Equations In Two-dimensional Space Research Articles

Multi-USV cooperative oil spill source seeking via extremum seeking combined with information fusion

This paper addresses the issue of cooperative source seeking for oil spill pollution by multiple unmanned surface vehicles (USVs), where the oil spill plume propagation is modeled by a linear advection-diffusion equation in two-dimensional space. The main objective is to construct a cooperative source seeking scheme for a group of USVs in the framework of extremum seeking combined with information fusion to solve such cooperative source seeking problem. To do this, both a gradient-based control law and a formation control law are included in the cooperative source seeking scheme. The gradient-based control law that is constructed by extremum-seeking control combined with information fusion drives the multiple USVs to approach the oil spill source, where the gradient is estimated online via the extremum-seeking control and the fused concentration value that is derived via a linear H∞ filter from the local concentration measurements at the USV’s current position. In the source seeking process, the USVs driven by the formation control law based on the virtual structure method maintain a preset circular formation. The convergence performance of the cooperative source seeking scheme is analyzed rigorously by the Lyapunov method. Finally, extensive numerical simulations are included to validate the theoretical algorithm.

Three-Dimensional Stress Fields in Thick Orthotropic Plates with Sharply Curved Notches under In-Plane and Out-of-Plane Shear.

In this paper, an analytical solution for the stress fields in the close neighbourhoods of radiused notches in thick orthotropic plates under shear loading and twisting is provided. In the first step, the equations of the three-dimensional theory of elasticity are successfully reduced to two uncoupled equations in two-dimensional space. Later, the 3D stress field solution for orthotropic plates with radiused notches is presented and its degree of accuracy is discussed by comparing theoretical results and numerical data from 3D FE analyses. The solution proposed can be satisfactorily used to characterise the stress field in plates made with polymeric composite materials, such as fibre-reinforced polymers and natural composites.

The study of approximate solution of one class of hypersingular integral equations of first kind

This work presents the method for calculating the approximate solution to the integral equation of the first kind of the Neumann boundary value problems for the Helmholtz equation in two-dimensional space

Lie analysis and laws of conservation for the two-dimensional model of Newell–Whitehead–Segel regarding the Riemann operator fractional scheme in a time-independent variable

The framework of fractional partial differential models is the first-rate hyperlink between mathematics and applied physics. This article project intends to utilize outcomes for the time-fractional Newell–Whitehead–Segel model in 2 D -space including conservation laws, Lie point symmetry analysis, and series solutions. We introduce a particular fractional model that is free of the type of approximate methods, whilst, environmental flow and magnetohydrodynamics processes are considered to be the main real-world phenomena treated with such a model. Herein, the method of power series is exercised to provide an analytical solution to the current governing model. The idea and undertaking of the method lie in the assumption that the solution is a power series of coefficients that are determined by a recurrence relation obtained by substituting the series solution in the considered model. Also, the Riemann approach is utilized as a total derivative. Simulation effects are systematically demonstrated through a chain of check cases. Strong proofs with some related plots are performed to confirm the accuracy and fitness of the model version and the presented approach. Moreover, the laws of conservation depend on the existence of a Lagrangian of the fractional Newell–Whitehead–Segel model utilized. Ultimately, a few related comments and future proposals are epitomized. HIGHLIGHT The Lie point symmetries of the time-fractional Newell–Whitehead–Segel equation in two-dimensional space is utilized and derived. The technique of the power series is applied to conclude the explicit solutions for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space for the first time. The conservation laws for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space are built using a novel conservation theorem. Several graphical countenances were utilized to award a visual performance of the obtained solutions.

Analysis of Lie Symmetry, Explicit Series Solutions, and Conservation Laws for the Nonlinear Time-Fractional Phi-Four Equation in Two-Dimensional Space

Analysis of Lie Symmetry, Explicit Series Solutions, and Conservation Laws for the Nonlinear Time-Fractional Phi-Four Equation in Two-Dimensional Space

Compact Difference Method for Time-Fractional Neutral Delay Nonlinear Fourth-Order Equation

In this paper, we present a compact finite difference method for a class of fourth-order nonlinear neutral delay sub-diffusion equations in two-dimensional space. The fourth-order problem is first transformed into a second-order system by a reduced-order method. Next by using compact operator to approximate the second order space derivatives and L2-1σ formula to approximate the time fractional derivative, the difference scheme which is fourth order in space and second order in time is obtained. Then, the existence and uniqueness of solution, the convergence rate of and the stability of the scheme are proved. Finally, numerical results are given to verify the accuracy and validity of the scheme.

Lie symmetry analysis, explicit solutions, and conservation laws of the time-fractional Fisher equation in two-dimensional space

In these analyses, we consider the time-fractional Fisher equation in two-dimensional space. Through the use of the Riemann-Liouville derivative approach, the well-known Lie point symmetries of the utilized equation are derived. Herein, we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries. The diminutive equation's derivative is in the Erdélyi-Kober sense, whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time. The conservation laws for the dominant equation are built using a novel conservation theorem. Several graphical countenances were utilized to award a visual performance of the obtained solutions. Finally, some concluding remarks and future recommendations are utilized.

A linearized compact ADI numerical method for the two-dimensional nonlinear delayed Schrödinger equation

We develop a linearized compact alternating direction implicit (ADI) numerical method to solve the nonlinear delayed Schrödinger equation in two-dimensional space. By discrete energy estimate method, we analyse the convergence of the fully-discrete numerical method, and show that the numerical scheme is of order O(Δt2+h4) with time stepsize Δt and space stepsize h. At last, we present several numerical examples to confirm theoretical analyses.

The behaviour of the maximum and minimum error for Fredholm-Volterra integral equations in two-dimensional space

In this paper, we study the behaviour of the maximum ( Max.) and minimum (Min.) error for Fredholm-Volterra integral equations (F-VIEs) of the second kind using Collocation (CM) and Galerkin (GM) methods by choosing N-linearly independent functions. The approximate solution is obtained by two techniques; the first technique (1st TM) depends on representing F-VIE as a system of Fredholm integral equations (FIEs) of the second kind where the approximate (Appr.) solution is obtained as functions of x at fixed times. In the second technique (2nd TM), we represent the approximate solution as a sum of functions of x and t. Furthermore, the comparisons between the results which are obtained by two techniques in each method are devoted and the results are represented in group of figures and tables.

Bound state solution and thermodynamic properties of the screened cosine Kratzer potential under influence of the magnetic field and Aharanov–Bohm flux field

In this article, approximate analytical bound state solutions of the Schrödinger equation in two-dimensional space for screened cosine Kratzer potential (SCKP) under the influence of the magnetic field and Aharanov–Bohm flux field have been investigated. We obtained energy eigenvalues and wave functions for SCKP with external fields (magnetic field and Aharanov–Bohm flux field) via parametric Nikiforov–Uvarov (pNU) method using the approximation method suggested by Greene–Aldrich for handling centrifugal barriers. We deduced energy eigenvalues for screened Kratzer potential (SKP) and Hellmann potential (HP) from the obtained energy spectrum of SCKP with an external field. We extended our results in D dimensions for Hellmann potential in the absence of external fields. Thermodynamic properties such as partition function Z(B→,ΦAB,β), mean energy U(B→,ΦAB,β), mean free energy F(B→,ΦAB,β), entropy S(B→,ΦAB,β), specific heat capacity Cs(B→,ΦAB,β), magnetization at finite temperature (B→,ΦAB,β) and magnetic susceptibility χm(B→,ΦAB,β) at finite temperature are presented. The obtained numerical resultsfor SKP and Hellmann potential are in very good agreementwith numerical results available in the literature with and without external fields respectively.

Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation

In this paper, two alternating direction implicit Galerkin-Legendre spectral methods for distributed-order differential equation in two-dimensional space are developed. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(Δt + σ2 + N−m) and O(Δt2 + σ2 + N−m), respectively, where Δt, σ, N, and m are the time step size, step size in distributed-order variable, polynomial degree, and regularity in the space variable of the exact solution, respectively. Moreover, the applicability and accuracy of the two schemes are demonstrated by numerical experiments to support our theoretical analysis.

Partial differential equation calculation and visualization

The calculation of partial differential equations has wide applied background. Many of the problems in scientific research and engineering can be the mathematical model of partial differential equations. Differential equation containing multiple independent variables is called as PDE (partial differential equation). As for PDE problem, it is very difficult to solve. Apart from a few special cases, the vast majority of cases are difficult to obtain exact solutions. For the majority of applications workers, from the point of partial differential equations, using the finite element method or finite difference method has to spend a lot of effort to get numerical solution. By MATLAB PDEToolbox, the article introduced the accurate solution process of two-dimensional problem with high-speed. In order to see the relationship between graph, function value and independent variable from the mathematical expression of partial differential equations by MATLAB programming, numerical solution, and the results can be visualized. PDE Toolbox provides a powerful and flexible as well as practical environment for research and problem solving about partial differential equations in two-dimensional space.

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples $(h,\Delta t)$ of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words, for each fixed value of $\Delta t$ below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

Semi-discretization Difference Approximation for a Cauchy Problem of Heat Equation in Two-dimensional Space

Semi-discretization Difference Approximation for a Cauchy Problem of Heat Equation in Two-dimensional Space

Interface and vortex motion in the two-component complex dissipative Ginzburg-Landau equation in two-dimensional space.

We study interface and vortex motion in the two-component dissipative Ginzburg-Landau equation in two-dimensional space. We consider cases where the whole system is divided into several domains, and we assume that these domains are separated by interfaces and each domain contains quantized vortices. The equations for interface and vortex motion will be derived by means of a variational approach by Kawasaki. These equations indicate that, along an interface, the phase gradient fields of the complex order parameters is parallel to the interface. They also indicate that the interface motion is driven by the curvature and the phase gradient fields along the interface, and vortex motion is driven by the phase gradient field around the vortex. With respect to the static interactions between defects, we find an analogy between quantized vortices in a domain and electric charges in a vacuum domain surrounded by a metallic object in electrostatic. This analogy indicates that there is an attractive interaction between an interface and a quantized vortex with any charge. We also discuss several examples of interface and vortex motion.

Application of low-dimensional finite element method to fractional diffusion equation

Classical finite element method (FEM) has been applied to solve some fractional differential equations, but its scheme has too many degrees of freedom. In this paper, a low-dimensional FEM, whose number of basis functions is reduced by the theory of proper orthogonal decomposition (POD) technique, is proposed for the time fractional diffusion equation in two-dimensional space. The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved. Moreover, error estimation of the method is obtained. Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

New Homotopy Perturbation Method for a Special Kind of Volterra Integral Equations in Two-Dimensional Space

In this work, a new modified homotopy perturbation (NHPM) is applied to solve a special kind of nonlinear Volterra integral equations in two-dimensional space. The two most important steps in the application of the new Homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. In the present paper, convergence of the new approach is proved. Comparison of our solution with the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) shows that the NHPM is effective and accurate in solving these kind of problems.

Green Function on a Quantum Disk for the Helmholtz Problem

In this work, we present a new result which concerns the derivation of the Green function relative to the time-independent Schrodinger equation in two-dimensional space. The system considered in this work is a quantal particle that moves in an axi-symmetric potential. At rst, we have assumed that the potential V (r) to be equal to a constant V0 inside a disk (radius a) and to be equal to zero outside the disk. We have used, to derive the Green function, the continuity of the solution and of its rst derivative, at r = a (at the edge). Secondly, we have assumed that the potential V (r) is equal to zero inside the disk and is equal to V0 outside the disk (the inverted potential). Here, also we have used the continuity of the solution and its derivative to obtain the associate Green function showing the discrete spectra of the Hamiltonian. DOI: 10.12693/APhysPolA.124.636

A new CUDA-based GPU implementation of the two-dimensional Athena code

Abstract We present a new version of the Athena code, which solves magnetohydrodynamic equations in two-dimensional space. This new implementation, which we have named Athena-GPU, uses CUDA architecture to allow the code execution on Graphical Processor Unit (GPU). The Athena-GPU code is an unofficial, modified version of the Athena code which was originally designed for Central Processor Unit (CPU) architecture. We perform numerical tests based on the original Athena-CPU code and its GPU counterpart to make a performance analysis, which includes execution time, precision differences and accuracy. We narrowed our tests and analysis only to double precision floating point operations and two-dimensional test cases. Our comparison shows that results are similar for both two versions of the code, which confirms correctness of our CUDA-based implementation. Our tests reveal that the Athena-GPU code can be 2 to 15-times faster than the Athena-CPU code, depending on test cases, the size of a problem and hardware configuration.

Finite Volume Multilevel Approximation of the Shallow Water Equations

The authors consider a simple transport equation in one-dimensional space and the linearized shallow water equations in two-dimensional space, and describe and implement a multilevel finite-volume discretization in the context of the utilization of the incremental unknowns. The numerical stability of the method is proved in both cases.