In this paper, a new concept, i.e., ultra-chaos, is proposed for the first time. Unlike a normal-chaos, statistical properties such as the probability density functions (PDF) of an ultra-chaos are sensitive to tiny disturbances. We illustrate that ultra-chaos is widely existed and thus has general scientific meanings. It is found that statistical non-reproducibility is an inherent property of an ultra-chaos so that an ultra-chaos is at a higher-level of disorder than a normal-chaos. Thus, it is impossible in practice to replicate experimental/numerical results of an ultra-chaos even in statistical meanings, since random environmental noises always exist and are out of control. Thus, the ultra-chaos should be an insurmountable obstacle of reproducibility and replicability. Similar to Gödel's incompleteness theorem, such kind of "incompleteness of reproducibility" reveals a limitation of our traditional scientific paradigm based on reproducible experiments, which can be traced back to Galileo. The ultra-chaos opens a new door and possibility to study chaos theory, turbulence theory, computational fluid dynamics (CFD), the statistical significance, reproducibility crisis, and so on.

This research work deals with a study on dynamic behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams under various types of dynamic loads. Carbon nanotubes (CNTs) are used as the reinforcing materials that distribute continuously across the beam thickness. By using third order shear deformable theory (TSDT) in this current study, the straightness and normality of the transverse normal after deformation are unconstrained. The equations of motion based on TSDT are solved by Gram-Schmidt-Ritz method in which the displacement functions are generated via Gram-Schmidt procedure. Additionally, the time-integration of Newmark is also employed to carry out dynamic response of the beams under dynamic loads. Several effects such as material distributions, types of dynamic loads, boundary conditions and so on are taken into account. According to numerical results, it can be revealed that adding small amount of CNTs can reduce considerably the dynamic amplitude of FG-CNTRC beams. Moreover, the dynamic analysis of beam-like structures plays an important role in structural design because mass inertia matrix of the beam being involved in the equations of motion, which yields much larger deflection than that predicted by simple static analysis.

In this paper, we propose an immersed finite volume element method for solving the semi-linear elliptic interface problems with non-homogeneous jump conditions. Furthermore, two-grid techniques are used to improve the computational efficiency. In this way, we only need to solve a non-linear system on the coarse grid, and a linear system on the fine grid. Numerical results illustrate that the proposed method can solve the semi-linear elliptic interface problems efficiently. Approximate second-order accuracy for the solution in the $L^{\infty}$ norm can be obtained for the considered examples.

In this paper, a new technique for analysing functionally graded material (FGM) beams using the Chebyshev polynomials and Lagrange multipliers with various beam theories is presented. By utilizing the inner products and the Chebyshev polynomials' orthogonality properties incorporated with Lagrange multipliers, we can combine the governing equation and boundary conditions to yield the matrix equations with explicit weighting coefficients. Numerical examples are provided for vibration analysis of various beam theories and assumptions. Based on numerical evaluations, it is revealed that the proposed technique can efficiently achieve good agreement with those of the references.

We employ the Riemann-Hilbert (RH) method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions. Through the spectral analysis, the asymptoticity, symmetry, and analysis of the Jost functions are obtained, which play a key role in constructing the RH problem. Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem. Choosing some appropriate parameters of the resulting solutions, we further derive the soliton solutions with different order poles, including four cases of a fourth-order pole, two second-order poles, a third-order pole and a first-order pole, and four first-order points. Finally, the dynamical behavior of these solutions are analyzed via graphic analysis.

In this paper, a lattice Boltzmann model with BGK operator (LBGK) for solving time-fractional nonlinear wave equations in Caputo sense is proposed. First, the Caputo fractional derivative is approximated using the fast evolution algorithm based on the sum-of-exponentials approximation. Then the target equation is transformed into an approximate form, and for which a LBGK model is developed. Through the Chapman-Enskog analysis, the macroscopic equation can be recovered from the present LBGK model. In addition, the proposed model can be extended to solve the time-fractional Klein-Gordon equation and the time-fractional Sine-Gordon equation. Finally, several numerical examples are performed to show the accuracy and efficiency of the present LBGK model. From the numerical results, the present model has a second-order accuracy in space.

This paper is concerned with the weak Galerkin mixed finite element method (WG-MFEM) for the second-order hyperbolic acoustic wave equation in velocity-pressure formulation. In this formulation, the original second-order differential equation in time and space is reduced to first-order differential equations by introducing the velocity and pressure variables. We employ the usual discontinuous piecewise-polynomials of degree $k\geq 0$ for the pressure and $k+1$ for the velocity. Furthermore, the normal component of the pressure on the interface of elements is enhanced by polynomials of degree $k+1$. The time derivative is approximated by the backward Euler difference. We show the stability of the semi-discrete and fully-discrete schemes, and obtain the suboptimal order error estimates for the velocity and pressure variables. Numerical experiment confirms our theoretical analysis.

Image segmentation is a significant problem in image processing. In this paper, we propose a new two-stage scheme for segmentation based on the Fischer-Burmeister total variation (FBTV). The first stage of our method is to calculate a smooth solution from the FBTV Mumford-Shah model. Furthermore, we design a new difference of convex algorithm (DCA) with the semi-proximal alternating direction method of multipliers (sPADMM) iteration. In the second stage, we make use of the smooth solution and the K-means method to obtain the segmentation result. To simulate images more accurately, a useful operator is introduced, which enables the proposed model to segment not only the noisy or blurry images but the images with missing pixels well. Experiments demonstrate the proposed method produces more preferable results comparing with some state-of-the-art methods, especially on the images with missing pixels.

Heating or cooling one-dimensional inviscid compressible flow can be modeled by the Euler equations with energy sources. A tricky situation is the sudden appearance of a single-point energy source term. This source is discontinuous in both the time and space directions, and results in multiple discontinuous waves in the solution. We establish a mathematical model of the generalized Riemann problem of the Euler equations with source term. Based on the double CRPs coupling method proposed by the authors, we determine the wave patterns of the solution. Theoretically, we prove the existence and uniqueness of solutions to both "heat removal" problem and "heat addition" problem. Our results provide a theoretical explanation for the effect of instantaneous addition or removal of heat on the fluid.