ko (18.07.1927–04.07.1996) including, in particular, the interpolation of functions of several variables by branched continued fractions, the determination of efficient criteria of convergence and computational stability of these fractions, the correspondence between multiple power series and functional branched continued fractions, the investigation of various classes of functional fractions, and the application of branched continued fractions.

On the convergence criterion for branched continued fractions with independent variables

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.