Puiseux Series Research Articles

Existence and Uniqueness of Solution Represented as Fractional Power Series for the Fractional Advection–Dispersion Equation

The fractional advection–dispersion equation is used in groundwater hydrology for modeling the movements of contaminants/solute particles along with flowing groundwater at the seepage velocity in porous media. This model is used for the prediction of the transport of nonreactive dissolved contaminants in groundwater. This paper establishes the existence and the uniqueness of solutions represented as fractional bi-variate power series of some initial-value problems and boundary-value problems for the fractional advection–dispersion equation. Moreover, a method to approximate the solutions using fractional polynomials in two variables and to evaluate the errors in a suitable rectangle is designed. Illustrative examples showing the applicability of the theoretical results are presented.

A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].

Hahn series and Mahler equations: Algorithmic aspects

AbstractMany articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well‐ordered receptacle for the supports of the potential Hahn series solutions.

Fractional analysis of non-linear fuzzy partial differential equations by using a direct procedure

In this study, an accurate analytical solution is presented for fuzzy FPDEs. It is done by using a novel method called the Laplace-residual power series (LRPSM) to build a series solution to the given problems. The fundamental instruments of the employed method are the Laplace transform, fractional Laurent, and fractional power series. Using the idea of a limit at infinity, we provide a series solution to a fuzzy FPDE with quick convergence and simple coefficient finding. We analyze three cases to obtain approximate and exact solutions to show the effectiveness and reliability of the Laplace- residual power series approach. To demonstrate the accuracy of the suggested procedure, we compare the findings to the real data.

Iterative solutions for nonlinear equations via fractional derivatives: adaptations and advances

In recent years, the utilization of fractional calculus has witnessed a notable surge across various scientific and engineering domains. This manuscript delves into the exploration of adapted iterative techniques tailored for solving nonlinear equations, capitalizing on the diverse range of derivatives available for addressing different problem contexts. We scrutinize previously developed iterative methods, enhancing their efficacy by introducing an auxiliary parameter to the root search process for nonlinear equations (NLE), alongside a fixed order of fractional derivatives. The selection of the auxiliary parameter is confined to the interval (0,1] for convenience. A thorough convergence analysis is conducted, employing a fractional power series expansion of f(x) in terms of fractional derivatives. Subsequently, a series of NLEs is solved to showcase and contrast the efficiency of our proposed methods with established iterative techniques. This refined abstract aims to succinctly elucidate the objectives and contributions of our study, providing readers with a clearer understanding of the manuscript's scope and significance.

Some Properties of the Functions Representable as Fractional Power Series

The α-fractional power moduli series are introduced as a generalization of α-fractional power series and the structural properties of these series are investigated. Using the fractional Taylor’s formula, sufficient conditions for a function to be represented as an α-fractional power moduli series are established. Beyond theoretical formulations, a practical method to represent solutions to boundary value problems for fractional differential equations as α-fractional power series is discussed. Finally, α-analytic functions on an open interval I are defined, and it is shown that a non-constant function is α-analytic on I if and only if 1/α is a positive integer and the function is real analytic on I.

On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations

We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as a fractional power series. We use, in the real field, the method of majorants and the analytic version of the implicit-function theorem, in a way that circumvents difficulties associated to fractional calculus. Some corollaries on continuity are derived, with computational examples for illustration, and a discussion on fractional partial differential equations is included with a case study and counterexamples. Open problems are raised at the end.

On the periodicity of singular vectors and the holomorphic block-circulant SVD on the unit circumference

We investigate the singular value decomposition of a rectangular matrix that is analytic on the complex unit circumference, which occurs, e.g., with the matrix of transfer functions representing a broadband multiple-input multiple-output channel. Our analysis is based on the Puiseux series expansion of the eigenvalue decomposition of analytic para-Hermitian matrices on the complex unit circumference. We study the case in which the rectangular matrix does not admit a full analytic singular value factorization, either due to partly multiplexed systems or to sign ambiguity. We show how to find an SVD factorization in the ring of Puiseux series where each singular value and the associated singular vectors present the same period and multiplexing structure, and we prove that it is always possible to find an analytic pseudo-circulant factorization, meaning that any arbitrary arrangements of multiplexed systems can be converted into a parallel form. In particular, one can show that the sign ambiguity can be overcome by allowing non-real holomorphic singular values.

Heat Transfer in Hybrid Nanofluid Flow Past an Infinite Orthogonal Plate with Biot Number and Velocity Slip Effects

Nanofluids are made up of nanoscale particles such as copper, carbides, graphite, and alumina, which help base fluids transmit heat more efficiently. These nanofluids have a broad range of applications in today’s framework of cooling and heating, solar-powered cells, hybrid-powered engines, new fuel generation, cancer therapy, and pharmaceuticals. This present investigation emphasizes the importance of a specific type of fluid called a hybrid nanofluid, which consists of (Cu and TiO2) nanoparticles suspended in H2O (water). This fluid is subjected to a combination of several complex phenomena of heat transfer in hybrid nanofluid flow past an infinite orthogonal plate, including velocity slip and Biot number using fractional calculus. The system of governing partial differential equations (PDEs) are transformed into a set of first-order ordinary differential equations (ODEs) using appropriate mathematical transformations. These equations were then solved numerically using fractional power series method (FPSM). FPSM is a very powerful method in solving fractional differential equations arising from different types of scientific problems. The study investigated the behaviour of velocity profiles, temperature, skin friction and heat transfer for various values of the parameters involved.The rate of heat transfer decreases with increasing the hybrid nanofluid parameter but it increases with increasing the fractional order, velocity slip and Biot number Additionally, the skin friction decreases with increasing both the hybrid nanofluid parameter and velocity slip, but it increases with increasing of the fractional order. However, there is no change in skin friction when the Biot number increases. It is also clear that the velocity increases for increasing both fractional order, velocity slip, it decreases for increasing the nanofluid parameter. The temperature profile rises when both the nanofluid parameter and Biot number increase. Also temperature profile decreases when the values of the fractional order and slip parameters increase.

A Fuzzy Fractional Power Series Approximation and Taylor Expansion for Solving Fuzzy Fractional Differential Equation

Fuzzy fractional differential has the strength to capture the senses of memory and uncertainty simultaneously involved in dynamical systems. However, a solution for fuzzy fractional differential equations is not always found regularly. This paper discusses a numerical solution approach for the fuzzy fractional differential equation using power series approximation with a fuzzy fractional counterpart of Taylor’s theorem. Caputo’s definition of the fractional derivative and generalized Hukuhara difference are used to describe the fuzzy differential equation in this paper. Utilization of the generalized Hukuhara difference for the fuzzy valued function ensures the uniqueness and boundedness of the fuzzy solution in parametric form.

Simulation of exceptional-point systems on quantum computers for quantum sensing

There has been debate around applicability of exceptional points (EPs) for quantum sensing. To resolve this, we first explore how to experimentally implement the non-Hermitian non-diagonalizable Hamiltonians, which exhibit EPs, in quantum computers that run on unitary gates. We propose to use an ancilla-based method in this regard. Next, we show how such Hamiltonians can be used for parameter estimation using quantum computers and analyze its performance in terms of the quantum Fisher information (QFI) at EPs, both without noise and in the presence of noise. It is well known that QFI of a parameter to be estimated is inversely related to the variance of the parameter by the quantum Cramer–Rao bound. Therefore, the divergence of the QFI at EPs promises sensing advantages. We experimentally demonstrate in a cloud quantum architecture and theoretically show, using Puiseux series, that the QFI indeed diverges in such EP systems that were earlier considered to be non-divergent.

THE FRACTAL STRUCTURE OF ANALYTICAL SOLUTIONS TO FRACTIONAL RICCATI EQUATION

Analytical solutions to the fractional Riccati equation are considered in this paper. Solutions to fractional differential equations are expressed in the form of fractional power series in the Caputo algebra. It is demonstrated that solutions to higher-order Riccati fractional equations inherit some solutions from lower-order Riccati equations under special initial conditions. Such nested and fractal-like structure of solutions is investigated by means of analytical fractional differentiation operator techniques and computational experiments.

From formal to actual Puiseux series solutions of algebraic differential equations of first order

The existence and uniqueness of formal Puiseux series solutions of non-autonomous algebraic differential equations of the first order at a nonsingular point of the equation is proven. The convergence of those Puiseux series is established. Several new examples are provided. Relationships to the celebrated Painleve theorem and lesser-known Petrovic's results are discussed in detail.

Solution of fractional modified Kawahara equation: a semi-analytic approach

The present study examines a semi-analytical method known as the Fractional Residual Power Series Method for obtaining solutions to the non-linear, time-fractional Kawahara and modified Kawahara equations. These equations are fifth-order, non-linear partial differential equations that arise in the context of shallow water waves. The analytical process and findings are compared with those obtained from the well-known Variational Iteration Method (VIM) and Homotopy Perturbation Method (HPM). The results obtained from the Fractional Residual Power Series Method are found to be more efficient, reliable, and easier to implement compared to other analytical and semi-analytical methods.

Numerical investigation of a fractional model of a tumor-immune surveillance via Caputo operator

Fractional calculus has become a potent tool for simulating the complexity of interactions in tumor-immune system dynamics. This paper investigates the existence and uniqueness of its approximation solutions of a fractional model of tumor-immune surveillance. This analysis is essential for proving the validity and dependability of the model and provides a more in-depth understanding of the dynamics of the tumor-immune surveillance system. Second, we examine the fractional model's numerical features. We utilize a numerical technique called the Laplace residual power series method to address the equations' complexity and nonlinearity. By defining the answers as a fractional power series, this method enables us to efficiently approximate the solutions. The use of this technique enables us to investigate the temporal evolution of the tumor-immune system, offering important insights into the stability and behavior of the system. We assess the effectiveness of the Laplace residual power series method in locating approximations through a series of thorough numerical simulations. To ensure the correctness and dependability of our method, we compare the numerical findings whenever possible with well-known analytical solutions.

Anisotropic Fractional Cosmology: K-Essence Theory

In the particular configuration of the scalar field k-essence in the Wheeler–DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is β=2α2α−1. This fractional equation belongs to different intervals depending on the value of the barotropic parameter; when ωX∈[0,1], the order belongs to the interval 1≤β≤2, and when ωX∈[−1,0), the order belongs to the interval 0<β≤1. In the quantum scheme, we introduce the factor ordering problem in the variables (Ω,ϕ) and its corresponding momenta (ΠΩ,Πϕ), obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time T(τ); however, in the dust era we found a closed solution in the gauge time τ.

A Systematic Approach to Delay Functions

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions.

Complex solutions for nonlinear fractional partial differential equations via the fractional conformable residual power series technique and modified auxiliary equation method

The fractional-order nonlinear Gardner and Cahn–Hilliard equations are often used to model ultra-short burst beams of light, complex fields of optics, photonic transmission systems, ions, and other fields of mathematical physics and engineering. This study has two main objectives. First, the main objective of this investigation is to solve the fractional-order nonlinear Gardner and Cahn–Hilliard equations by using the modified auxiliary equation method, which is not found in the literature. Second, the exact and approximate solutions of these equations are obtained by utilizing the fractional conformable residual power series algorithm and the modified auxiliary equation method. For the analytical and numerical solutions to two equations, we employ two separate techniques and establish consistency between the precise answers that are derived and the compatible numerical solution. To the best of our knowledge, this method of solving equations has never been investigated in this manner. The 2D and 3D contours have been defined using appropriate parametric values to support the physical compatibility of the results. The assessed findings suggested that the approach used in this study to recover inclusive and standard solutions is approachable, efficient, and faster in computing and can be considered a useful tool in resolving more complex phenomena that arise in the field of engineering, mathematical physics, and optical fiber.

On Abel’s Problem and Gauss Congruences

Abstract A classical problem due to Abel is to determine if a differential equation $y^{\prime}=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta $ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta $, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta $ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty \}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y^{\prime}=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta (x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y^{\prime}=\eta y$ with an algebraic solution when $x\eta (x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks.

New solutions of fractional 4D chaotic financial model with optimal control via He-Laplace algorithm

The objective of current investigation is to propose a solution to predict the interest rate, investment demand, and price index with optimal control in a fractional financial 4D chaotic model. He-Laplace method (HLM) is introduced with fractional derivative in Caputo sense to characterize the memory effect of the 4D chaotic model. For validation and comparison purposes, the given financial model is also solved through fractional residual power series algorithm. Analysis revealed that HLM provide improved results as compared to RPSA. Model is also analyzed graphically for interest rate, investment demand, price index and input control in fractional environment to understand the physical behavior of the model. The impact of variations in saving amount, cost per investment, and elasticity in demand are also presented through contours. It is reported that initially the interest rate, investment demand and price index are uniform, but later on drastic increase have been observed. Analysis also revealed that proposed methodology is stable and performed exceptionally well in chaotic scenarios, and hence can be extended to other complex models.