Taylor Power Series Research Articles

Generalized conformable fractional Newton-type method for solving nonlinear systems

In a recent paper, a conformable fractional Newton-type method was proposed for solving nonlinear equations. This method involves a lower computational cost compared to other fractional iterative methods. Indeed, the theoretical order of convergence is held in practice, and it presents a better numerical behaviour than fractional Newton-type methods formerly proposed, even compared to classical Newton-Raphson method. In this work, we design a generalization of this method for solving nonlinear systems by using a new conformable fractional Jacobian matrix, and a suitable conformable Taylor power series; and it is compared with classical Newton’s scheme. The necessary concepts and results are stated in order to design this method. Convergence analysis is made and a quadratic order of convergence is obtained, as in classical Newton’s method. Numerical tests are made, and the Approximated Computational Order of Convergence (ACOC) supports the theory. Also, the proposed scheme shows good stability properties observed by means of convergence planes.

Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

High-temperature effect on the material constants and elastic moduli for solid rocks

Abstract Thermally coupled constitutive relations are generally used to determine material constants and elastic moduli (Young's modulus and shear modulus) of solid media. Conventional studies on this issue are mainly based on the linear temperature dependence of elastic moduli, whereas analytical difficulties are often encountered in theoretical studies on nonlinear temperature dependence, particularly at high temperatures. This study investigates the thermally coupled constitutive relations for elastic moduli and material constants using the assumption of axisymmetric fields, with applications to geologic materials (marble, limestone and granite). The Taylor power series of the Helmholtz free energy function within dimensionless temperatures could be used to develop the thermally coupled constitutive relations. The thermoelastic equivalent constitutive equations were formulated under the generalized Hooke's law. The material constants of solid rocks were determined by fitting experimental data using axisymmetric stress and strain fields at different temperatures, based on their thermomechanical properties. For these geologic materials, the resultant equivalent elastic moduli and deformations were in good agreement with those from the experimental measurements. Thermal stresses, internal moisture evaporation and internal rock compositions significantly affected the experimental results. This study provides a profound understanding of the thermally coupled constitutive relations that are associated with the thermomechanical properties of solid rocks exposed to high temperatures.

THE RAY METHOD FOR SOLVING THE DYNAMICS OF THE ELASTIC-VISCOUS-PLASTIC SHELLS

The processes of dynamic deformation of shells are actively studied by both domestic and foreign scientists . The work describes the ray method for solving systems of partial differential equations of hyperbolic type. This method consists in constructing the equations of discontinuity transfer along the propagation of perturbations on moving surfaces, as well as in representing the solution in the form of a Taylor power series in the variable distance behind the perturbation front while preserving a sufficient number of terms. The implementation of the method is shown by the example of a system of partial differential equations of hyperbolic type, which describe the process of dynamic deformation of thin cylindrical shells of revolution from elastoviscoplastic materials . An algorithm for constructing a solution up to the required order is given.

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.

Restarted holomorphic embedding load-flow model based on low-order Padé approximant and estimated bus power injection

The holomorphic embedding load-flow model (HELM) is a noniterative method based on the computation of a Taylor power series that is a function of an embedded parameter. In this method, a continuous function based on a Padé approximant is used to replace this series, extending the radius of convergence. However, depending on the problem and how it is solved, the convergence can demand a high-order series or have a stagnated solution for the case of the continued equivalent function. In this paper, a restarted HELM (RHELM) is introduced. The method works with a very much reduced number of coefficients of the bus voltage Taylor series, different from HELM. Additionally, in the problem formulation, the power balance equations (PBEs) are embedded, considering a parameter and an estimated bus-injected power. This injected power is computed from a voltage estimate whose value is updated at every restarting point. The restart process allows for an approximate solution of the voltages computed via Padé approximation to be used as a new starting point. It is expected that at a new restarting point, the estimated power tends to converge toward the effective power injection of the bus. Tests to accept a given result for a voltage are evaluated for each of two consecutive new coefficient power series that are calculated. The coefficients are calculated until a given specified power mismatch is achieved. In the event that this tolerance is not reached up to a user-defined number of coefficients, the process is then restarted. The method’s performance is evaluated for a wide variety of test systems including a 13659-bus network model. The obtained results reveal that in comparison with the traditional iterative methods based on Newton’s method and the HELM, the proposed formulation is very efficient for computations involving large-scale systems and presents fast convergence.

The Borel map in locally integrable structures

Given a locally integrable structure {mathcal {V}} over a smooth manifold varOmega and given pin varOmega we define the Borel map of{mathcal {V}} atp as the map which assigns to the germ of a smooth solution of {mathcal {V}} at p its formal Taylor power series at p. In this work we continue the study initiated in Barostichi et al. (Math. Nachr. 286(14–15):1439–1451, 2013), Della Sala and Lamel (Int J Math 24(11):1350091, 2013) and present new results regarding the Borel map. We prove a general necessary condition for the surjectivity of the Borel map to hold and also, after developing some new devices, we study some classes of CR structures for which its surjectivity is valid. In the final sections we show how the Borel map can be applied to the study of the algebra of germs of solutions of {mathcal {V}} at p.

S-Numbers of Weighted Shift Operators on P-Summable Formal Entire Functions of M-Variables

The idea of multiplying a formal Taylor power series by z to make a right shift operator on the space of all square summable sequences of real numbers was due to A.L. Shield. In this work, we consider Taylor power series in m-variables and we give upper and lower estimations of s-numbers for multiplication of m- right weighted shift operators. This allowed us to estimate upper bounds for s-numbers of infinite series of m-right weighted shift operators and give some applications.

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.

Numerical investigations for time-fractional nonlinear model arise in physics

Abstract In this work, we suggest a numerical scheme to find analytically a solution of Caputo-time-fractional nonlinear model arise in physics. This model is called Belousov-Zhabotinsky (BZ) and reads as D t α u ( x , t ) = u ( x , t ) ( 1 - u ( x , t ) - rv ( x , t ) ) + u xx ( x , t ) , D t α v ( x , t ) = - au ( x , t ) v ( x , t ) + v xx ( x , t ) , where 0 α ⩽ 1 , 0 t R 1 . Also, a ≠ 1 and r are positive parameters. A modified version of generalized Taylor power series method will be used in this work. Graphical justifications on the reliability of the proposed method are provided. Finally, the effects of the fractional order on the solution of Belousov-Zhabotinsky model is also discussed.

A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application

The current work contributes to find a supportive analytical solution of Caputo time-fractional nonlinear equations of the form $$\begin{aligned}&{\mathcal {D}}^{\alpha }_{t}\big [u(x,t)\big ]+\left( \sum _{k=1}^{N_1}\lambda _{k}u^k(x,t)\right) u_{x}(x,t)\nonumber \\&\quad +\sum _{k=2}^{N_2}\delta _{k}\frac{\partial ^ku(x,t)}{\partial x^k}=0,\ \ \ \ 0<\alpha \le 1. \end{aligned}$$ We proposed a modified version of the generalized Taylor power series method to extract a reliable approximate solution of this problem. Theorems regarding the convergence of the obtained solution are provided. The method is tested on two models of the proposed equation, namely the Caputo time-fractional Gardner and Kawahara equations. Finally, both graphical explanations and tabular analysis are performed to study the accuracy of our suggested scheme, and to study the effects of $$\alpha $$ on the behavior of the obtained solution.

On the Analysis and Design of Fractional-Order Chebyshev Complex Filter

This paper introduces the concept of fractional-order complex Chebyshev filter. A fractional variation of Chebyshev differential equations is introduced based on Caputo fractional operator. The proposed equation is solved using fractional Taylor power series method. The condition for fractional polynomial solutions is obtained and the first four polynomials scaled using an appropriate scaling factor. The fractional-order complex Chebyshev low-pass filter based on the obtained fractional polynomials is developed. Two methods for obtaining the transfer functions of the complex filter are discussed. Circuit implementations are simulated using Advanced Design System (ADS) and compared with MATLAB numerical simulation of the obtained transfer functions to prove the validity of the two approaches.

Solution of conformable fractional ordinary differential equations via differential transform method

Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in (Khalil et al., 2014). Then, the fractional versions of chain rules, exponential functions, Gronwall’s inequality, integration by parts, Taylor power series expansions is developed in (Abdeljawad, 2015). In this paper, we give conformable fractional differential transform method and its application to conformable fractional differential equations.

Existence and uniqueness theorems for sequential linear conformable fractional differential equations

Recently, a new fractional derivative called the conformable fractional derivative is given on based basic limit definition derivative in [4]. Then, the fractional versions of chain rules, exponential functions, Gronwalls inequality, integration by parts, Taylor power series expansions is developed in [5]. In this paper, we present existence and uniqueness theorems for sequential linear conformable fractional differential equations.

On conformable fractional calculus

Recently, the authors Khalil et al. (2014) introduced a new simple well-behaved definition of the fractional derivative called conformable fractional derivative. In this article we proceed on to develop the definitions there and set the basic concepts in this new simple interesting fractional calculus. The fractional versions of chain rule, exponential functions, Gronwall’s inequality, integration by parts, Taylor power series expansions, Laplace transforms and linear differential systems are proposed and discussed.

Dynamics of three Toda oscillators with nonlinear unidirectional coupling

We study the dynamics of three unidirectionally coupled Toda oscillators with nonlinear coupling function in the form of first three terms of Taylor power series. We analytically investigate how the coupling influence the stability of steady state. Basing on calculation of the first Lyapunov coefficient, we show that destabilization may occur by the sub- or supercritical Andronov-Hopf bifurcation. Born periodic solutions are calculated using path-following as a function of coupling strength and Taylor series coefficients. We present that initially stable or unstable branch of periodic solutions may undergo a sequence of bifurcations including: period doubling, Neimark-Saker and fold.

Taylor expansion of the inverse function with application to the Langevin function

A Taylor power series is a powerful mathematical tool, which can be used to express an inverse function especially if it is given in an implicit form. This is for example the case for the inverse Langevin function, which is an indispensable ingredient of full-network rubber models. In the present paper, we propose a simple recurrence procedure for calculating Taylor series coefficients of the inverse function. This procedure is based on the Taylor series expansion of the original function and results in a simple recurrence formula. This formula is further applied to the inverse Langevin function. The convergence radius of the resulting series is evaluated. Within this convergence radius the obtained approximation of the inverse Langevin function demonstrates better agreement with the exact solution in comparison to different Padé approximants.

Decomposition of matrices over a two-dimensional local field

G(Fq((x))((t))) = G(Fq((x))[[t]])G(Fq [[x]]((t))). Here G is an arbitrary connected linear reductive group, Fq is a finite field, and for an arbitrary field k by k((x)) and k[[x]] we denote the fields of Laurent and Taylor power series, respectively, with coefficients in k. In the present paper, we prove this lemma for G = GL(n) by elementary methods. Note that the assertion is not always true for nonreductive groups G. For example, if G is a onedimensional additive group, then it is not true that

Computation of the exponential and other isotropic tensor functions and their derivatives

In the present paper we focus on numerical aspects of the computation of isotropic tensor functions and their derivative. In the general case of non-symmetric tensor arguments only two numerical algorithms appear to be appropriate. The first one represents a recurrent procedure resulting from the Taylor power series expansion of an isotropic tensor function. The second algorithm is based on a recently proposed closed-form representation which can be obtained from the definition of an isotropic tensor function either by the tensor power series or by the Dunford–Taylor integral. To improve the accuracy in the case of nearly equal eigenvalues a series expansion of this closed formula is proposed. Both algorithms are finally illustrated by an example of the exponential tensor function where emphasis is placed on the precision issue.

Error performance of matched and partially matched one-shot detectors for doubly selective Rayleigh fading channels

In this paper, optimum one-shot detection over known and partially known doubly selective Rayleigh fading channels is investigated. Reduced complexity channel models based on Gauss quadrature rules and Taylor power series are derived and are employed to develop novel analytical tools for the performance analysis of one-shot detectors. Numerical results allow to assess the implicit diversity gain provided by both channel multipath and signal fading and the energy loss due to the mismatch of the receiver filter.