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In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD 4 + T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host’s blood is considered a separate class. It is reported that when HIV enters a human’s body, it infects a large amount of CD 4 + T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD 4 + T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2 × 2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems.

Recent advances in the field of uncertainty quantification are based on achieving suitable functional representations of the solutions to random systems. This aims at improving the performance of Monte Carlo simulation, at least for low-dimensional problems and moderately large independent variable. One of these functional representations are the so-called generalized polynomial chaos (gPC) expansions, based upon orthogonal polynomial decompositions. When the input random parameters are independent (a germ), a Galerkin projection technique applied to the truncated gPC expansion is usually employed. This approach exhibits fast mean-square convergence for smooth dynamics, whenever applicable. However, the main difficulty arises when solving the Galerkin system for the gPC coefficients, which may rely on different solvers (algorithms and codes) than those for the original system. A recent contribution noticed that, for random Hamiltonian systems, the Galerkin system is Hamiltonian too. Thus, the well-known symplectic integrators can be applied. The present paper investigates random Hamiltonian systems in general, when the input random parameters may be non-independent. In such a case, polynomial expansions based on the canonical basis and an imitation of the Galerkin projection technique are proposed. The Hamiltonian structure of the original system is unfortunately not conserved, but volume preservation is. Hence volume-preserving integrators are of use. Numerical experiments suggest that the proposed polynomial expansions may be useful for fast and accurate uncertainty quantification.

With analytical and numerical methods, dynamic response of a composite laminated buckled beam with a lumped mass and an axial excitation is investigated in this paper. The dynamic equation of the first mode for this model, which is a perturbation of a Hamilton system with both quadratic and cubic nonlinearities as well as external and parametric excitations, is obtained with the Galerkin method. The distribution of the equilibriums, homoclinic orbits and periodic orbits for the unperturbed system under different system parameters are obtained analytically. Chaotic vibrations arising from homoclinic intersections are studied with Melnikov method. Chaotic feature on the location of the lumped mass is investigated in detail. It is presented that there exist controllable frequencies decreasing monotonously with the location of the lumped mass for this system. Subharmonic bifurcation is also studied with the subharmonic Melnikov method. It is numerically proved that the system may undergo chaotic motions through finite subharmonic bifurcations. Numerical simulations including the phase portraits, time histories, Poincar e ´ sections, Lyapunov spectrums and bifurcation diagram are given, which confirm the analytical results.

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincare map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.

The Fractional-order model of fin problem in the presence of a temperature-dependent rate of internal heat generation in fin material has been studied in the proposed paper. Fractional-order Legendre functions (F-OLFs) with the Galerkin approach has been used to compute the result of the proposed fin problem. The exact solution is also calculated in a particular case of our fractional-order fin problem. The proposed fin problem has been presented in the non-dimension form. Temperature distribution in the straight fin is calculated for different values of used variables or parameters along space coordinate. The impact of different parameters such as fractional order, fin parameter, internal heat generation, generation number, etc. on the temperature profile in the fin is described in detail. The cooling process is faster in fractional order fin problem than the integer-order.

This paper investigated the nonlinear dynamics of a cracked vibration platform subjected to a concentrated transverse excitation force considering the structural and supported nonlinearities. To precisely predict vibration properties of the continuum structure, the dynamic model is established based on mechanical relations and the governing differential equations of motion are obtained with the nonlinear regime via an energy method according to the Lagrange's equation, which are discretized by utilizing the Galerkin truncation method. A comparative analysis via the finite element method (FEM) for verification accuracy and convergence is performed. To investigate the influences of the key parameters on the nonlinear behaviors of cracked platform, a wide range of excitation, nonlinear support and crack parameters are analyzed and discussed by bifurcation, time domain for nonlinear mechanics analysis. The numerical results show that a considerable effect of excitation amplitude on the global dynamics for the vibration platform is discovered, namely the higher excitation is able to suppress the chaotic behaviors but increase the vibration amplitude of the system. The increasing nonlinear support stiffness may enhance the chaotic characteristics of the vibration platform, the support position can effectively control the chaotic motion and nonlinear vibration as the support is close to the middle of vibration platform. The crack parameters play an effective role in dynamic behaviors.