Volume 18, Issue 5 -

Abstract: In this paper, we study the nonlinear matrix equation Xs±∑i=1mAiTXδiAi=Q$X^{s}\pm\sum^{m}_{i=1}A^{T}_{i}X^{\delta_{i}}A_{i}=Q$, where Ai(i=1,2,…,m)$A_{i}\;(i=1,2,\ldots,m)$ is n×n$n\times n$ nonsingular real matrix and Q$Q$ is n×n$n\times n$ Hermitian positive definite matrix. It is shown that the equation has an unique Hermitian positive definite solution under some conditions. Iterative algorithms for obtaining the Hermitian positive definite solution of the equation are proposed. Finally, numerical examples are reported to illustrate the effectiveness of algorithms.

An energy stable conservative method is developed for the Cahn–Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the fully discretized CH equation. Numerical results for the quartic double-well and the logarithmic potential functions with constant and degenerate mobility confirm the theoretical convergence rates, accuracy and the performance of the proposed approach.

In this article, we introduce fractional-order Solow-type models as a new tool for modeling and analysis in mathematical finance. Sufficient conditions for the Mittag–Leffler stability of their states are derived. The main advantages of the proposed approach are using of fractional-order derivatives, whose nonlocal property makes the fractional calculus a suitable tool for modeling actual financial systems as well as using of impulsive perturbations which give an opportunity to control the dynamic behavior of the model. The modeling approach proposed in this article can be applied to investigate macroeconomic systems.

This article emphasizes the influence of convective boundary condition on mixed convection flow of a micropolar fluid over a truncated cone with Soret and viscous dissipation effects. The governing micropolar fluid flow equations are non-dimensionalized using suitable non-similarity transformations. Several authors have applied the spectral quasilinearization method to solve the ordinary differential equations, but here the resulting nonlinear partial differential equations are solved for non-similarity solution by using a newly developed method called the spectral quasilinearization method (SQLM). The comparison of convection process namely free, forced and mixed convection on the micropolar fluid is provided in detail. The convergence and error analysis are also discussed to test the accuracy of the spectral method. From the results, it perceived that with the rise in viscous dissipation parameter, the wall couple stress coefficient and Nusselt number reduce, but velocity, temperature, concentration, skin friction coefficient and Sherwood number increase for both in the absence and in the presence of Soret number.

The implicit coupling method is applied to model the 0.8 m disk-band-gap parachute at Mach 2.0. The fluid and structure governing equations are solved by the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) algorithm and Newmark scheme, respectively. By exchanging the numerical results of the coupling surface with Gauss-Seidel algorithm, high accuracy solutions at every physical time step are obtained. The numerical results of the canopy drag coefficient and projected area fit well with the wind tunnel test results. The simulation reproduces the shock oscillation and breathing phenomenon of the canopy that are usually observed in these systems at Mach 2.0. Furthermore, it is found that the unstable saddle point is the main reason for the shock oscillation of the canopy. And the unsynchronized phases of the canopy area and shock oscillation curves lead to the drag of the canopy oscillate in irregular state.

Many turbulent flow simulations require the use of hybrid methods because LES methods are computationally too expensive and RANS methods are not sufficiently accurate. We consider a recently suggested hybrid RANS-LES model that has a sound theoretical basis: it is systematically derived from a realizable stochastic turbulence model. The model is applied to turbulent swirling and nonswirling jet flow simulations. The results are shown to be in a very good agreement with available experimental data of nonswirling and mildly swirling jet flows. Compared to commonly applied other hybrid RANS-LES methods, our RANS-LES model does not seem to suffer from the ’modeled-stress depletion’ problem that is observed in DES and IDDES simulations of nonswirling jet flows, and it performs better than segregated RANS-LES models. The results presented contribute to a better physical understanding of swirling jet flows through an explanation of conditions for the onset and the mechanism of vortex breakdown.

In this article, extended complex Lü models (ECLMs) are proposed. They are obtained by substituting the real variables of the classical Lü model by complex variables. These projections, spanning from five dimensions (5D) and six dimensions (6D), are studied in their dynamics, which include phase spaces, calculations of eigenvalues and Lyapunov’s exponents, Poincaré maps, bifurcation diagrams, and related analyses. It is shown that in the case of a 5D extension, we have obtained chaotic trajectories; meanwhile the 6D extension shows quasiperiodic and hyperchaotic behaviors and it exhibits strange nonchaotic attractor (SNA) features.

Abstract: We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis.

The linear superposition principle is applied to hyperbolic and trigonometric function solutions to generalized bilinear equations. We determine sufficient and necessary conditions for the existence of linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations. By using weights, three examples are given to show applicability of our theory.

Heavy-gauge Q235 steel plate was successfully roll bonded, and this process was simulated using MARC software. The test results revealed that the chosen ultrasonic testing method satisfies Ⅰ level requirements and the mechanical properties of the materials fully meet the necessary standards. There were uniform ferrite+pearlite microstructures and sound bond areas at the bond interface. Simultaneously, fracture analysis revealed ductile fracture. Using EDS analysis, oxide+MnS inclusions were detected in sample indentations. The MARC simulation showed that after the first rolling pass, little equivalent strain occurred at the center of slab. At the third hot rolling pass, the equivalent strain of the center increased to 0.5. When rolling the last pass, the equivalent strain tended to be uniform in the roll gap region.

This paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel technique takes advantage of the properties of shifted Jacobi polynomials and is applied for solving multi-dimensional VIEs. Several numerical examples demonstrate the efficiency of the method and an error analysis verifies the correctness and feasibility of the proposed method when solving VIE.

In this paper, the classic Lorenz system is revisited. Some dynamical behaviors are shown with the Rayleigh number ρ$\rho $ somewhat smaller than the critical value 24.06 by studying the basins characterization of attraction of attractors and tracing the one-dimensional unstable manifold of the origin, indicating some interesting clues for detecting the existence of hidden transient chaos. In addition, horseshoes chaos is verified in the famous system for some parameters corresponding to the hidden transient chaos by the topological horseshoe theory.

In the current manuscript, an attempt has been made to understand the dynamics of a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses for large initial data. In Ref. Upadhyay and Agrawal, 83(2016) 821–837, it was shown that the model possesses globally bounded solutions, for small initial conditions, under certain parametric restrictions. Here, we show that actually solutions to this model system can blow-up in finite time, for large initial condition, even under the parametric restrictions derived in Ref. Upadhyay and Agrawal, 83(2016) 821–837. We prove blow-up in the delayed model, as well as the non-delayed model, providing sufficient conditions on the largeness of data, required for finite time blow-up. Numerical simulations show that actually the initial data does not have to be very large, to induce blow-up. The spatially explicit system is seen to possess non-Turing instability. We have also studied Hopf-bifurcation direction in the spatial system, as well as stability of the spatial Hopf-bifurcation using the central manifold theorem and normal form theory.