Fractional Power Series Expansion Research Articles

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.

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.

Exceptional point in a degenerate system made of a gyrator and two unstable resonators

We demonstrate that a circuit comprising two unstable LC resonators coupled via a gyrator supports an exceptional point of degeneracy (EPD) with purely real eigenfrequency. Each of the two resonators includes either a capacitor or an inductor with a negative value, showing a purely imaginary resonance frequency when not coupled to the other via the gyrator. With external perturbation imposed on the system, we show analytically that the resonance frequency response of the circuit follows the square-root dependence on perturbation, leading to possible sensor applications. Furthermore, the effect of small losses in the resonators has been investigated, and we show that losses lead to instability. In addition, the EPD occurrence and sensitivity are demonstrated by showing that the relevant Puiseux fractional power series expansion describes the eigenfrequency bifurcation near the EPD. The EPD has the great potential to enhance the sensitivity of a sensing system by orders of magnitude.

High-sensitivity in various gyrator-based circuits with exceptional points of degeneracy

Exceptional points of degeneracy (EPD) can enhance the sensitivity of circuits by orders of magnitude. We show various configurations of coupled LC resonators via a gyrator that support EPDs of second and third-order. Each resonator includes a capacitor and inductor with a positive or negative value, and the corresponding EPD frequency could be real or imaginary. When a perturbation occurs in the second-order EPD gyrator-based circuit, we show that there are two real-valued frequencies shifted from the EPD one, following a square root law. This is contrary to what happens in a Parity-Time (PT) symmetric circuits where the two perturbed resonances are complex valued. We show how to get a stable EPD by coupling two unstable resonators, how to get an unstable EPD with an imaginary frequency, and how to get an EPD with a real frequency using an asymmetric gyrator. The relevant Puiseux fractional power series expansion shows the EPD occurrence and the circuit's sensitivity to perturbations. Our findings pave the way for new types of high-sensitive devices that can be used to sense physical, chemical, or biological changes.

Exceptional degeneracies in traveling wave tubes with dispersive slow-wave structure including space-charge effect

The interaction between a linear electron beam and a guided electromagnetic wave is studied in the context of exceptional points of degeneracy (EPD) supported by such an interactive system, focusing on a linear beam traveling wave tube (TWT) with a realistic helix waveguide slow-wave structure (SWS). The interaction is formulated by an analytical model that is a generalization of the Pierce model, assuming a one-dimensional electron flow along a dispersive single-mode guiding SWS and taking into account space-charge effects. The augmented model using phase velocity and characteristic impedance obtained via full-wave simulations is validated by calculating gain vs frequency and comparing it with that from more complex electron beam simulators. This comparison also shows the accuracy of our new model compared with respect to the nondispersive Pierce model. EPDs are then investigated using the augmented model, observing the coalescence of complex-valued wavenumbers and the system's eigenvectors. The point in the complex dispersion diagram at which the TWT-system starts/ceases to exhibit a convection instability, i.e., a mode starts/ceases to grow exponentially along the TWT, is the EPD. We also demonstrate the EPD existence by showing that the Puiseux fractional power series expansion well approximates the bifurcation of the dispersion diagram at the EPD. This latter concept also explains the “exceptional” sensitivity of the TWT-system to changes in the beam's electron velocity when operating near an EPD.

Exceptional Points of Degeneracy Directly Induced by Space–Time Modulation of a Single Transmission Line

We demonstrate how exceptional points of degeneracy (EPDs) are induced in a single transmission line (TL) directly by applying periodic space-time modulation to the per-unit-length distributed capacitance. In such space-time modulated (STM)-TL, two eigenmodes coalesce into a single degenerate one, in their eigenvalues (wavenumbers) and eigenvectors (voltage-current states) when the system approaches the EPD condition. The EPD condition is achieved by tuning a parameter in the space-time modulation, such as spatial or temporal modulation frequency, or the modulation depth. We unequivocally demonstrate the occurrence of the EPD by showing that the bifurcation of the wavenumber around the EPD is described by the Puiseux fractional power series expansion. We show that the first order expansion is sufficient to approximate well the dispersion diagram, and how this "exceptional" sensitivity of an STM-TL to tiny changes of any TL or modulation parameter enables a possible application as a highly sensitive TL sensor when operating at an EPD.

Solving Parametric Fractional Differential Equations Arising from the Rough Heston Model Using Quasi-Linearization and Spectral Collocation

The rough Heston model has recently attracted the attention of many finance practitioners and researchers as it maintains the basic structure of the classical Heston model while having descriptive capabilities in terms of microstructural foundations of the market. Using the fact that the characteristic function of log-price in this model could be expressed in terms of the solution of a nonlinear parametric fractional Riccati differential equation not admitting a closed-form solution, devising efficient numerical schemes for pricing and calibration under this model has become a crucial need in the computational finance community. Although the fractional Adams method has been used in most of the recent studies on the rough Heston model, this method suffers from some stability and convergence issues in treating the problem. In this paper, we present a numerical method based on Newton--Kantorovich quasi-linearization to solve the nonlinearity issue followed by spectral collocation based on “polyfractonomials" to approximate the fractional derivative in an accurate and efficient manner. We provide sufficient conditions under which our method is convergent and the order of convergence is also obtained. In order to guarantee the specified convergence rate, we first prove some regularity results on the linearized problem and then employ the proposed scheme to solve a practical calibration problem from the SPX options market. The efficiency of the proposed method is illustrated by comparing the obtained results with those of the fractional Adams method as well as a fast hybrid scheme based on fractional power series expansion.

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.

The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions

In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable fractional derivative of order $0 < \alpha<1.$ We then apply the Frobenius method with the fractional power series expansion to obtain two linearly independent solutions of the problem. For certain eigenvalues, the infinite series solutions truncate to obtain the singular and non-singular fractional Laguerre functions. We obtain the fractional Laguerre functions in closed forms, and establish their orthogonality result. The applicability of the new fractional Laguerre functions is illustrated.

Two computational approaches for solving a fractional obstacle system in Hilbert space

The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0<alphaleq1, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

A new iterative method based solution for fractional Black–Scholes option pricing equations (BSOPE)

In this manuscript, a new expansion technique namely residual power series method is used for finding the analytical solution of the Fractional Black–Scholes equation with an initial condition for European option pricing problem. The Black–Scholes formula is important for estimating European call and put option on a non-dividend paying stock in particular when it contains time-fractional derivatives. The fractional derivative is defined in Caputo sense. This technique is based on fractional power series expansion. The convergence analysis of the present method is also deliberated. Example problems are given to examine the efficacy of the proposed method. Obtained solutions are compared with exact solutions solved by other techniques which demonstrate that the present method is robust and easy to implement for other fractional problems arising in science and engineering.

An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering

The purpose of the current work is to provide an analytical solution framework based on extended fractional power series expansion to solve 2D temporal–spatial fractional differential equations. For this purpose, we first present a new trivariate expansion endowed with twofold Caputo-fractional derivatives ordering, namely $$\alpha ,\,\beta \in (0,1]$$ , to study the combined effect of fractional derivatives on both temporal and spatial coordinates. Then, by virtue of this expansion, a parallel scheme of the Taylor power series solution method is utilized to extract both closed-form and supportive approximate series solutions of 2D temporal–spatial fractional diffusion, wave-like, telegraph, and Burgers’ models. The obtained closed-form solutions are found to be in harmony with the exact solutions exist in the literature when $$\alpha =\beta =1$$ , which exhibits the legitimacy and the validity of the proposed method. Moreover, the accuracy of the approximate series solutions is validated using graphical and tabular tools. Finally, a version of Taylor’s Theorem that associated with our proposed expansion is derived in terms of mixed fractional derivatives.

An analytical study of physical models with inherited temporal and spatial memory

Du et al. (Sci. Reb. 3, 3431 (2013)) demonstrated that the fractional derivative order can be physically interpreted as a memory index by fitting the test data of memory phenomena. The aim of this work is to study analytically the joint effect of the memory index on time and space coordinates simultaneously. For this purpose, we introduce a novel bivariate fractional power series expansion that is accompanied by twofold fractional derivatives ordering $\alpha$ , $\beta\in(0,1]$ . Further, some convergence criteria concerning our expansion are presented and an analog of the well-known bivariate Taylor’s formula in the sense of mixed fractional derivatives is obtained. Finally, in order to show the functionality and efficiency of this expansion, we employ the corresponding Taylor’s series method to obtain closed-form solutions of various physical models with inherited time and space memory.

Theory and applications of a more general form for fractional power series expansion

The latent potentialities and applications of fractional calculus present a mathematical challenge to establish its theoretical framework. One of these challenges is to have a compact and self-contained fractional power series representation that has a wider application scope and allows studying analytical properties. In this letter, we introduce a new more general form of fractional power series expansion, based on the Caputo sense of fractional derivative, with corresponding convergence property. In order to show the functionality of the proposed expansion, we apply the corresponding iterative fractional power series scheme to solve several fractional (integro-)differential equations.

Analytic Solution of Spatial-Temporal Fractional Klein-Gordon Equation Arising in Physical Models

Making use of bivariate (α, β)-fractional power series expansion scheme, we provide an analytical solution to the spatial–temporal fractional Klein–Gordon equation that has many applications in physics such as quantum mechanics, solitons, and condensed matter physics. The twofold fractional derivative orders, namely α, β ∈ (0, 1], are described in the Caputo sense. Some examples are given to show the ability and efficiency of the proposed method for solving (non)linear (non)homogeneous Klein–Gordon equation in fractal space. The obtained closed-form solutions are found to be in a consensus with the exact solutions exist in the literature when α, β → 1, which demonstrates the feasibility and validity of the proposed method.

A novel expansion iterative method for solving linear partial differential equations of fractional order

In this manuscript, we implement a relatively new analytic iterative technique for solving time–space-fractional linear partial differential equations subject to given constraints conditions based on the generalized Taylor series formula. The solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations. This method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics, physics, and engineering. Some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality, generality, and accuracy for solving such equations with different constraints conditions. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm.

ANALYTICAL SOLUTION OF TIME-FRACTIONAL TWO-COMPONENT EVOLUTIONARY SYSTEM OF ORDER 2 BY RESIDUAL POWER SERIES METHOD

In this paper, we introduce and formulate a novel study of obtaining the approximate solutions to the generalized time-fractional two-component evolutionary system of order 2 subject to given constraints conditions based on the generalized Taylor series formula. Here, a very recent technique based on the so called residual power series method is extended to handle such system. The solution methodology is based on generating the multiple fractional power series expansion solutions in the form of a rabidly convergent series with minimum size of calculations. A detailed description of the method is given and the obtained results reveal that the technique is a new signi cant method for exploring linear and nonlinear fractional models.

Asymptotic analysis for Merton's problem with transaction costs in power utility case

We revisit the optimal investment and consumption problem with proportional transaction costs. We prove that both the value function and the slopes of the lines demarcating the no-trading region are analytic functions of cube root of the transaction cost parameter. Also, we can explicitly calculate the coefficients of the fractional power series expansions of the value function and the no-trading region.

Sommation effective d’une somme de Borel par séries de factorielles

In this article, we consider the effective resummation of a Borel sum by its associated factorial series expansion. Our approach provides concrete estimates for the remainder term when truncating this factorial series. We then generalize a theorem of Nevanlinna which gives us the natural framework to extend the factorial series method for Borel-resummable fractional power series expansions.

Fractional Power Series Solutions for Systems of Equations

In this paper we extend the explorations in [8] to include the fractional power series expansions of k equations in d variables, where d>k . An analog of Newton's polygon construction which uses the Minkowski sum P of the Newton polytopes P 1 ,...,P k of the k equations is given for computing such series expansions. If the Newton polytopes of these equations are the same, then the common domains of convergence for the solutions correspond to the vertices of a certain fiber polytope Σ(P) . In general, our results suggest the existence of a ``mixed fiber polytope'' of k polytopes. It is also indicated that there may be a relationship between these mixed fiber polytopes and a generalization of the discriminant, which we call the mixed discriminant.