Power Series Solution Research Articles

Nonlocal zigzag analytical solution for Laplacian hygrothermal stress analysis of annular sandwich macro/nanoplates with poor adhesions and 2D-FGM porous cores

A zigzag nonlocal model is presented for annular/circular sandwich macro- and nanoplates with bidimensional graded porous cores and weak interfacial adhesions. This is the first time that a stress analysis is performed in the framework of Eringen's nonlocality concept. In this regard, after determination of the displacements by utilizing a power series solution, the governing Laplacian equations of the stress components in terms of the displacement components are solved in a post-processing stage, using a power series solution in terms of stress parameters. Effects of the thermal stresses and material degradation and deterioration due to the temperature rise and moisture absorption are also taken into account. To guarantee continuity of the transverse shear and normal stresses, the bonding layers are modeled by spring elements. To present a general model, it is assumed that the core porosity changes in both radial and transverse directions. It is the first time that the effects of the porosity are investigated on transverse distributions of the displacement and stress components. The results have captured the significant effects of the nonlocality, moisture absorption, and bond strength on the stress and lateral deflection results, and especially, the abrupt changes in the in-plane displacement and stress components at the interfaces.

Sound transmission through truncated conical shells

A theoretical model is presented to investigate the sound transmission through a thin-walled truncated circular conical shell excited by an oblique incident plane sound wave, which impinges on the outer surface of the shell. Love’s theory is used to implement the governing equations of the shell motion, and a convergent power series solution is applied to obtain the exact dynamic response of the shell. In order to calculate the acoustic pressure loadings acting on the shell surfaces, the shell is divided into several narrow segments which can be considered to be locally cylindrical. At first, the model predictions are compared with their experimental and analytical counterparts reported in the literature. Then, the effects of crucial design parameters including different boundary conditions at the ends of the shell, cone angle, incident sound wave angle, length, radius, thickness and material properties of the shell, and the material properties of the fluid media are studied on the characteristics of the sound transmission loss. The proposed model can provide an effective tool in the acoustic design stage of the truncated circular conical shells.

Maillet type theorem for nonlinear totally characteristic partial differential equations

The paper discusses a holomorphic nonlinear singular partial differential equation $$(t \partial _t)^mu=F(t,x,\{(t \partial _t)^j \partial _x^{\alpha }u \}_{j+\alpha \le m, j<m})$$ under the assumption that the equation is of nonlinear totally characteristic type. By using the Newton polygon at $$x=0$$ , the notion of the irregularity at $$x=0$$ of the equation is defined. In the case where the irregularity is greater than one, it is proved that every formal power series solution belongs to a suitable formal Gevrey class. The precise bound of the order of the formal Gevrey class is given, and the optimality of this bound is also proved in a generic case.

On the Legendre differential equation with uncertainties at the regular‐singular point 1: L p ( Ω ) random power series solution and approximation of its statistical moments

is the peer reviewed version of the following article: Calatayud, J, Cortes, J-;C, Jornet, M. On the Legendre differential equation with uncertainties at the regular-singular point 1: Lp random power series solution and approximation of its statistical moments. Comp and Math Methods. 2019; 1:e1045. https://doi.org/10.1002/cmm4.1045 , which has been published in final form at https://doi.org/10.1002/cmm4.1045. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

Symmetry reductions, exact solutions, stability analysis and conservation laws of time-fractional Sharma–Tasso–Olever equation

In this paper, Lie group analysis method is applied to consider a vector field and symmetry reductions for time-fractional Sharma–Tasso–Olver equation, exact hyperbolic wave solutions, power series solutions and its convergence are investigated. Stability and boundedness analysis of trivial solution to the reduced ordinary differential equation is shown by constructing appropriate Lyapunov function. Conservation laws of the equation are well constructed with a detailed derivation making use of Noether’s operator.

The Deflection of Rotating Composite Tapered Beams with an Elastically Restrained Root in Hygrothermal Environment

Abstract This article aims to study the static deflection of a rotating composite Timoshenko beam subjected to the laterally distributed load and restrained by the elastic root and affected by the various cross-section, installation mode, and hygrothermal environment. The governing equation is established according to the force equilibrium condition and solved by a semianalytical power series solution. To verify the correctness, the results of differential quadrature method are introduced to make a comparison. Then, several parameters that can affect the static deflection of the beam, such as the rotating speed, temperature variation, elastic root, and so on, are investigated. Results indicate that (1) pitch angle, rotating speed, and hub radius can result in the centrifugal stiffening effect; (2) setting angle, fibre orientation angle, taper ratio, and elastic root affect the static deflection by changing the rigidity of the rotating composite tapered beam; and (3) temperature variation and moisture concentration can cause the expansion deformation and the change of material properties.

A New Approach for the Approximate Analytical Solution of Space-Time Fractional Differential Equations by the Homotopy Analysis Method

The motivation of this study is to construct the truncated solution of space-time fractional differential equations by the homotopy analysis method (HAM). The first space-time fractional differential equation is transformed into a space fractional differential equation or a time fractional differential equation before the HAM. Then the power series solution is constructed by the HAM. Finally, taking the illustrative examples into consideration we reach the conclusion that the HAM is applicable and powerful technique to construct the solution of space-time fractional differential equations.

Vlasov-Maxwell system with physical reasonableness constraints

Complete mathematical description on plasmas contains not only some coupled partial differential equations (PDEs) reflecting physics laws but also some inequalities reflecting physical reasonableness requirement. The phrase ‘physical reasonableness’ refers to that solutions should always correspond to non-negative-valued probability distribution function and non-negative-valued particle density. This work displays a universal strict method on the V-M system with a constraint inequality f ≥ 0∀ r, p, t. It treats the Vlasov equation as a recurrence formula relating expansion coefficient functions of the power series solution of the equation, and strictly demonstrates that the constraint determines the shape, or geometric characteristics, of space-time contours of the fluid velocity field. Consequently, the constraint inequality makes those PDEs reflecting physics laws to be expressed finally as an ordinary equation of the fluid velocity field with respect to its contour.

A Note on the Convergence of Multivariate Formal Power Series Solutions of Meromorphic Pfaffian Systems

Here, we present some complements to theorems of R. Gerard and Y. Sibuya, on the convergence of multivariate formal power series solutions of nonlinear meromorphic Pfaffian systems. Their most known results concern completely integrable systems with non-degenerate linear parts, whereas we consider some cases of non-integrability and degeneracy.

State-vector equation method for the frequency domain spectral element modeling of non-uniform one-dimensional structures

This paper presents a state-vector equation method (SVEM)-based spectral element method (SEM) to formulate frequency domain spectral element models for non-uniform 1-D structures. In the SVEM-based SEM, the frequency domain homogeneous governing differential equations for non-uniform 1-D structures are first transformed into state-vector equations. The system matrix and state-vector are represented with power series. These power series representations are then substituted into the state-vector equation, and the differential transformation method is used to efficiently derive a recurrence formula for the coefficients of the power series solution of the state-vector equation. By computing these coefficients from the recurrence formula, the transfer function that relates the state-vector at an arbitrary position to that at the initial position of a finite element is derived. The transfer matrix is then obtained from the transfer function. Finally, the spectral element matrix or exact dynamic stiffness matrix is obtained from the transfer matrix. The accuracy and computational efficiency of the proposed SVEM-based SEM are verified in due course through comparisons with other solution methods for non-uniform axial rods, Bernoulli−Euler beams, and Timoshenko beams with different spatial variations.

Stability analysis, optical solitons and complexitons of the two-dimensional complex Ginzburg-Landau equation

ABSTRACTIn this paper, the two-dimensional complex Ginzburg-Landau equation is investigated, which describes phase transitions in superconductors near their critical temperature in the field of electromagnetic behavior dynamics and in the study of external magnetic fields. We employ the hypothetical method to find the bright soliton, dark soliton and complexitons of the equation. We also find its power series solution with its convergence analysis. Moreover, some constraint conditions are provided which can guarantee the existence of solitons. By use of the Hamiltonian description, we analyze the modulation instability and stable solutions. In order to further understand the dynamic behavior, the graphics analysis is provided of these solutions.

Series solutions and bifurcation of traveling waves in the Benney–Kawahara–Lin equation

In this paper, the Lie group method is performed on the BKL equation. By reducing the BKL equation to four classes of ODEs, we obtain new power series solutions of it. Furthermore, as an important reduced equation, the traveling wave equation is investigated in detail. Treating it as a singular perturbation system in $${\mathbb {R}}^4$$ , we apply the geometric singular perturbation theory and dynamical system methods to study its phase space geometry. By using some techniques, such as tracking the unstable manifold of the saddle, discussing transversality of the stable and unstable manifolds and investigating some complicated nonlocal bifurcation, we obtain wave speed conditions to guarantee the existence of various bounded traveling waves of the BKL equation.

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

A powerful analytical approach, namely the fractional residual power series method (FRPS), is applied successfully in this work to solving a class of fractional stiff systems. The methodology of the FRPS method gets a Maclaurin expansion of the solution in rapidly convergent form and apparent sequences based on the Caputo sense without any restriction hypothesis. This approach is tested on a fractional stiff system with nonlinearity ranging. Meanwhile, stability and convergence study are presented in the domain of interest. Illustrative examples justify that the proposed method is analytically effective and convenient, and it can be implemented in a large number of engineering problems. A numerical comparison for the experimental data with another well-known method, the reproducing kernel method, is given. The graphical consequences illuminate the simplicity and reliability of the FRPS method in the determination of the RPS solutions consistently.

Shifted Chebyshev polynomials based solution of partial differential equations

This paper presents an efficient numerical method based on shifted Chebyshev polynomials for solving Partial Differential Equations (PDEs). In this method, a power series solution in terms of shifted Chebyshev polynomials has been chosen such that it satisfies the given conditions. Plugging this series solution into the given PDE and using appropriate collocation points a system of linear equations with unknown Chebyshev coefficients is obtained. Then, unknowns are found with the help of Gauss elimination or Newton’s method. Next, different discretization patterns have also been considered to understand the behavior of the results depending upon the collocation points in the domain. These are the two main modifications and novelty of the procedure and this small contribution in the assumption of power series solution in terms of shifted Chebyshev polynomials results in obtaining the approximate solution with less number of terms with good accuracy. Several numerical examples are provided to confirm the reliability and effectiveness of the proposed method.

Time-fractional Drinfeld-Sokolov-Wilson system: Lie symmetry analysis, analytical solutions and conservation laws

In this paper, under the background of the Riemann-Liouville fractional differential, the Lie point symmetries are obtained by using the Lie symmetry method. The symmetry reductions are also derived ulteriorly. Next, the power series solution and its convergence proof are given. Finally, conservation laws are well constructed based on the Noether theorem. Especially, the approximate analytical solution is studied by employing the q-homotopy analysis method under the background of Caputo fractional differential. What is more, the dynamic behaviour of all these exact solutions of the equation are described with changing the value of $ \alpha$ .

EXACT SOLUTION AND CONSERVATION LAWS OF FRACTIONAL COUPLED WAVE INTERACTION EQUATION

In this paper, we consider a time-fractional coupled KdV equations describing the interaction of equatorial and midlatitude Rossby waves. From the application of Lie group analysis, the governing system of fractional partial differential equations (FPDEs) is reduced to a system of fractional ordinary differential equations (FODEs). Further, we construct the group-invariant solution as well as the power series solution for the given coupled equations. Next, the evolutionary behavior of the waves under the influence of the fractional order derivative [Formula: see text] is studied graphically through the group-invariant solution. Finally, the conservation laws for the given system of FPDEs are obtained.

On the summability of divergent power series satisfying singular PDEs

The aim of this note is to apply the Borel–Laplace summation method studied by H. Chen, Z. Luo and C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) to the divergent power series solutions to two families of nonlinear PDEs. The first one contains particularly a two-dimensional version of the so-called Euler equation (ODE), while the second is called totally characteristic type PDE by H. Chen and H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).

Symmetries and conservation laws of the Yao\u2013Zeng two-component short-pulse equation

This paper uses the Lie group method to analyze the symmetries of the Yao–Zeng two-component short-pulse equation which describes the propagation of polarized ultrashort light pulses in cubically nonlinear anisotropic optical fibers. Similarity reductions and exact solutions are obtained by constructing an optimal system of one-dimensional subalgebras. Moreover, the explicit solutions are constructed by the power series method and the convergence of power series solutions is proved. In addition, nonlinear self-adjointness and conservation laws of this system are discussed.

Power Series Solution forStrongly Non-linear Conservative Oscillators

Power Series Solution forStrongly Non-linear Conservative Oscillators

Time fractional Biswas–Milovic equation: Group analysis, soliton solutions, conservation laws and residual power series solution

In this work, we studied time fractional Biswas–Milovic equation (FBME) which describes the long distance optical communication with Kerr law nonlinearity. We analyzed group analysis of equation by using Lie symmetry analysis and reduced it to fractional ordinary differential equation. We obtained analytic soliton solutions for the equation by means of DtαGG expansion method in form of hyperbolic, trigonometric and rational function form. The conservation laws for fractional governing equation are constructed by using non linear self adjointness and Noether's operators. The numerical approximated solution are also obtained by using residual power series method. Some figures are also added to understand the solutions.