Narrow Results

The objective of this work is to study a frictionless contact scenario involving a thermo-viscoelastic body and a thermally conductive foundation. The history-dependent constitutive law involves time-fractional derivatives of the Caputo type to describe displacement behavior. The modeling of contact involves the utilization of the normal response condition. We formulate the problem variationally and prove the existence of its weak solution by employing a combination of techniques, such as the theory of monotone operators, Caputo derivatives, the Galerkin method, and the Banach fixed-point theorem. Further, we introduce a fully discrete scheme to solve the problem numerically. Under certain solution regularity assumptions, we derive an optimal order error estimate of the discretization. Finally, we provide numerical simulations which show the efficiency of the method and illustrate the theoretical error estimate.

Our main interest in this study is to obtain approximate time-dependent solutions of the Lotka–Volterra predator–prey model given the populations of both species at a certain time. We achieve this by considering an earlier scheme by Shinohara and Yamamoto and modifying it so as to address the initial conditions. An important feature of this scheme is that it treats the system’s period as an unknown variable, making it possible to calculate it as a side product. In this method, the Galerkin procedure is utilized to convert the problem to a system of nonlinear algebraic equations, where sine and cosine waves are chosen as the basis functions in order to ensure the periodicity of solutions. The resulting overdetermined system is then solved in a least squares sense, which yields the coefficients making up the Galerkin approximations. After a detailed presentation of the numerical scheme, we apply it to two example problems and evaluate the results according to several accuracy criteria.

In this paper, we investigate a novel model problem involving a viscoelastic, frictionless contact model characterized by history-dependent operators. The constitutive relation is formulated based on a time-fractional Kelvin–Voigt model. We represent the contact by incorporating normal compliance within the framework of a time-fractional derivative. In addressing the contact problem, we begin by formulating its weakness and subsequently verify the existence of a solution within this framework. Finally, we study the solution’s behavior concerning the integral term and its relation to the solution, while also presenting convergence results.

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus T^d of any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍⁿ(T^d), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a difference with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their ℍⁿ distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

In this paper we explore one of the most important features of the Galerkin method, which is to achieve high accuracy with a relatively modest computational effort, in the dynamics of Robinson–Trautman spacetimes.

This paper is concerned with a numerical spectral solution to a one-dimensional linear telegraph type equation with constant coefficients. An efficient Galerkin algorithm is implemented and analyzed for treating this type of equations. The philosophy of utilization of the Galerkin method is built on picking basis functions that are consistent with the corresponding boundary conditions of the telegraph type equation. A suitable combination of the orthogonal shifted Gegenbauer polynomials is utilized. The proposed method produces systems of especially inverted matrices. Furthermore, the convergence and error analysis of the proposed expansion are investigated. This study was built on assuming that the solution to the problem is separable. The paper ends by checking the applicability and effectiveness of the proposed algorithm by solving some numerical examples.

Herein, we propose new efficient spectral algorithms for handling the fractional diffusion wave equation (FDWE) and fractional diffusion wave equation with damping (FDWED). In these algorithms, we employ new basis functions of the shifted fifth-kind Chebyshev polynomials that satisfy all the initial and boundary conditions of the equation. The key idea of the presented algorithms depends on transforming the FDWE and FDWED with their underlying conditions into systems of algebraic equations in the unknown expansion coefficients. Our study is supported by a careful convergence analysis of the suggested shifted fifth-kind Chebyshev expansion. Finally, some numerical examples are presented to confirm the accuracy and efficiency of the proposed algorithms.

This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order $\rho$ in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

The time-fractional diffusion equation is applied to a wide range of practical applications. We suggest using a potent spectral approach to solve this equation. These techniques’ main objective is to efficiently solve the linear time-fractional problem by transforming it into a system of linear algebraic equations in the expansion coefficients, together with the problem’s initial and boundary conditions. The main advantage of our technique is that the resulting linear systems have special structures which facilitate their computational solution. The numerical methods are supported by a thorough convergence study for the suggested Chebyshev expansion. Some test problems are offered to demonstrate the suggested methods’ broad applicability and a high degree of accuracy.

We present the first numerical code based on the Galerkin method to integrate the field equations of the Bondi problem. The Galerkin method is a spectral method whose main feature is to provide high accuracy with moderate computational effort. Several numerical tests were performed to verify the issues of convergence, stability and accuracy with promising results. This code opens up several possibilities of applications in more general scenarios for studying the evolution of spacetimes with gravitational waves.

The Forchheimer-extended Brinkman’s Darcy-flow model was used to investigate the initiation of ferroconvection in a flat porous layer while accounting for effective viscosity. The rigid ferromagnetic, rigid paramagnetic and stress-free isothermal boundary conditions are the three categories. The eigenvalue issue can be properly addressed for stress-free boundaries; the Galerkin approach is utilized to find the critical stability constraints quantitatively for other barriers. It was discovered that the boundary types had a strong influence on the system’s stabilization. Ferromagnetic boundaries are less preferred than paramagnetic boundaries in control of convection. The dependence of many physical limitations on the linear stability of the system is intentionally given, and it is demonstrated that increasing the value of the viscosity ratio delays the beginning of convection.

This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge–Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM.

In this paper, we investigate the intermediate-frequency oscillation in a SEPIC power-factor-correction (PFC) converter under one-cycle control. The converter operates in continuous conduction mode (CCM). A systematic method is proposed to analyze the bifurcation behavior and explain the inherent physical mechanism of the intermediate-frequency oscillation. Based on the nonlinear averaged model, the approximate analytical expressions of the nominal periodic equilibrium state are calculated with the help of Galerkin approach. Then, the stability of the system is judged by the Floquet theory and the Floquet multiplier movement of the monodromy matrix is analyzed to reveal the underlying mechanism of the intermediate-frequency oscillation behavior. In addition, Floquet multiplier sensitivity is proposed to facilitate the selection of key parameters with respect to system stability so as to guide the optimal design of the system. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical ones.

This paper shows how to approximate the solution to a nonlinear conformable space-time fractional partial differential equations. The proposed method is based on the Cubic B-spline polynomials and Galerkin method. Two test problems show that the approach we use to approximate the proposed equation is accurate and efficient. We apply the Von Neumann approach to show that stability requires some conditions.

In this study, dynamic response of a micro- and nanobeams under electrostatic actuation is investigated using strain gradient theory. To solve the governing sixth-order partial differential equation, mode shapes and natural frequencies of beam using Euler–Bernoulli and strain gradient theories are derived and then compared with classical theory. Galerkin projection is utilized to convert the partial differential equation to ordinary differential equations representing the system mode shapes. Accuracy of proposed one degree of freedom model is verified by comparing the dynamic response of the electrostatically actuated micro-beam with analogue equation and differential quadrature methods. Moreover, the static pull-in voltages of micro-beams found by one DOF model are compared with the reported data in literature. The main advantage of proposed method based on the Galerkin method is its simplicity and also its low computational cost in analyzing the dynamic and static responses of micro- and nanobeams. Additionally, effect of axial force, beam thickness and applied voltage are analyzed. The results obtained based on strain gradient theory, are compared with classical and modified couple stress theories which are the special cases of the strain gradient theory. It is shown that strain gradient theory leads to higher frequency and lower amplitude in comparison with two other theories.

This paper is concerned with the buckling and post-buckling behaviors of a simply supported symmetric functionally graded (FG) microplate lying on a nonlinear elastic foundation. The modified couple stress theory is used to capture the size effects of the FG microplate, and the Mindlin plate theory with von Karman’s geometric nonlinearity taken into account is adopted to describe its deflection behavior. Based on these assumptions and the principle of minimum potential energy, the equilibrium equations of the FG microplate and associated boundary conditions are derived. By applying the Galerkin method to the equilibrium equations, closed-form solutions for the critical buckling load and the load–displacement relation in the post-buckling stage are obtained. Furthermore, the effects of the power law index, the material length scale parameter to thickness ratio, the stiffness of the elastic foundation, and in-plane boundary conditions on the buckling and post-buckling behaviors of the FG microplate are discussed in detail.

Dynamic analysis of an Euler–Bernoulli beam with nonlinear supports is receiving greater research interest in recent years. Current studies usually consider the boundary and internal nonlinear supports separately, and the system rotational restraint is usually ignored. However, there is little study considering the simultaneous existence of axial load, lumped mass and internal supports for such nonlinear problem. Motivated by this limitation, the dynamic behavior of an axially loaded beam supported by a nonlinear spring-mass system is solved and investigated in this paper. Modal functions of an axially loaded Euler–Bernoulli beam with linear elastic supports are taken as trail functions in Galerkin discretization of the nonlinear governing differential equation. Stable steady-state response of such axially loaded beam supported by a nonlinear spring-mass system is solved via Galerkin truncation method, which is also validated by finite difference method. Results show that parameters of nonlinear spring-mass system and boundary condition have a significant influence on system dynamic behavior. Moreover, appropriate nonlinear parameters can switch the system behavior between the single-periodic state and quasi-periodic state effectively.

In this study, the free vibrational behavior of a thin-walled functionally graded conical shell with intermediate ring support is investigated. Theoretical formulations were established based on the first-order shear deformation theory. The governing equations of motion were solved using the Galerkin method. Applying a set of displacement functions, the equations of motion result in an eigenvalue problem, by solving which, the natural frequencies of vibration are determined. Material properties are assumed to be varied in the thickness direction according to the power-law volume fraction function. It has been attempted to examine the effects of ring support position on the natural frequencies of vibration and to introduce the optimal scenarios of the support placement to achieve a higher frequency. In addition, a 3D FE model was built in the ABAQUS CAE software in order to validate the results of the analytical model. The analytical results were in close agreement with the literature and also the numerical ones. Moreover, the effects of some commonly used end conditions, variations in the shell geometrical parameters, changes in the ring support placement have been investigated on the vibrational behavior.

For some transmission shafting, nonlinear supports are connected to beam structures through the pattern of the contact surface. Nonlinear foundations are typically installed to the beam structure for a limited range. The unsuitable installation mode of nonlinear foundations will make the parameters of nonlinear foundations no longer uniform. Most existing studies ignore the boundary rotational restraints of beam structures and concentrated mass introduced by the nonlinear supports or foundations. To improve the engineering acceptance of beam structures with nonlinearity, it is of great significance to study the dynamic behavior of the generally restrained pre-pressure beam structure with a partial non-uniform foundation of nonlinear stiffness. This study establishes the nonlinear vibration model of the beam structure with a local non-uniform nonlinear stiffness distribution. Nonlinear dynamic responses of the beam structure are predicted by the Galerkin truncation method (GTM). Mode functions of the generally restrained pre-pressure beam structure are set as the trail and weight function. The correctness of the GTM for dynamic prediction of the beam structure with a partial non-uniform nonlinear foundation is verified by using the harmonic balance method (HBM). The influence of sweeping ways of excitations and parameters of the partial non-uniform nonlinear foundation on nonlinear dynamic responses of the beam structure are investigated. Dynamic responses of the beam structure with a partial non-uniform nonlinear foundation are sensitive to their initial values. Vibration states of the beam structure are transformed effectively by changing parameters of the partial non-uniform nonlinear foundation. The vibration at both ends of the beam structure can be suppressed by applying suitable parameters of the partial non-uniform nonlinear foundation.

The sloped bottom tank generates higher sloshing-induced base shear with less liquid mass compared to rectangular tanks. Due to the improved sloshing characteristics, sloped bottom TLD achieves better performance than its conventional flat-bottom counterpart. These improvements in sloshing characteristics are due to the modified tank configuration, which reduces the impulsive liquid mass and results in higher convective participation of liquid mass during sloshing. To further increase the convective characteristics, a sloped bottom tank with internal object is proposed. A velocity potential-oriented finite element model has been adopted for the numerical analysis where a combination of three-node triangular and four-node quadrilateral finite elements is used to discretize the liquid domain. Seismic characteristics of liquid sloshing inside a ground-supported sloped bottom tank with internal object are investigated under six different seismic motions categorized as low-, intermediate-, and high-frequency contents. Also, the influence of the internal object on slosh dynamics is investigated suitably for different heights of the object under all earthquakes considered. The dynamic impulsive and convective components of base shear force and hydrodynamic pressure are quantified successfully. The developed numerical model in this study can be utilized in tuning and designing structures coupled with sloped-bottom TLD.