Two-dimensional Equations Research Articles

Biquadratic element discrete duality finite volume method for solving elliptic equations on quadrilateral mesh

Nowadays numerical methods for handling partial differential equations is so vast that it is almost taken for granted. In this paper, we propose a discrete duality finite volume scheme with biquadratic elements and its application in solving two-dimensional elliptic equation on quadrilateral meshes. To test the robustness and efficiency of this method, we investigate several examples from physical and engineering fields with different parameters and coefficients. Numerical experiments, encompassing linear and nonlinear elliptic equations with constant or variable coefficients, reveal that the new method achieves optimal convergence in the continuous H1-norm, with a 2nd-order convergence, and in the L2-norm, with a 3rd-order convergence. Moreover, comparison with the existing biquadratic element finite volume scheme shows that the above method has an advantage of accuracy in the discrete L∞-norm on several distorted meshes.

Trans-Net: A transferable pretrained neural networks based on temporal domain decomposition for solving partial differential equations

Physics-Informed Neural Networks (PINNs) has provided a novel direction for solving partial differential equations (PDEs) and has achieved significant advancements in the field of scientific computing. PINNs effectively incorporate the physical constraints of equations into the loss function, enabling neural networks to learn and approximate the behavior of physical systems by optimizing the loss function. According to the existing research, we propose a transferable pretrained neural networks framework based on temporal domain decomposition to solve partial differential equations. Specifically, we divide the domain into multiple subdomains according to time, each of which is solved using an individual neural network. Subsequently, the trained subnetwork and the predicted values at the common boundary are used as the pretrained model and initial values for the next subdomain, respectively. This not only improves the prediction accuracy of the subnetwork, but also reduces the training time. Finally, we present a series of classical numerical experiments including one-dimensional, two-dimensional, and three-dimensional partial differential equations. The experimental results indicate that the proposed method outperforms existing approaches in terms of accuracy and efficiency. Relevant data and code can be accessed on GitHub via https://github.com/karry-Zhang/Trans-Net.

High order conservative Lagrangian schemes for two-dimensional radiation hydrodynamics equations in the equilibrium-diffusion limit

Radiation hydrodynamics equations (RHE) refer to the study of how interactions between radiation and matter influence thermodynamic states and dynamic flow, which has been widely applied to high temperature hydrodynamics, such as inertial confinement fusion (ICF) and astrophysical gaseous stars. Solving RHE accurately and robustly even under the equilibrium diffusion approximation is a challenging task. To address this, we develop two types of high order conservative Lagrangian schemes for RHE in the equilibrium-diffusion limit for the two dimensional case on the Lagrangian moving mesh. Based on the multi-resolution WENO reconstruction for the spatial discretization and strong stability preserving Runge-Kutta (SSP-RK) time discretization, we first develop an explicit Lagrangian scheme with the HLLC numerical flux to achieve high order accuracy in space and time. We also discuss the positivity-preserving property of the high order explicit Lagrangian scheme. To overcome the severe time step restriction arising from the nonlinear radiation diffusion term in the explicit scheme, we further present a high order explicit-implicit-null (EIN) Lagrangian scheme. By adding a sufficiently large linear diffusion term on both sides of the scheme, we treat the complicated nonlinear parts explicitly and efficiently, and treat the added linear diffusion term on the right-hand side implicitly with a relaxed time step restriction. According to our numerical experiments, these two types of Lagrangian schemes are high order accurate, conservative and can capture the interfaces automatically. Additionally, the explicit scheme is found to be non-oscillatory and can preserve positivity while maintaining the original high order accuracy.

Determination of a Coefficient and Kernel in a Two-dimensional Fractional Integrodifferential Equation

Determination of a Coefficient and Kernel in a Two-dimensional Fractional Integrodifferential Equation

Fourth-order modon in a rotating self-gravitating fluid

We present a two-dimensional nonlinear equation to govern the dynamics of disturbances in a rotating self-gravitating fluid. The nonlinear term of the equation has the form of a Poisson bracket (Jacobian), and the linear part contains, along with the Laplacian, a biharmonic operator. A solution was found in the form of a dipole vortex (modon). The solution and all its derivatives up to the fourth order are continuous on the separatrix.

Solution to a Two-Dimensional Nonlinear Parabolic Heat Equation Subject to a Boundary Condition Specified on a Moving Manifold

Solution to a Two-Dimensional Nonlinear Parabolic Heat Equation Subject to a Boundary Condition Specified on a Moving Manifold

A space-time spectral approximation for solving two dimensional variable-order fractional convection-diffusion equations with nonsmooth solutions

The study focuses on the numerical solutions of two-dimensional variable-order fractional convection-diffusion equations, which combine the principles of diffusion and convection to describe the movement of particles, energy, other physical quantities within a system. The numerical solution is obtained using shifted Legendre Gauss–Lobatto and shifted Chebyshev Gauss–Radau collocation techniques. The convection-diffusion equation is transformed into a system of algebraic equations utilizing shifted Chebyshev Gauss–Radau and shifted Legendre Gauss–Lobatto nodes. Additionally, numerical test examples are presented to demonstrate the method’s efficacy to handle nonsmooth solutions to the given problems.

Learning domain-independent Green’s function for elliptic partial differential equations

Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

Solvable Two-Dimensional Dirac Equation with Matrix Potential: Graphene in External Electromagnetic Field

It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of a massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now, such two-component wave functions were calculated for different forms of external potentials, though as a rule depending on only one spatial variable. Here, we shall find analytically the solutions for a wide class of combinations of matrix and scalar external potentials which physically correspond to applied mutually orthogonal magnetic and longitudinal electrostatic fields, both depending really on two spatial variables. The main tool for this progress is provided by supersymmetrical (SUSY) intertwining relations, specifically, by their most general—asymmetrical—form proposed recently by the authors. This SUSY-like method is applied in two steps, similar to the second order factorizable (reducible) SUSY transformations in ordinary quantum mechanics.

On The Double Integral Transform (Complex EE Transform) and Their Properties and Applications

Due to the importance of solutions of partial differential equations, linear, nonlinear, homogeneous, and non-homogeneous, in important life applications, including engineering applications, physics and astronomy, medical sciences, and life technology, and their importance in solutions to heat transfer equations, wave, Laplace equation, telegraph, etc. In this paper, a new double integral transform has been proposed. In this work, we have introduced a new double transform ( Double Complex EE Transform ). In addition, we presented the convolution theorem and proved the properties of the proposed transform, which has an effective and useful role in dealing with the solution of two-dimensional partial differential equations. Moreover, two examples of important mathematical equations are solved to illustrate method. This double integral transformation has a complex kernel.

Heat transfer analysis of single-walled carbon nanotubes in Ellis's fluid model: Comparative study of uniform and non-uniform channels

BackgroundThe essential features of nanoparticles in nanofluids play a vital role in nanotechnology. The nanosized particles of carbon are known as carbon nanotubes (CNTs). The properties of CNT like thermal and electrical conductivity, lightweight, flexibility, and mechanical strength support its applications in the medical field i.e., diagnostic and therapeutic agents (drugs, antibodies, biosensors, vaccines etc.). These are not only excellent vehicles for drug delivery and gene therapies but also useful in biosensor diagnostic, enantiomer separation of chiral drugs, tissue regeneration, extraction, and analysis of pollutants and drugs. ObjectiveThe focus of this research work is to open the essential features of peristaltic transport of Ellis nanofluid through uniform and non-uniform channels with suspension of single-walled carbon nanotubes (SWCNT). The consideration of blood flow with SWCNT under the effects of heat generation and electroosmosis is one of the key factors of this investigation. MethodThe complexity of the system is reduced with the implementation of the concepts of longer wavelength and lower Reynolds number. The governing two-dimensional equations are simplified by introducing dimensionless variables and parameters. The mathematical tool “MATHEMATICA 13.3” is used to compute the exact solution and graphical demonstration of velocity, temperature, heat transfer rate, and streamlines. Computational resultsIt is noticed that velocity, temperature, and heat transfer rate are prominent in non-uniform channels as compared to uniform channels. Similarly, the size and number of boluses during the trapping phenomenon increase or decrease quickly in non-uniform channels. The volumetric addition of SWCNT diminishes the velocity and temperature while the heat transfer rate is enhanced in both channels but in the nonuniform channel effects are more prominent. ApplicationsThis study explores the applications of SWCNT in medical fields like angiography, angioplasty, and cancer therapy. NoveltyThe present study is new and has never been addressed before.

Asymptotic long-wave model for a high-contrast two-layered elastic plate

The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models.

Two-dimensional vector solitons in Bose-Einstein-condensate mixtures

We derive two decoupled KP-I equations from the system of two-dimensional (2D) Gross-Pitaevskii equations for a two-component Bose-Einstein condensate (BEC), using the multiple-scale expansion method. We produce asymptotic analytical vector-soliton solutions, viz., dark-dark (DD) and dark-antidark (DAD) one-soliton and two-soliton states, by tuning coupling constants and norms of species, and address their evolution numerically under the action of the harmonic-oscillator (HO) trap, in the local-density approximation. We find that shallow single-line DD and DAD solitons are stable, while single-lump DAD solutions (weakly localized truly-2D states) split and lead to nucleation of two half-vortices. We also find that the BEC mixture placed in the HO trap admits stable asymptotic multi-soliton solutions, e.g., two-line DD and DAD solitons and two-lump DD ones, which were not reported before. In particular, the two-lump solutions describe inelastic collisions between the lumps. The analysis developed in this work may also be applied to systems with spin-orbit coupling and gauge fields, which have been realized in atomic BEC.

Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G=defu+y2〈t〉φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ, which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data.

Physics-based modeling of surface potential and leakage current for vertical Ga2O3 FinFET

Gallium oxide (Ga2O3) is a promising ultra-wide bandgap material offering a large bandgap (>4.7 eV) and high critical electric fields. The increasing demand for electronic devices for high-power applications in electric automobiles, high-performance computing, green energy technologies, etc., requires higher voltages and currents with enhanced efficiency. Vertical transistors, such as fin-shaped field-effect transistors (FinFETs) have emerged to meet the growing need with improved current handling capabilities, reduced resistance, and enhanced thermal performance. However, to fully exploit the Ga2O3 power transistors, precise and reliable physics-driven models are crucial. Therefore, a comprehensive surface potential model has been developed in this work for a vertical Ga2O3 FinFET. The electric potential across the channel is explained by analyzing the two-dimensional (2D) Poisson equation employing parabolic approximation. Such a surface potential model is instrumental in determining the performance of the Ga2O3 FinFET as it affects the threshold voltage, the drain current, and fringing capacitance. Exploiting the surface potentials, a fringing capacitance model is derived which is crucial in analyzing the speed of the device in compact integrated circuits. In addition, statistical analysis of the Ga2O3 FinFET using the Monte Carlo simulation technique is performed to determine the leakage current fluctuation due to doping variations. The validation of the analytical model with experimental results confirms the effectiveness and prospects of the developed models in the rapid development and characterization of next-generation high-performance vertical Ga2O3 power transistors.

Heavy metal transport with adsorption for instantaneous and exponential attenuation of concentration

Heavy metal pollutant is a serious problem in environmental pollution, and it is very difficult to eradicate once it enters the soil. As heavy metal adsorption has been proven to occur, the heavy metal’s behaviour can be modeled as a transport equation with adsorption. Previous adsorption term mostly due to the concentration alone, while in here, the desorption effect given by the rate of change of the concentration is also included. Also, the heavy metals are frequently considered to enter the soil after being dumped into the soil for a certain period of time. But, quick dumping onto the soil can introduce heavy metal instantaneously. Heavy metals entering the soil through leaching or when their concentration in the soil is influenced by chemical reactions, can all lead to the exponential decay of heavy metals entering the soil. Based on two-dimensional advection diffusion equation (ADE) with the new adsorption term, analytical solutions are obtained for the cases of instantaneous and exponential attenuation of heavy metals emission to soil by the method of Laplace transform. The results highlight the significant influence of emission type on the peak concentrations. If heavy metals are instantaneously enter the soil, the peak occurs in the range of 1–3 m radius from the point of emission on the first day, while for exponential attenuation the peak occurs close to the point of emission. Furthermore, there exists a correlation between retardation factors and heavy metal concentrations, where a decrease in retardation factors leads to an increase in heavy metal concentration. It is essential to investigate both types of heavy metals emission to provide valuable information for proper pollution management, effective environmental regulations and enforcement.

Stationary and Non-Stationary Periodic Flows Mathematical Modelling using Various Vortex Viscosity Models

Introduction. Mathematical modelling of currents is an urgent research topic in the field of hydrodynamics and oceanography. Despite ongoing research in the field of developing accurate and efficient numerical methods for solving Navier-Stokes equations that take into account vortex viscosity, the problems of accurate prediction and control of turbulence remain unresolved. The influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions also remains relevant. The aim of the study is to study the influence of linearized and quadratic bottom friction and two turbulence models on the numerical solution of stationary and non-stationary periodic flows. Special emphasis is placed on comparing numerical results with analytical solutions within the framework of using various models of bottom friction.Materials and Methods. The computational models used in this study are based on a simplified two-dimensional wave model and full three-dimensional Navier-Stokes equations. The classical model of shallow water motion and the 2D model without taking into account dynamic changes in the geometry of the reservoir surface are derived from a system of equations for a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects – nonlinearities caused by higher-order terms in the equations of motion, i. e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.Results. The results of modeling stationary and non-stationary periodic flows in a schematized rectangular basin using linearized bottom friction are presented. The influence of linearization on the numerical solution is investigated in comparison with analytical profiles using models calculating bottom friction in a quadratic formulation. In combination with quadratic bottom friction, two turbulence models are studied: the constant vortex viscosity and the Prandtl mixing length model. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modeling and analytical solutions averaged in depth.Discussion and Conclusion. New approaches to modelling and studying flows with variable vortex viscosity are proposed, including analysis of the influence of linearization and the use of various turbulence models. For the linearized and quadratic formulations of bottom friction, it is proved that the numerical results for the case of stationary flow show great similarity with analytical solutions, since the surface height is much less than the water depth and advection can be neglected. The numerical results for the unsteady flow also show a good agreement with the theory. Unlike analytical solutions, numerical modelling has minor deviations in the long run. The study of flows, within the framework of using various turbulence models, will make it possible to take into account the influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions. The results obtained make it possible to better understand and describe the physical processes occurring in shallow waters. This opens up new possibilities for applying mathematical modelling to predict and analyze the impact of human activities on the marine environment and to solve other problems in the field of oceanology and geophysics.

SOLITON SOLUTIONS FOR THE TWO-DIMENSIONAL LOCAL FRACTIONAL BOUSSINESQ EQUATION

In this work we study the two-dimensional local fractional Boussinesq equation. Based on the basic definitions and properties of the local fractional derivatives and bilinear form, we studied the soliton solutions of non-differentiable type with the generalized functions defined on Cantor sets by using bilinear method. Meanwhile, we discuss the result when fractal dimension is 1, and compare it with the result when fractal dimension is [Formula: see text].

On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena

<abstract><p>In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.</p></abstract>

Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

<abstract><p>In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.</p></abstract>