Puiseux Series Research Articles

Series solutions for the Laguerre and Lane-Emden fractional differential equations in the sense of conformable fractional derivative

In the present paper, we use efficient and simple algorithms of the fractional power series and Adomain polynomial methods that provide effective tools for solving such linear and nonlinear fractional differential equations in the sense of conformable derivative. The first method succeeded in finding the fractional power series solution for the Laguerre linear fractional differential equation with some important special cases. The second method is applied to predict and construct the fractional Adomain approximation solution of the general conformable Lane-Emden fractional differential equation with some interesting linear and nonlinear special cases with their graphs. Finally, the results show that the Adomain approximation solutions of those problems are stable at different values of α and assure that this way can be applied successfully for solving many linear and nonlinear fractional differential equations in conformable sense.

Fractional calculus-based analysis of soil electrical properties

Purpose This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations. Design/methodology/approach This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations. Findings The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result. Originality/value This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation.

Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order

The Newell–Whitehead–Segel equation is one of the most nonlinear amplitude equations that plays a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion, and convection system. In this analysis, a recent numeric-analytic technique, called the fractional residual power series (FRPS) approach, was successfully employed in obtaining effective approximate solutions to the Newell–Whitehead–Segel equation of the fractional sense. The proposed algorithm relies on a generalized classical power series under the Caputo sense and the concept of an error function that systematically produces an analytical solution in a convergent fractional power series form with accurately computable structures, without the need for any unphysical restrictive assumptions. Meanwhile, two illustrative applications are included to show the efficiency, reliability, and performance of the proposed technique. Plotted and numerical results indicated the compatibility between the exact and approximate solution obtained by the proposed technique. Furthermore, the solution behavior indicates that increasing the fractional parameter changes the nature of the solution with a smooth sense symmetrical to the integer-order state.

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

The differential transformation, an approach based on Taylor’s theorem, is proposed as convenient for finding an exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem gives a reliable and expected outcome including the phenomenon of symmetry in choice of orders of the differential transformation of the multi-order system.

SOME COROLLARY FACTS OF THE N-POINT GRAVITATIONAL LENS EQUATION IN A COMPLEX FORM

In the theory of the N-point gravita- tional lens equation, two groups of problems can be dis- tinguished. These are the so-called primal and inverse problems. Primal problems include problems of image definition in a specified lens for a specified source. In- verse problems include problems of determining a lens, source, or multiple images from one or more specified images. Inverse problem have an important applica- tions. We studied the equation of the N-point gravitational lens in a complex form. These studies became the basis for the solution of the inverse problem in the following formulation. N-point gravitational lens has specified. It is necessary to determine all other images from one of the images of a point source in N-point gravitational lens. Determine the necessary and sufficient conditions under which this problem has solutions. The algebraic formulation of the problem has the following form. The equation (of N-point gravitational lens) has specified. It is necessary to solve the problem of solutions unification (to express unequivocally all of the equation solutions through one parameter). To solve the inverse problem, we used methods of al- gebraic geometry and function theory. Branches equa- tions of any algebraic function admit unequivocal pa- rameterization by Puiseux series. The solutions of the N-point gravitational lens equation are algebraic functions defined by a certain irreducible polynomial. That polynomial has unequivocally defined by the N- point gravitational lens equation. Thus, the polyno- mial roots also admits parameterization by Puiseux se- ries. In simple cases, for lenses with a small number of point masses, the solution can be obtained in a sim- pler form. In particular, for the Schwarzschild lens and binary lens, the inverse problem has a solution in radicals.

The complex life of hydrodynamic modes

We study analytic properties of the dispersion relations in classical hydrody- namics by treating them as Puiseux series in complex momentum. The radii of convergence of the series are determined by the critical points of the associated complex spectral curves. For theories that admit a dual gravitational description through holography, the critical points correspond to level-crossings in the quasinormal spectrum of the dual black hole. We illustrate these methods in N = 4 supersymmetric Yang-Mills theory in 3+1 dimensions, in a holographic model with broken translation symmetry in 2+1 dimensions, and in con- formal field theory in 1+1 dimensions. We comment on the pole-skipping phenomenon in thermal correlation functions, and show that it is not specific to energy density correlations.

On the Estimation of Coefficients of Irreducible Factors of Polynomials over a Field of Formal Power Series in Nonzero Characteristic

We discuss some results and problems related to the Newton–Puiseux algorithm and its generalization for nonzero characteristic obtained by the author earlier. A new method is suggested for obtaining effective estimates of the roots of a polynomial in the field of fractional power series in the case of arbitrary characteristic.

Generalized fractional power series solutions for fractional differential equations

The extension of fractional power series solutions for linear fractional differential equations with variable coefficients is considered. Generalized series expansions involving integer powers and fractional powers in the independent variable have recently been shown to provide solutions to certain linear fractional order differential equations. In these studies the generalized series could be formulated as the Cauchy product of a standard power series with integer powers and a fractional power series with powers as multiples of a fractional exponent. Here we show that generalized fractional powers series can provide solutions to a wider class of problems if there is no requirement that the series can be formulated as a Cauchy product.

On the global dynamics and integrability of the Chemostat system

We study a Chemostat system of the form ẋ=−qx+R̃K+yxy,ẏ=(c̃−y)q−R̃ã(K+y)xy,where q>0, R̃>0, K>0, c̃>0 and ã≠0. This system appears in competition modelling in biology. We describe its global dynamics on the Poincaré disc and study its Liouvillian integrability. For the first topic we use the well-known Poincaré compactification theory and for the second one we make use of the Puiseux series to derive the structure of all the irreducible invariant algebraic curves.

A new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients

The main purpose of this paper is to present a new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients using the fractional residual power series method (FRPSM). The fractional derivative is considered in the Caputo sense. This method is based on the generalized Taylor series formula and residual error function. Unlike other analytical methods, FRPSM has a special advantage, that it solves the nonlinear problems without using linearization, discretization, perturbation or any other restrictions. By numerical examples, it is shown that the FRPSM is a simple, effective, and powerful method for finding approximate analytical solutions of nonlinear fractional partial differential equations.

Fractional Gauss hypergeometric differential equation

In this paper, we use the new concept of fractional regular singular point and use the technique of fractional power series to solve the fractional Gauss hypergeometric differential equation. Also, we introduce the forms of the conformable fractional derivative and the integral representation of the fractional Guassian function.

A fractional power series neural network for solving a class of fractional optimal control problems with equality and inequality constraints

ABSTRACTThis paper solved fractional order optimal control problems, in which the dynamic control system involves integer and fractional order derivatives with equality and inequality constraints. According to the Pontryagin minimum principle (PMP) for fractional optimal control problem (FOCP) with fractional derivative in the Caputo sense and by constructing a suitable error function, an unconstrained minimization problem is defined. In the optimization problem, trial solutions are used for the states, Lagrange multipliers and control functions where these trial solutions are constructed by fractional power series neural network models. An error function is then minimized using a numerical optimization scheme where weight parameters (or coefficients of the series) and biases associated with all neurons are unknown. Some computational simulations are discussed in details.

An efficient method for solving nonlinear time-fractional wave-like equations with variable coefficients

In this paper, fractional residual power series method (FRPSM) is effectively applied for finding the approximate analytical solutions of general nonlinear time-fractional wave-like equations with variable coefficients. This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without need of linearization, discretization, perturbation or unrealistic assumptions. Numerical results are given and then they are compared with the exact solutions both numerically and graphically. By numerical examples, it is shown that the FRPSM is very simple, efficient and convenient for solving different forms of nonlinear fractional partial differential equations.

Toward computational algorithm for time-fractional Fokker–Planck models

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

A New Approach for the Solution of Space-Time Fractional Order Heat-Like Partial Differential Equations by Residual Power Series Method

The main concern of this article has been to apply the Residual Power Series Method (RPSM) effectively to find the exact solutions of fractional-order space-time dependent nonhomogeneous partial differential equations in the Caputo sense. Our first step is to reduce fractional-order space-time dependent non-homogeneous partial differential equations to fractional-order space-time dependent homogeneous partial differential equations before applying the proposed method. Obtaining fractional power series solutions of the problem and reproducing the exact solution is the main step. The illustrative examples reveal that RPSM is a very significant and powerful method for obtaining the solution of any-order time-space fractional non-homogeneous partial differential equations in the form of fractional power series.

New Fractional Analytical Study of Three-Dimensional Evolution Equation Equipped With Three Memory Indices

Abstract Herein, analytical solutions of three-dimensional (3D) diffusion, telegraph, and Burgers' models that are equipped with three memory indices are derived by using an innovative fractional generalization of the traditional differential transform method (DTM), namely, the threefold-fractional differential transform method (threefold-FDTM). This extends the applicability of DTM to comprise initial value problems in higher fractal spaces. The obtained solutions are expressed in the form of a γ¯-fractional power series which is a fractional adaptation of the classical Taylor series in several variables. Furthermore, the projection of these solutions into the integer space corresponds with the solutions of the classical copies for these models. The results detect that the suggested method is easy to implement, accurate, and very efficient in (non)linear fractional models. Thus, research on this trend is worth tracking.

Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative.

In this paper, the time-fractional nonlinear dispersive (TFND) partial differential equations (PDEs) in the sense of conformable fractional derivative (CFD) are proposed and analyzed. Three types of TFND partial differential equations are considered in the sense of CFD, which are the TFND Boussinesq, TFND Klein-Gordon, and TFND B(2, 1, 1) PDEs. Solitary pattern solutions for this class of TFND partial differential equations based on the residual fractional power series method is constructed and discussed. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions. The results suggest that the algorithm presented here offers solutions to problems in a rapidly convergent series leading to ideal solutions. Furthermore, the results obtained are like those in previous studies that used other types of fractional derivatives. In addition, the calculations used were much easier and shorter compared with other types of fractional derivatives.

On Average Behaviour of Regular Expressions in Strong Star Normal Form

For regular expressions in (strong) star normal form a large set of efficient algorithms is known, from conversions into finite automata to characterisations of unambiguity. In this paper we study the average complexity of this class of expressions using analytic combinatorics. As it is not always feasible to obtain explicit expressions for the generating functions involved, here we show how to get the required information for the asymptotic estimates with an indirect use of the existence of Puiseux expansions at singularities. We study, asymptotically and on average, the alphabetic size, the size of the [Formula: see text]-follow automaton and of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions.

Basic matrix perturbation theory

In this expository note, we give proofs of several results in finite-dimensional matrix perturbation theory: continuity of the spectrum, regularity of the total eigenprojectors, existence and computation of one-sided directional derivatives of semi-simple eigenvalues, and Puiseux expansions of coalescing eigenvalues. These results are all classical, at least in the case of one-dimensional, analytical perturbations; a standard reference is the treatise of T. Kato, Perturbation theory for linear operators (Springer, 1980). In contrast with Kato, we consider perturbations which are not necessarily smooth, in arbitrary finite dimension, and for coalescing eigenvalues we do not use the notion of multi-valued function. The proofs use Rouché’s theorem, representations of projectors as contour integrals, and the description of conjugacy classes of connected covering maps of the punctured disk.

On (2 + 1)-dimensional physical models endowed with decoupled spatial and temporal memory indices⋆

The current work concerns the development of an analytical scheme to handle (2 + 1) -dimensional partial differential equations endowed with decoupled spatial and temporal fractional derivatives (abbreviated by $ (\alpha,\beta)$ -models). For this purpose, a new bivariate fractional power series expansion has been integrated with the differential transform scheme. The mechanism of the submitted scheme depends mainly on converting the $ (\alpha,\beta)$ -model to a recurrence-differential equation that can be easily solved by virtue of an iterative procedure. This, in turn, reduces the computational cost of the Taylor power series method and consequently introduces a significant refinement for solving such hybrid models. To elucidate the novelty and efficiency of the proposed scheme, several $ (\alpha,\beta)$ -models are solved and the presence of remnant memory, due to the fractional derivatives, is graphically illustrated.