Results in Applied Mathematics最新文献

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.

This paper presents a positivity-preserving discontinuous Galerkin (DG) scheme for the linear hyperbolic problem with variable coefficients on structured Cartesian domains. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the unmodulated augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. A simple scaling limiter can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients. To improve efficiency, an inexact augmented basis function can be obtained rather than a analytic non-polynomial solution to the auxiliary problem from the method of characteristics.

The Lotka–Volterra predator–prey system for dynamic sea turtle population is solved using r-PINN-Adam method, a novel approach which combines Physics-Informed Neural Network (PINN) with restarting strategy. This method allows us to monitor the loss function values of PINN such that when there is no progress made, we stop the process and pick a good value to be used in the next process. Subsequently, the training time decreases and the accuracy increases. The numerical solutions are compared to the popular Runge–Kutta method in terms of correctness which presented graphically. Simulation results also displayed in terms of trainable parameters and optimal loss function performance. The research highlights the robustness and superiority of the proposed method, positioning it as a valuable tool for sea turtle conservation efforts.

In this paper, we employ a nonclassical sinc-collocation method to compute numerical solutions for singularly perturbed singular third-order boundary value problems prevalent in various scientific and engineering domains. Utilizing the sinc approximation offers a strategic advantage in navigating singularities, thus enabling an efficient computational strategy. Our method streamlines the solution process by converting singular boundary value problems into sets of linear equations, thereby improving computational efficiency. Moreover, its straightforward implementation adds to its robustness. We explore the convergence properties and error estimation of our proposed methods in detail. Finally, we provide two illustrative examples that demonstrate the effectiveness of our approach.

This study introduces a hybrid procedure based on a block-by-block scheme (created by the Gauss–Lobatto integration formula) and a set of the hybrid functions (defined by the Legendre polynomials and block-pulse functions) to solve a class of systems of mixed Volterra–Fredholm integral equations. More precisely, the proposed scheme combines the Gauss–Lobatto quadrature rule for the temporal variable and the hybrid functions for the spacial direction. In the established procedure, several values of the problem solution are elicited simultaneously, without employing any starting value for beginning. The convergence, along with the analysis of error for the method are proved. Some numerical examples are solved to show the efficiency and accuracy of the proposed strategy.

A prior error estimate is considered for the finite element (FE) approximation of a parabolic optimal control (POC) with spatial measurement data. We use conforming linear finite element to discretize the space for the state, piecewise constant for the control, and Euler method to discretize the time. The convergence order $O(h^{2-\frac{s}{2}}+k^{\frac{1}{2}})$ in the $L^2(0,T,L^2\left(\Omega \right))$-norm of state variable, co-state, and control variable are obtained. To validate our theory, numerical tests are executed.

In this piece of work, a family of compact implicit numerical algorithms for (∂u/∂n) of order of accuracy six are proposed on a 9- and 19-point compact cell for two- and three- dimensional Poisson equations ∆²u=f which are quite often useful in mathematical physics and engineering, where ∆² is either two or three dimensional Laplacian operator. First, we propose a family of new numerical algorithms of order of accuracy six for the computation of the solution of 2D and 3D Poisson equations on 9- and 27-points compact stencil, respectively. Then with the aid of the numerical solution of u, we propose a new family of compact sixth order implicit numerical algorithms for the estimates of (∂u/∂n). The proposed algorithms are free from derivatives of the source functions, which makes our algorithms more efficient for computation. Suitable iteration techniques are used for computation to demonstrate the sixth order convergence of the proposed algorithms. Numerical results are tabulated, confirming the usefulness of the suggested numerical algorithms.

In this work, we couple a high-accuracy phase-field fracture reconstruction approach iteratively to fluid–structure interaction. The key motivation is to utilise phase-field modelling to compute the fracture path. A mesh reconstruction allows a switch from interface-capturing to interface-tracking in which the coupling conditions can be realised in a highly accurate fashion. Consequently, inside the fracture, a Stokes flow can be modelled that is coupled to the surrounding elastic medium. A fully coupled approach is obtained by iterating between the phase-field and the fluid–structure interaction model. The resulting algorithm is demonstrated for several numerical examples of quasi-static brittle fractures. We consider both stationary and quasi-stationary problems. In the latter, the dynamics arise through an incrementally increasing given pressure.

This letter presents an extended analysis and a novel upper bound of the subclass of Linear Quadratic Near Potential Differential Games (LQ NPDG). LQ NPDGs are a subclass of potential differential games, for which there is a distance between an LQ exact potential differential game and the LQ NPDG. LQ NPDGs exhibit a unique characteristic: The smaller the distance from an LQ exact potential differential game, the more closer their dynamic trajectories. This letter introduces a novel upper bound for this distance. Moreover, a linear relation between this distance and the resulting trajectory errors is established, opening the possibility for further application of LQ NPDGs.