Power Series Solution Research Articles

On the convergence of generalized power series solutions of q-difference equations

A sufficient condition for the convergence of a generalized formal power series solution to an algebraic q-difference equation is provided. The main result leans on a geometric property related to the semi-group of (complex) power exponents of such a series. This property corresponds to the situation in which the small divisors phenomenon does not arise. Some examples illustrating the cases where the obtained sufficient condition can or cannot be applied are also depicted.

Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients

We are interested in the Gevrey properties of the formal power series solution in time of the inhomogeneous semilinear heat equation with a power-law nonlinearity in $1$-dimensional time variable $t\in\mathbb{C}$ and $n$-dimensional spatial variable $x\in\mathbb{C}^n$ and with analytic initial condition and analytic coefficients at the origin $x=0$. We prove in particular that the inhomogeneity of the equation and the formal solution are together $s$-Gevrey for any $s\geq1$. In the opposite case $s<1$, we show that the solution is $1$-Gevrey at most while the inhomogeneity is $s$-Gevrey, and we give an explicit example in which the solution is $s'$-Gevrey for no $s'<1$.

Summability of the Formal Power Series Solutions of a Certain Class of Inhomogeneous Partial Differential Equations with a Polynomial Semilinearity and Variable Coefficients

In this article, we investigate the summability of the formal power series solutions in time of a certain class of inhomogeneous partial differential equations with a polynomial semilinearity, and with variable coefficients. In particular, we give necessary and sufficient conditions for the k-summability of the solutions in a given direction, where k is a positive rational number entirely determined by the linear part of the equation. These conditions generalize the ones given by the author for the linear case (Remy in J Dyn Control Syst 22(4):693–711, 2016; J Dyn Control Syst 23(4):853–878, 2017) and for the semilinear heat equation (Remy in J Math Anal Appl 494(2):124619, 2021). In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof our main theorem.

Lie Symmetry Analysis, Conservation Laws, Power Series Solutions, and Convergence Analysis of Time Fractional Generalized Drinfeld-Sokolov Systems

In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some figures are presented for explicit solutions.

Quantum Implications of Non-Extensive Statistics

Exploring the analogy between quantum mechanics and statistical mechanics, we formulate an integrated version of the Quantropy functional. With this prescription, we compute the propagator associated to Boltzmann–Gibbs statistics in the semiclassical approximation as K=F(T)exp(iScl/ℏ). We determine also propagators associated to different nonadditive statistics; those are the entropies depending only on the probability S± and Tsallis entropy Sq. For S±, we obtain a power series solution for the probability vs. the energy, which can be analytically continued to the complex plane and employed to obtain the propagators. Our work is motivated by the work of Nobre et al. where a modified q-Schrödinger equation is obtained that provides the wave function for the free particle as a q-exponential. The modified q-propagator obtained with our method leads to the same q-wave function for that case. The procedure presented in this work allows to calculate q-wave functions in problems with interactions determining nonlinear quantum implications of nonadditive statistics. In a similar manner, the corresponding generalized wave functions associated to S± can also be constructed. The corrections to the original propagator are explicitly determined in the case of a free particle and the harmonic oscillator for which the semiclassical approximation is exact, and also the case of a particle with an infinite potential barrier is discussed.

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.

Conservation laws and series solutions of variable coefficient time fractional Kawahara equation

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory solitary wave propagation in dispersive media. The similarity reductions and explicit solutions are derived by using Lie symmetry method. Further, using Ibragimov's theorem conservation laws for this equation are discussed.

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.

Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.

Optimal algebra and power series solution of fractional Black-Scholes pricing model

In the present article, we consider the time fractional Black-Scholes pricing model; governing in the fractal transmission system and describing the transaction costs in fractal market. From the application of symmetry analysis, we derive the infinitesimal transformations involving some arbitrary parameters; under which the equation remain invariant. Next, we write the set of operators corresponding to each of the parameter and under the full adjoint action, we construct and classify the finite dimensional optimal system of Lie algebras. Further, the similarity variables associated with one of the optimal algebra are used to reduce the governing fractional partial differential equations (FPDEs) to fractional ordinary differential equation (FODEs). Subsequently, we obtain the power series solution for the reduced FODEs which in turn produces the solution for the given FPDEs. Finally, we analyze its convergence and studied its behavior graphically under the influence of the fractional derivative.

Stability analysis, solitary wave and explicit power series solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation in a multicomponent plasma

PurposeThe purpose of this paper is to study the stability analysis and optical solitary wave solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation, which are derived from a multicomponent plasma with nonextensive distribution.Design Methodology ApproachBased on the ansatz and sub-equation theories, the authors use a direct method to find stability analysis and optical solitary wave solutions of the (2 + 1)-dimensional equation.FindingsBy considering the ansatz method, the authors successfully construct the bright and dark soliton solutions of the equation. The sub-equation method is also extended to find its complexitons solutions. Moreover, the explicit power series solution is also derived with its convergence analysis. Finally, the influences of each parameter on these solutions are discussed via graphical analysis.Originality ValueThe dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional nonlinear Schrödinger equation type nonlinear wave fields.

Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation

• Nonlinear Modified Gardner Equation • Lie group analysis • Power series solution • Conservation laws • quantum electron-positron-ion magneto plasmas • Ion-acoustic waves In this paper, the nonlinear modified Gardner (mG) equation is under consideration which represents the super nonlinear proliferation of the ion-acoustic waves and quantum electron-positronion magneto plasmas. The considered model is investigated with the help of Lie group analysis. Lie point symmetries are computed under the invariance criteria of Lie groups and symmetry group for each generator is reported. Furthermore, the one-dimensional optimal system of subalgebras is developed by adjoint technique and then we compute the similarity reductions corresponding to each vector field present in the optimal system, with the help of similarity reduction method we have to convert the PDE into the ODE. Some exact explicit solutions of obtained ordinary differential equations were constructed by the power series technique. With the aid of the Galilean transformation, the model is transformed into a planer dynamical system and the bifurcation behaviour is recorded. All practicable types of phase portraits with regard to the parameters of the problem considered are plotted. Meantime, sensitivity is observed by utilizing sensitivity analysis. In addition, the influence of physical parameters is studied by the application of an extrinsic periodic power. With additional perturbed term, quasi-periodic and quasi-periodic-chaotic behaviours is reported.

Power series solution and conservation laws of fractional Whitham-Broer-Kaup equation

Power series solution and conservation laws of fractional Whitham-Broer-Kaup equation

Power series solution and conservation laws of fractional Whitham-Broer-Kaup equation

Power series solution and conservation laws of fractional Whitham-Broer-Kaup equation

On the Formal Solutions of a Type of Singular Partial Differential Equations

In this paper, we research a partial differential equation with three complex variables. According to the theory of singular differential equations, it is proved by direct calculation that it has a special form of formal power series solution.

Arquile Varieties – Varieties Consisting of Power Series in a Single Variable

Abstract Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part. Our results are established simultaneously for algebraic, convergent and formal power series, as well as convergent power series with prescribed radius of convergence. The key technical tool is a linearisation theorem, already used implicitly by Greenberg and Artin, showing that analytic maps between power series spaces can be essentially linearised by automorphisms of the source space. Instead of stratifying arquile varieties, one may alternatively consider formal neighbourhoods of their regular points and reprove with similar methods the Grinberg–Kazhdan–Drinfeld factorisation theorem for arc spaces in the classical setting and in the more general setting.

Solving the problem of overdetermination of quasisymmetric equilibrium solutions by near-axis expansions. I. Generalized force balance

It is well known that the process of construction of quasisymmetric magnetic fields in magnetostatic equilibrium with isotropic pressure suffers from the problem of overdetermination. This has led to the widespread belief that global quasisymmetric solutions are likely not to exist. We develop a general near-axis expansion procedure that does not rely on the assumption of magnetostatic equilibria with isotropic pressure. We then demonstrate that in equilibria with anisotropic pressure, it is possible to circumvent the problem of overdetermination and carry out the power-series solutions to higher order. This suggests, contrary to current belief, that the existence of globally quasisymmetric fields is likely if one relaxes the assumption of magnetostatic equilibria with isotropic pressure.

Power-series solution of compartmental epidemiological models.

In this work, power-series solutions of compartmental epidemiological models are used to provide alternate methods to solve the corresponding systems of nonlinear differential equations. A simple and classical SIR compartmental model is considered to reveal clearly the idea of our approach. Moreover, a SAIRP compartmental model is also analyzed by using the same methodology, previously applied to the COVID-19 pandemic. Numerical experiments are performed to show the accuracy of this approach.

Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.