Power Series Expansion Method Research Articles

Invariant analysis, exact solutions, and conservation laws of time fractional thin liquid film equations

This present paper investigates Lie symmetry analysis, one-dimensional optimal system, exact solutions and conservation laws of the (2 + 1)-dimensional time fractional thin liquid film equations (TFTLFE) with Riemann–Liouville fractional derivative. Explicitly, we obtain six vector fields and the one-dimensional optimal system admitted by TFTLFE. Then, we perform the symmetry reductions with the help of Erdélyi–Kober fractional differential operator and (2 + 1)-dimensional TFTLFE is reduced into (1 + 1)-dimensional fractional partial differential equations (FPDEs). Additionally, by means of compound variable transformation and the power series expansion method, the solution of reduced FPDEs is obtained and its convergence is verified. Moreover, we derive other solutions for the reduced equations taking advantage of the invariant subspace method. Furthermore, the conservation laws are also established utilizing generalized Noether's theorem. Finally, we construct the exact solution using the method of conservation laws.

Dynamical analysis of a fractional-order nonlinear two-degree-of-freedom vehicle system by incremental harmonic balance method

At present, there are few studies considering both nonlinear and fractional characteristics of suspension in vehicle systems. In this paper, a fractional nonlinear model of a quarter vehicle with two-degree-of-freedom (2-DOF) is innovatively proposed to describe the suspension system containing the viscoelastic material metal rubber. Given the lack of a general calculation scheme for the multi-degree-of-freedom fractional-order incremental harmonic balance method (IHBM), a general calculation scheme for the 2-DOF incremental harmonic balance method for nonlinear systems with fractional order is derived. The nonlinear dynamical properties of the presented model are acquired using this method. The accuracy of the proposed method is verified through a comparison with the power series expansion method. Afterward, the effects of the various parameters on the dynamic performance are analyzed. The vibration peak value of the fractional-order model is significantly higher than that of the integer-order model (IOM). Therefore, the suspension parameters should be designed with a margin when using IOM.

A fractional-order model to study the dynamics of the spread of crime

Numerous crucial factors and parameters influence the dynamic process of the spread of crime. Various integer-order differential models have been proposed to capture crime spread. Most of these introduced dynamic systems have not considered the history of the criminal and the impact of crime on society. To address these shortcomings, a fractional-order crime transmission model is proposed in this manuscript considering five different classes viz law-abiding citizens, non-incarcerated criminals, incarcerated criminals, prison-released and recidivists. The primary focus of the proposed model is to study the effect of recidivism in society and decide the adequate imprisonment for repeat offenders. The existence, uniqueness, non-negativity and boundedness of the solution of the proposed model are examined. The local stability of the equilibrium points is also analysed using Routh–Hurwitz Criteria with Matignon conditions. Further, the threshold condition for the uniform asymptotic stability of the system is evaluated by the Lyapunov stability method. Moreover, the long-term impact of the imprisonment of criminals on society is also examined in the current study. The numerical simulations of the model for a range of fractional orders are obtained using power series expansion method to strengthen the theoretical results.

A study on singular boundary integrals and stability of 3D time domain boundary element method

The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem.

Blocking Mechanism and Evaluation Model of Insensitive Effect Under Multi-Frequency Electromagnetic Radiation

Due to the insensitive blocking effect on frequency equipment, the current effect evaluation model based on the third-order power series expansion is disabled to the insensitive blocking effect evaluation under multi-frequency electromagnetic radiation. We adopt the fifth-order power series expansion to analyze the insensitive blocking interference effect. In addition to revealing the mechanism of the insensitive blocking effect under multi-frequency electromagnetic radiation, we also build an improved effect evaluation model in this research. It defines the effect evaluation index. Subsequently, a ranging radar and a communication radio station are employed to verify the effectiveness of the proposed effect evaluation model separately. The results show that the proposed model can be applied to the electromagnetic radiation effect evaluation, regardless of whether the multi-frequency insensitive jamming effect exists on tested equipment or not. All effect evaluation indexes of the new model are smaller than that of the previous model based on the third-order power series expansion method. The absolute value of relative errors could be less than 1 dB. Not only is the theoretical analysis result of the multi-frequency insensitive blocking effect mechanism verified, but also the problem of poor universality of the current electromagnetic radiation effect evaluation model is figured out.

Dynamic analysis of piecewise nonlinear systems with fractional differential delay feedback control

This paper mainly analyzes a class of piecewise nonlinear systems with fractional differential delay feedback control. The average method is used to solve the nonlinear system to obtain the amplitude-frequency relationship of the system. The analytical solution of the system can quantitatively analyze the system. At the same time, the correctness of the analytical solution of the system is verified by numerical solution. The effects of linear damping, linear stiffness and nonlinear stiffness of the system and the proportion, fractional order coefficient and fractional order number of the feedback controller on the amplitude-frequency characteristics of the system are analyzed. Finally, the dynamic analysis of the system is carried out. Taking the fractional order coefficient as the bifurcation parameter, the global characteristics of the bifurcation diagram, time history diagram and phase diagram system under different parameter perturbations are studied by power series expansion method. It is found that the system exhibits periodic motion, period doubling motion and chaos with the change of perturbation parameters.

RI-IGABEM for 3D viscoelastic problems with body force

A novel radial integration IGABEM (RI-IGABEM) is proposed to solve the 3D viscoelastic problems with body force in this paper. The derivation of boundary integral equations for the viscoelastic problems is proposed. To reduce computing costs, the time-dependent shear modulus is expressed in the form of Prony series, and the memory stress is calculated by using the genetic integral. Since the fundamental solution of elastostatics is applied to derive the boundary integral equations, which leads to the existence of domain integrals. To ensure that this algorithm only needs to discretize the boundary, we use radial integration method (RIM) to transform the domain integrals related to body force and memory stress into equivalent boundary integrals through applied points. Moreover, the strongly singular integral of the boundary integral equation is solved by the rigid-body displacement technique after a simple transformation between control points and collocation points, and the power series expansion method is used to solve the weakly singular integrals in RI-IGABEM. Furthermore, the 3D surface traction recovery method (TRM) of the viscoelastic mechanics is proposed to solve the strain and stress of the boundary points, and the regularized strain and stress integral equations are given in this work to solve the strain and stress of the internal points. Some 3D examples are applied to prove the effectiveness of the present method for the viscoelastic problems with body force.

Series expansion of the overlap reduction function for scalar and vector polarizations for gravitational wave search with pulsar timing arrays

In our previous work [Phys. Rev. D 103, 064044 (2021)] we calculated the overlap reduction function for the tensor polarization without employing the short wavelength approximation, this was done by obtaining a power series of nested sums which is valid for all gravitational wave frequencies and pulsar distances. In this work we generalize the power-series expansion method to vector and scalar polarizations. We have compared our expression for the breathing and vector modes with previous literature. We present for the first time analytic expressions for the overlap reduction function of the longitudinal mode for all angles between the pulsar pairs.

Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

A vortex sheet based analytical model of the curled wake behind yawed wind turbines

Motivated by the need for compact descriptions of the evolution of non-classical wakes behind yawed wind turbines, we develop an analytical model to predict the shape of curled wakes. Interest in such modelling arises due to the potential of wake steering as a strategy for mitigating power reduction and unsteady loading of downstream turbines in wind farms. We first estimate the distribution of the shed vorticity at the wake edge due to both yaw offset and rotating blades. By considering the wake edge as an ideally thin vortex sheet, we describe its evolution in time moving with the flow. Vortex sheet equations are solved using a power series expansion method, and an approximate solution for the wake shape is obtained. The vortex sheet time evolution is then mapped into a spatial evolution by using a convection velocity. Apart from the wake shape, the lateral deflection of the wake including ground effects is modelled. Our results show that there exists a universal solution for the shape of curled wakes if suitable dimensionless variables are employed. For the case of turbulent boundary layer inflow, the decay of vortex sheet circulation due to turbulent diffusion is included. Finally, we modify the Gaussian wake model by incorporating the predicted shape and deflection of the curled wake, so that we can calculate the wake profiles behind yawed turbines. Model predictions are validated against large-eddy simulations and laboratory experiments for turbines with various operating conditions.

Model and performance analysis of non-uniform piezoelectric semiconductor nanofibers

In order to evidently improve the working performance of piezotronic devices, non-uniform piezoelectric semiconductor fibers with contoured profiles are designed. However, apart from the finite element method, it's hard to achieve analytical descriptions of the electromechanical fields in these non-uniform piezoelectric semiconductor fibers because of the governing equations with variable coefficients. For solving this bottleneck and exploring improved performance and new phenomena caused by the non-uniform profile, a power series expansion method is proposed based on the framework of the one-dimensional linearized model and is further applied to analyze the piezotronic performances of n-type non-uniform piezoelectric semiconductor fibers and PN junctions. It is revealed via systematical numerical simulations that the antisymmetry of the electromechanical fields in piezoelectric semiconductor fibers is broken because of the contoured profile and the piezotronic coupling. Meanwhile, the barrier configuration in a non-uniform PN junction is sensitive to the variation of cross-sectional area. Furthermore, the current-voltage relation of a necking heterogeneous piezoelectric semiconductor PN junction can be manipulated more conveniently by external mechanical loadings, meaning that the sensitivity of the device is improved. Not limited by non-uniform piezoelectric semiconductor fibers with contoured profiles, this method exhibits broad applicability, which is still available for solving multiple-coupled properties of functional graded piezoelectric semiconductor media.

Analysis of Dynamic Characteristics of a Fractional-Order Spur Gear Pair With Internal and External Excitations

Abstract The nonlinear frequency response characteristics of a spur gear pair with fractional-order derivative under combined internal and external excitations are investigated based on the incremental harmonic balance (IHB) method. First, a pure torsional vibration model is proposed that contains various complex factors, such as the time-varying mesh stiffness, transmission error, the fluctuation of input torque, backlash. Then, the IHB method is developed to calculate the higher-order approximate solution of the system, and the correctness of the results is verified by comparing with numerical simulation results obtained by the power series expansion (PSE) method. Furthermore, the types of various impact situations and their judgment conditions are discussed, and the different impact behaviors are analyzed in detail when ω∈[0, 1.5] by using phase diagrams and amplitude–frequency response curves. The influence of important parameters on the dynamic characteristics of gear pair is analyzed at last. The results indicate that the analytical solution derived by IHB method is sufficiently precise. Significantly, the dynamic characteristics of the system could be effectively controlled by adjusting time-varying mesh stiffness coefficient, the order and coefficient of fractional-order term, and the amplitude of internal excitation or external excitation. As a part of the theory of fractional-order mechanical system, the impact performance of fractional-order gear pair is approached for the first time by analytical method.

On the solutions and conservation laws of the Yu–Toda–Sasa–Fukuyama equation of plasma physics

In this work, we investigate the two-dimensional Yu–Toda–Sasa–Fukuyama equation, which has many applications in plasma physics and other fields of study in natural sciences. Exact analytical solutions of this model are presented by invoking Lie symmetry analysis technique. Moreover, the power series expansion method is also used to construct exact analytical solutions. We present the 2D and 3D graphical representations of the obtained solutions for some parametric values so as to demonstrate the dynamic behaviour of the solutions. Furthermore, conserved vectors are derived by making use of two methods, namely the multiplier method and Noether’s theorem.

IGABEM of 2D and 3D liquid inclusions

Randomly distributed liquid inclusions in elastic matrix exist in nature, biology, material science and other fields. In this paper, the isogeometic analysis boundary element method (IGABEM) is applied to study the mechanical properties of the elastic matrix with liquid inclusions, in which the liquid inclusion is assumed to be linearly compressible and the interface tension is neglected. Singularity subtraction technique (SST) and Tells transformation method are respectively used to solve the strongly and weakly singular integrals in two-dimensional (2D) problems, while the power series expansion method is used to carry out various singular integrals in three-dimensional (3D) problems. The oriented bounding box (OBB) is adopted to generate the randomly distributed liquid inclusions, which can be formed easily by the control points of the IGA. Numerical examples show the accuracy and effectiveness of the present method in the study of liquid inclusions.

Spinning test body orbiting around a Schwarzschild black hole: Comparing spin supplementary conditions for circular equatorial orbits

The Mathisson-Papapetrou-Dixon (MPD) equations describe the motion of an extended test body in general relativity. This system of equations, though, is underdetermined and has to be accompanied by constraining supplementary conditions, even in its simplest version, which is the pole-dipole approximation corresponding to a spinning test body. In particular, imposing a spin supplementary condition (SSC) fixes the center of mass of the spinning body, i.e., the centroid of the body. In the present study, we examine whether characteristic features of the centroid of a spinning test body, moving in a circular equatorial orbit around a massive black hole, are preserved under the transition to another centroid of the same physical body, governed by a different SSC. For this purpose, we establish an analytical algorithm for deriving the orbital frequency of a spinning body, moving in the background of an arbitrary, stationary, axisymmetric spacetime with reflection symmetry, for the Tulczyjew-Dixon, the Mathisson-Pirani, and the Ohashi-Kyrian-Semer\'ak SSCs. Then, we focus on the Schwarzschild black hole background, and a power series expansion method is developed in order to investigate the discrepancies in the orbital frequencies expanded in power series of the spin among the different SSCs. Lastly, by employing the fact that the position of the centroid and the measure of the spin alters under the centroid's transition, we impose proper corrections to the power expansion of the orbital frequencies, which allows to improve the convergence between the SSCs. Our concluding argument is that when we shift from one circular equatorial orbit to another in the Schwarzschild background, under the change of a SSC, the convergence between the SSCs holds only up to certain powers in the spin expansion, and it cannot be achieved for the whole power series.

Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier–Stokes equations

In this paper, we investigate the exact solutions and conservation laws of (2 + 1)-dimensional time fractional Navier–Stokes equations (TFNSE). Specifically, Lie symmetries and corresponding one-dimensional optimal system for TFNSE in Riemann–Liouville sense are obtained. Then, based on the admitted symmetries and optimal system, we reduce these equations to one-dimensional equations or (1 + 1)-dimensional fractional partial differential equations (PDEs) with the help of Erdélyi–Kober fractional differential operator and compound variable transformation. In addition, we solve the reduced PDEs applying power series expansion method and invariant subspace method, respectively. Furthermore, the conservation laws of TFNSE are derived using new Noether theorem.

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

RI-IGABEM based on generalized-[formula omitted] method in 2D and 3D elastodynamic problems

The isogeometric analysis boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation and good approximation properties. In this paper, a novel radial integration IGABEM (RI-IGABEM) based on the generalized-α method is proposed to solve 2D and 3D elastodynamic problems of homogeneous and inhomogeneous materials. First of all, the elastostatics Kelvin fundamental solution is used as the fundamental solution of the problem. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral caused by the material heterogeneity and the inertia term into an equivalent boundary integral by means of applied points. In addition, using a simple transformation method, the rigid-body technique is applied to solve the strongly singular integrals, and the Telles scheme and the power series expansion method are used to solve the weakly singular integrals in RI-IGABEM respectively. Furthermore, the generalized-α method is adopted to solve the time domain problem, which can improve the stability of numerical results by effectively filtering out the false response of high frequency and minimizing the attenuation of low frequency response. A number of 2D and 3D examples, such as those with homogeneous materials, functionally gradient materials, and material defects and inclusions, are used to demonstrate the ability of the scheme to simulate the elastodynamic problems.

IG-DRBEM of three-dimensional transient heat conduction problems

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

Stress intensity factors for a centrally cracked Brazilian disk under non-uniformly distributed pressure

To simulate the real distribution of pressure on the specimen in the Brazilian test, non-uniform distribution is assumed to be the form of pressure distribution on the disk. The analytic solutions of the stress field for an uncracked Brazilian disk under several typical non-uniformly distributed pressures are obtained by the power series expansion method and mathematical method. Then the analytic solutions of the mode I and mode II stress intensity factors for a centrally cracked Brazilian disk under the same pressures are obtained by the weight function method. The results show that the influence of distribution form on stress intensity factors cannot be ignored for the case of large relative crack length and load distribution angle. The applicable ranges of the formulae to calculate stress intensity factors under concentrated force, uniformly and non-uniformly distributed pressure are given according to the analysis results. In addition, the loading conditions for pure mode I and II fracture are discussed when a disk with a central crack is subjected to non-uniformly distributed pressure.