Results in Applied Mathematics最新文献

In this work we study travelling wave solutions to bistable reaction–diffusion equations on bi-infinite $k$-ary trees in the continuum regime where the diffusion parameter is large. Adapting the spectral convergence method developed by Bates and his coworkers, we find an asymptotic prediction for the speed of travelling front solutions. In addition, we prove that the associated profiles converge to the solutions of a suitable limiting reaction–diffusion PDE. Finally, for the standard cubic nonlinearity we provide explicit formulas to bound the thin region in parameter space where the propagation direction undergoes a reversal.

A method classically used in the lower polynomial degree for the construction of a finite element basis of the space of divergence-free functions is here extended to any polynomial degree for a bounded domain without topological restrictions. The method uses graphs associated with two differential operators: the gradient and the divergence, and selects the basis using a spanning tree of the first graph. It can be applied for the two main families of degrees of freedom, weights and moments, used to express finite element differential forms.

In this work, the $\psi$-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, $\psi$, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the $\psi$-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the $\psi$-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

Based on the existing initial boundary value and variational problems of Lord–Shulman thermopiezoelectricity a transient analysis of the behavior of piezoelectric materials is performed numerically. For space discretization of the variational problem the finite element method is used and for time discretization a hybrid time integration scheme is constructed on the basis of Newmark scheme for hyperbolic equation and generalized trapezoidal rule for parabolic equations. The unconditional stability of the developed time integration scheme is proved for some specific values of the scheme parameters by making use of energy balance law for the obtained time-discretized variational problem. Finally, the applicability of the constructed numerical scheme is demonstrated by the results of the numerical experiment which are compared with the ones available in literature.

This paper addresses the problem of characterizing and localizing objects via through-the-wall radar imaging. We consider two separate problems. First, we assume a single object is located in a room and we use a convolutional neural network (CNN) to classify the shape of the object. Second, we assume multiple objects are located in a room and use a U-net CNN to determine the location of the object via pixel-by-pixel classification. For both problems, we use numerical methods to simulate the electromagnetic field assuming known room parameters and object location. The simulated data is used to train and evaluate both the CNN and U-net CNN. In the case of single objects, we achieve 90% accuracy in classifying the shape of the object. In the case of multiple objects, we show that the U-Net outputs an image segmentation heat map of the domain space, enabling visual analysis to identify the characteristics of multiple unknown objects. Given sufficient data, the U-net heat map highlights object pixels which provide the location and shape of the unknown objects, with precision and recall accuracy exceeding 80%.

In this paper, we introduce the flexible interpretable gamma (FIG) distribution, with origins in Weibulisation, power weighting, and a stochastic representation. The FIG parameters have been verified graphically, mathematically, and through simulation as having separable roles in influencing the left tail, body, and right tail shape. The generalised gamma (GG) distribution has become a standard model for positive data in statistics due to its interpretable parameters and tractable equations. Although there are many generalised forms of the GG that can provide a better fit to data, none of them extend the GG so that the parameters are interpretable. We conduct simulation studies on the maximum likelihood estimates and respective sub-models of the FIG. Finally, we assess the flexibility of the FIG relative to existing models by applying the FIG model to hand grip strength and insurance loss data.

Square matrices of the form $\tilde{A}=A+eD{f}^{\ast }$ are considered. An explicit expression for the inverse is given, provided $\tilde{A}$ and $D$ are invertible with $rank\left(\tilde{A}\right)=rank\left(A\right)+rank\left(eD{f}^{\ast }\right)$. The inverse is presented in two ways, one that uses singular value decomposition and another that depends directly on the components $A$, $e$, $f$ and $D$. Additionally, a matrix determinant lemma for singular matrices follows from the derivations.

In solving the coupled vapor and liquid unidirectional flows in micro heat pipes, we discovered numerically an integral identity. After asymptotic and polynomial expansions, the coupled flows yield two reciprocal systems of equations. In system A, a vapor velocity $U_A$ obeys the Poisson equation and drives, through an interfacial boundary condition, a liquid velocity $W_A$ that satisfies the Laplace equation. In reciprocal system B, a liquid velocity $W_B$ obeys the Poisson equation and drives, through another interfacial boundary condition, a vapor velocity $U_B$ that satisfies the Laplace equation. We found that the vapor volume flow rate of $U_B$ is numerically equal to the liquid volume flow rate of $W_A$ for seven different pipe shapes. Here, a general proof is presented for the integral identity, and some interesting implications of this identity are discussed.

With the continuous development in the field of deep learning, in recent years, it has also been widely used in the field of solving partial differential equations, especially the physics-informed neural networks (PINNs) method. However, the PINNs method has some limitations in solving coupled Klein–Gordon–Zakharov (KGZ) systems. To this end, in this article, inspired by the PINNs method and combined with the characteristics of the coupled KGZ systems, we design a neural network model, named multi-output physics-informed neural networks based on time differential order reduction (TDOR-MPINNs), to solve the coupled KGZ systems. Compared with the PINNs, the TDOR-MPINNs first reduces the time derivatives, and thus can increase supervised learning tasks. And through comparing the numerical results obtained by using TDOR-MPINNs and PINNs for solving the one-dimensional (1-D) and two-dimensional (2-D) coupled KGZ systems, we further validate the effectiveness, accuracy and reliability of the TDOR-MPINNs.