Puiseux Series Research Articles

Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

Solution of Fractional Partial Differential Equations Using Fractional Power Series Method

In this paper, we are presenting our work where the noninteger order partial differential equation is studied analytically and numerically using the noninteger power series technique, proposed to solve a noninteger differential equation. We are familiar with a coupled system of the nonlinear partial differential equation (NLPDE). Noninteger derivatives are considered in the Caputo operator. The fractional-order power series technique for finding the nonlinear fractional-order partial differential equation is found to be relatively simple in implementation with an application of the direct power series method. We obtained the solution of nonlinear dispersive equations which are used in electromagnetic and optics signal transformation. The proposed approach of using the noninteger power series technique appears to have a good chance of lowering the computational cost of solving such problems significantly. How to paradigm an initial representation plays an important role in the subsequent process, and a few examples are provided to clarify the initial solution collection.

Analysis of Solutions of Some Multi-Term Fractional Bessel Equations

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space Cp is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation $$\sum\limits_{i = 1}^m {{d_i}{x^{{\alpha _i}}}} {D^{{\alpha _i}}}u\left( x \right) + \left( {{x^\beta } - {v^2}} \right)u\left( x \right) = 0,\,{\alpha _i} >0,\,\beta >0,$$ which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients di. Our theory leads to a threshold admissible value for ν2, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.

Computing Puiseux series: a fast divide and conquer algorithm

Let F∈𝕂[X,Y] be a polynomial of total degree D defined over a perfect field 𝕂 of characteristic zero or greater than D. Assuming F separable with respect to Y, we provide an algorithm that computes all singular parts of Puiseux series of F above X=0 in an expected Ø ˜(Dδ ) operations in 𝕂, where δ is the valuation of the resultant of F and its partial derivative with respect to Y. To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in 𝕂[[X]][Y] up to an arbitrary precision X N with Ø ˜(D(δ +N)) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with Ø ˜(D 3 ) arithmetic operations and, if 𝕂=ℚ, with Ø ˜((h+1)D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F.

The Fractional Residual Power Series Method for Solving a System of Linear Fractional Fredholm Integro-differential Equations

In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations. The significant role of this system in various fields has attracted the attention of researchers for a decade. The definition of fractional derivative here is described in the Caputo sense. The proposed method relies on the generalized Taylor series expansion as well as the fact that the fractional derivative of stationary is zero. The process starts by constructing a residual function by supposing the finite order of an approximate power series solution that prescribes the initial conditions. Then, utilizing some conditions, the residual functions are converted to a linear system for the power series coefficients. Solving the linear system reveals the coefficients of the fractional power series solution. Finally, by substituting these coefficients into the supposed form of a solution, the approximate fractional power series solutions are derived. This technique has the advantage of being able to be applied directly to the problem and spending less time on computation. It is not only an easy method for implementation of the problem, but also provides productive results after a few iterations. Some problems with known solutions emphasize the procedure's simplicity and reliability. Moreover, the obtained exact solution demonstrated the efficiency and accuracy of the presented method.

Center problem and ν-cyclicity of polynomial zero-Hopf singularities with non-singular rotation axis

We consider three-dimensional polynomial families of vector fields parameterized by the admissible coefficients having a fixed zero-Hopf equilibrium and a non-singular rotation axis through it. We are interested in the periodic ν -orbits, that is, those orbits that makes a fixed arbitrary number ν (or a divisor of ν ) of revolutions about the rotation axis and then returns to the initial point closing the orbit. We develop a Bautin-type method to study the ν -cyclicity of the equilibrium, that is, the maximum number of small-amplitude ν -limit cycles (isolated periodic ν -orbits) that can be made to bifurcate from the equilibrium by moving the parameters of the family restricted to some open semi-algebraic sets. The method uses branching theory based on Newton-Puiseux Theorem to get a finite number of an analytic reduced one-dimensional Poincaré maps with associated Bautin ideal on certain Noetherian ring of rational functions on an extended parameter space. We derive global upper bounds on the number of bifurcated ν -limit cycles even in the infinite codimension case for which the perturbation of a local two-dimensional invariant manifold through the singularity completely foliated by periodic ν -orbits needs to be performed. A cubic normal form serves as an example of this procedure.

Anomalous-order exceptional point and non-Markovian Purcell effect at threshold in one-dimensional continuum systems

For a system consisting of a quantum emitter coupled near threshold (band edge) to a one-dimensional continuum with a van Hove singularity in the density of states, we demonstrate general conditions such that a characteristic triple level convergence occurs directly on the threshold as the coupling $g$ is shut off. For small $g$ values the eigenvalue and norm of each of these states can be expanded in a Puiseux expansion in terms of powers of $g^{2/3}$, which suggests the influence of a third-order exceptional point. However, in the actual $g \rightarrow 0$ limit, only two discrete states in fact coalesce as the system can be reduced to a $2 \times 2$ Jordan block; the third state instead merges with the continuum. Moreover, the decay width of the resonance state involved in this convergence is significantly enhanced compared to the usual Fermi golden rule, which is consistent with the Purcell effect. However, non-Markovian dynamics due to the branch-point effect are also enhanced near the threshold. Applying a perturbative analysis in terms of the Puiseux expansion that takes into account the threshold influence, we show that the combination of these effects results in quantum emitter decay of the unusual form $1 - C t^{3/2}$ on the key timescale during which most of the decay occurs. We then present two conditions that must be satisfied at the threshold for the anomalous exceptional point to occur: the density of states must contain an inverse square-root divergence and the potential must be non-singular. We further show that when the energy of the quantum emitter is detuned from threshold, the anomalous exceptional point splits into three ordinary exceptional points, two of which appear in the complex-extended parameter space. These results provide deeper insight into a well-known problem in spontaneous decay at a photonic band edge.

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.

Criteria for minimization of spectral abscissa of time-delay systems

Spectral abscissa (SA) is defined as the real part of the rightmost characteristic root(s) of a dynamical system, and it can be regarded as the decaying rate of the system, the smaller the better from the viewpoint of fast stabilization. Based on the Puiseux series expansion of complex-valued functions, this paper shows that the SA can be minimized within a given delay interval at values where the characteristic equation has repeated roots with multiplicity 2 or 3. Four sufficient conditions in terms of the partial derivatives of the characteristic function are established for testing whether the SA is minimized or not, and they can be tested directly and easily.

A hybrid augmented compact finite volume method for the Thomas–Fermi equation

A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm.

Universal axion backreaction in flux compactifications

We study the backreaction effect of a large axion field excursion on the saxion partner residing in the same mathcal{N} = 1 multiplet. Such configurations are relevant in attempts to realize axion monodromy inflation in string compactifications. We work in the complex structure moduli sector of Calabi-Yau fourfold compactifications of F-theory with four-form fluxes, which covers many of the known Type II orientifold flux compactifications. Noting that axions can only arise near the boundary of the moduli space, the powerful results of asymptotic Hodge theory provide an ideal set of tools to draw general conclusions without the need to focus on specific geometric examples. We find that the boundary structure engraves a remarkable pattern in all possible scalar potentials generated by background fluxes. By studying the Newton polygons of the extremization conditions of all allowed scalar potentials and realizing the backreaction effects as Puiseux expansions, we find that this pattern forces a universal backreaction behavior of the large axion field on its saxion partner.

A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations

Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the solutions. The two approaches are illustrated and compared by means of four numerical examples.

Explicit and Approximate Solutions for the Conformable-Caputo Time-Fractional Diffusive Predator–Prey Model

The aim of this work is twofold. First, we seek functional explicit solutions to the conformable-time fractional predator–prey model by adapting the extended Kudryashov method. Second, we find fractional power series solution to the same model when the fractional derivative is of Caputo type. Graphical analysis will be presented to serve two features; to recognize the physical shapes and the propagations of the predator–prey densities, and to observe the impact of the fractional derivatives acting in the model.

Semi-decoupling hybrid asymptotic and augmented finite volume method for nonlinear singular interface problems

An accurate semi-decoupling numerical method has been proposed for nonlinear singular differential equation with interface, which combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to decouple the singular interface problem and get high order accurate numerical solution. Key to our proposed new method is introducing three augmented variables involving the interface and the singularity, and reconstructing the representative of the solution as the Puiseux series expansions on the interface to decouple original problem as two standard nonlinear singular problems with the second jump condition. In this way, the augmented variables related with semi-analytic solutions near the interface and numerical solutions can be simultaneously solved in the remaining interval on both sides of the interface. It demonstrates that our method does not take more heavier works for handling jump conditions like other methods, and is independent of the interface and jump ratio. A rigorous error estimate for the solution of nonlinear singular differential equation with interface and augmented variables is obtained. Numerical experiments for those singular differential equations with interface confirm the theoretical analysis and accuracy of the new approach. In particular, an interesting example with blow-up coefficient at singular point shows that our approach can be extended to numerically solve strongly singular interface problem.

Efficient Estimation of Roots from the Field of Fractional Power Series of a Given Polynomial 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 efficient estimates of the roots of a polynomial in the field of fractional power series in the case of arbitrary characteristic.

Power series solution of the fractional logistic equation

Using a series of fractional powers we present a representation of the solution to the fractional logistic equation is presented. To simplify we consider the simplest case and prove that the power series is indeed the exact solution. Some numerical approximations are presented to show the good approximations obtained by truncating the fractional power series.

Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation

Abstract The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

The method of Puiseux series and invariant algebraic curves

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems [Formula: see text], [Formula: see text] with [Formula: see text]. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Liénard dynamical systems with [Formula: see text] can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.

Fractional power series and the method of dominant balances

This paper derives an explicit formula for a type of fractional power series, known as a Puiseux series, arising in a wide class of applied problems in the physical sciences and engineering. Detailed consideration is given to the gaps which occur in these series (lacunae); they are shown to be determined by a number-theoretic argument involving the greatest common divisor of a set of exponents appearing in the Newton polytope of the problem, and by two number-theoretic objects, called here Sylvester sets, which are complements of Frobenius sets. A key tool is Faà di Bruno’s formula for high derivatives, as implemented by Bell polynomials. Full account is taken of repeated roots, of arbitrary multiplicity, in the leading-order polynomial which determines a fractional-power expansion, namely the facet polynomial. For high multiplicity, the fractional powers are shown to have large denominators and contain irregularly spaced gaps. The orientation and methods of the paper are those of applications, but in a concluding section we draw attention to a more abstract approach, which is beyond the scope of the paper.