Rectangle Domain Research Articles

Patterns dynamics in a diffusive prey–predator system with ratio-dependent functional response in a planar domain

This paper is concerned with a classical two-species prey–predator reaction–diffusion system with ratio-dependent functional response and subject to homogeneous Neumann boundary condition in a two-dimensional rectangle domain. By analyzing the associated eigenvalue problem, the spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the constant coexistence equilibrium are established. Then when the bifurcation parameter is in the interior of range for Turing instability and near Turing bifurcation curve, the amplitude equations of the original system near the constant coexistence equilibrium are obtained by multiple-scale time perturbation analysis. On the basis of the obtained amplitude equations, the stability and classifications of spatiotemporal patterns of the original system at the constant coexistence equilibrium are discussed. Finally, to verify the validity of the obtained theoretical results, numerical simulations are also carried out.

Diffusion-driven instability of a predator–prey model with interval biological coefficients

Interval biological coefficients as the imprecise parameters should be considered in biological models since there are various imprecisions in the real world. In this paper, we deal with the pattern dynamics of the predator–prey model with interval biological coefficients and no-flux boundary conditions. We first give the boundedness of the solutions by comparison principle and constructing an invariant rectangle domain with different interval variable values. Then by treating the diffusion coefficient of the predator as the critical parameter, the stable and unstable intervals of the positive equilibrium are discussed. Also, in this manner, the emergence condition of the Turing instability is performed. In the sequel, the amplitude equations around the threshold of the Turing instability are deduced by employing the weakly nonlinear analysis method. In this fashion, the existence and stability of various pattern solutions are classified. Finally, non-symmetrical and symmetrical spatial patterns are displayed with the help of the numerical simulation results in 2D space by choosing different interval biological coefficients.

Adaptive meshless numerical method of solving 2D variable order time fractional mobile-immobile advection-diffusion equations

This paper inspires us to propose an adaptive meshless numerical method of solving a 2D variable order time fractional mobile-immobile advection-diffusion equation on arbitrary domain. The mentioned method has three main contributions. (i) The adaptive meshless method could decrease amounts of computations effectively. (ii) On arbitrary domain, we successfully apply Legendre multiwaves in reproducing kernel Hilbert space (RKHS) defined on rectangle domain to approximate the exact solution. (iii) We skillfully build a special spline space, convergence order is obtained in view of convergence theories of spline space. Numerical experiments further demonstrate the validity of the method. • The adaptive meshless method could decrease amounts of computations effectively. • On arbitrary domain, we apply Legendre multiwaves to approximate the exact solution using some approximation theorems. • We successfully give the error estimate in the case of low regularity of the function.

Positivity-preserving high order finite difference WENO schemes for compressible Navier-Stokes equations

In this paper, we construct a high order weighted essentially non-oscillatory (WENO) finite difference discretization for compressible Navier-Stokes (NS) equations, which is rendered positivity-preserving of density and internal energy by a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The novelty of this paper is WENO reconstruction performed on variables from a positivity-preserving convection diffusion flux splitting, which is different from conventional WENO schemes solving compressible NS equations. The core advantages of our proposed method are robustness and efficiency, which especially are suitable for solving tough demanding problems of both compressible Euler and NS equation including low density and low pressure flow regime. Moreover, in terms of computational cost, it is more efficient and easier to implement and extend to multi-dimensional problems than the positivity-preserving high order discontinuous Galerkin schemes and finite volume WENO scheme for solving compressible NS equations on rectangle domain. Benchmark tests demonstrate that the proposed positivity-preserving WENO schemes are high order accuracy, efficient and robust without excessive artificial viscosity for demanding problems involving with low density, low pressure, and fine structure.

Trefftz numerical functions for solving inverse heat conduction problems

This paper presents a concept of solving the inverse heat conduction problem with the use of a linear combination of functions satisfying the differential equation in terms of identity, that is Trefftz functions. Characteristic feature of Trefftz functions applied for solving the heat equation is lack of their analytical form. It is known that these functions satisfy the differential equation with known boundary conditions. It was proved that so defined Trefftz functions create a complete system of functions to develop the solution to the heat conduction equation. It means that solving the inverse problem is reduced to determining unknown coefficients of the linear combination of basis functions, obtained from the solution of the direct problem. Two examples of inverse problems were solved: using the conjugate gradient method and using the Tikhonov regularization. As the first example, the inverse heat conduction problem was solved in the rectangle domain with known analytical solution to this problem and with verified stability of this solution. The second example concerns determination of temperature distribution on internal boundaries of the multiply-connected domain for the material, the heat conduction coefficient of which is the function of temperature.

A fourth-order Cartesian grid method for multiple acoustic scattering on closely packed obstacles

In this paper, we present a fourth-order Cartesian grid-based boundary integral method (BIM) for a multiple acoustic scattering problem on closely packed obstacles. We reformulate the exterior Helmholtz boundary value problems (BVPs) as a Fredholm boundary integral equation (BIE) of the second kind for some unknown density function. Unlike the traditional boundary integral method, a distinctive feature of our scheme is that we do not require quadratures and direct evaluations of nearly singular, singular or hyper-singular boundary integrals in the solution of BIEs. Instead, we reinterpret boundary integrals as solutions to equivalent simple interface problems in an extended rectangle domain, which can be solved efficiently by a fourth-order finite difference method coupled with numerical corrections, FFT based solution and interpolations. Extensive numerical experiments show that our method is formally high-order accurate, fast convergent and in particular insensitive to complexity of scatterers.

Numerical method for a boundary value problem for a linear singularly perturbed parabolic delay differential equation of reaction-diffusion type with discontinuous source term

A singularly perturbed boundary value problem for a linear parabolic second order delay differential equation of reaction-diffusion type with discontinuous source term is considered in the rectangle domain Ω = {(x, t) : 0 < x < 2,0 < t ⩽ T}. Discontinuity occurs in the source term at (d, t) < ∈ Ω. As the highest order space derivative is multiplied by a singular perturbation parameter, the components of the solution exhibit boundary layers at (x,t) = (0,t) and (x,t) = (2,t) and interior layers at (x, t) = (1, t) and/or at (x,t) = (d,t) and (x,t) = (1 + d, t) with respect to the position of (d, t) in Ω. A numerical method which uses classical finite difference scheme on a Shishkin piecewise uniform mesh is suggested to approximate the solution. The method is proved to be first order convergent uniformly for all the values of singular perturbation parameters. Numerical illustrations are presented so that the theoretical results are supported.

Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition

This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of N_{r} times N_{t} elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

Interior jump and contact singularity for compressible flows with inflow jump datum

Initial jump datum at a point must generate a jump curve directing into the region by the hyperbolic character of the continuity equation and the pressure gradient (in the momentum equation) is not well-defined. For this we construct a velocity vector field directed by the pressure difference on the jump curve. In bounded domains the jump curve must meet some points of the boundary and brings us contact singularities. In a rectangle domain we split the velocity vector field into a non-smooth part plus a contact singular one plus a smoother one and similar for the density function. We show piecewise regularity for the Navier-Stokes equations of compressible viscous flows and derive the Rankine-Hugoniot conditions.

Dispersive Quantization of 2D Linear Dispersive Equations

The dispersive quantization of the 2D linear KdV equation and the 2D linear Schrödinger equation were studied over a bounded rectangle domain in the plane. The research shows that, for the KdV equation, if the period ratio is a rational number, at the rational moments, the solution to the periodic initial boundary value problem will be the linear combination of the initial value conditions; whereas, at the irrational moments, the solution will be continuous and nondifferentiable, and exhibit a fractallike profile. The same is true for the 2D linear Schrödinger equation.

The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions

A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable v=Δu, the fourth-order problem is reduced into a couple of second-order system. The L1 scheme on graded mesh is considered for the Caputo fractional derivatives of order α∈(0,1) by inserting more grid points near the initial time. By virtue of some properties, such as complementary discrete convolution kernel and discrete fractional Grönwall inequality, we establish unconditional stability and convergence for the original unknown u and auxiliary variable v. Some numerical experiments are provided to further verify our theoretical analysis.

Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains

In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules.

Choice of the regularization parameter for the Cauchy problem for the Laplace equation

PurposeIn this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy.Design/methodology/approachThe Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied.FindingsNumerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the normNmax.Practical implicationsCalculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary.Originality/valueThe paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

Numerical investigation of alternating-direction method for Poisson equation with weighted integral conditions

The present paper deals with a generalization of the alternating-direction implicit(ADI) method for a two dimensional Poisson equation in a rectangle domain with aweighted integral boundary condition in one coordinate direction. We consider the alternatingdirection method for a system of difference equations that approximates Poisson equationwith weighed integral boundary conditions with the fourth-order accuracy. Sufficient conditionsof stability for ADI method are investigated numerically. An analysis of results ofcomputational experiments is presented. &#x0D;

Stabilization of a heat-like equation in rectangle domain

In this paper, we discuss the stabilization of the heat-like equation in rectangle domain with boundary control. We firstly transfer the heat-like equation in 2D into series of heat equations in 1D via eigenfunction expansion, then, by using the normal backstepping transformation, we obtain the existence and estimate for series of kernel functions in transformations. Finally, combined with the states and controls estimate of series of heat equations, we finally obtain the exponential and finite time stabilization of the heat-like equation in 2D.

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill‐posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well‐posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method.

Supine to prone colon registration and visualization based on optimal mass transport

Registration and visualization of surfaces play a significant role in the engineering and medical fields. Especially in virtual colonoscopy (VC), an efficient and robust registration method between supine and prone computed tomography (CT) scans is highly desirable, due to the fact that it helps improve polyp detection rates. However, supine and prone colon registration is still a challenging task due to the large distortion in colon shape. In this work, we present a novel registration and visualization framework for supine and prone colon scans using the optimal mass transport (OMT) theory. The proposed novel method enables parameterization of the colon surface onto the planar rectangle domain for better registration and visualization. Also, we develop novel flattened visualization of the colon wall with magnification of polyps using color-coded distance change in the 2D map. Experimental results validate our proposed registration method and demonstrate the effectiveness and robustness of our method.

Topology optimization of conformal structures on manifolds using extended level set methods (X-LSM) and conformal geometry theory

In this paper, we propose a new method to systematically address the issue of structural shape and topology optimization on free-form surfaces. A free-form surface, also termed manifold, is conformally mapped onto a 2D rectangle domain where the level set function is defined. With the conformal mapping, the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar. Keeping this intrinsic relation in mind, we derive the Euclidean form for the Riemannian Hamilton–Jacobi equation governing the boundary evolution on the manifold, which can be solved on a 2D plane using classical level set methods, such as the upwind finite difference or fast marching method. By reducing the dimension of the problem, the topology optimization problem on the manifold embedded in the 3D space can be recast as a 2D topology optimization problem in the Euclidean space. Compared with other approaches which need project the Euclidean differential operators to the manifold, the proposed method can not only reduce the computational cost but also preserve all the advantages of conventional level set methods. The proposed method reveals the fundamental relation between topology optimization on manifolds and Euclidean planes. It provides a unified level-set-based computational framework for the generative design of conformal structures with increasing applications in different fields of interests.

Efficient Simulation of Electromagnetic Wave Propagation in Complex Shaped Domain by Hybrid Method of FDTD and MTDM Based on Interpolating Moving Least Squares Method

To utilize the advantages of both finite-difference time-domain (FDTD) method and meshless time-domain method (MTDM) based on the interpolating moving least squares method, a hybrid method of FDTD and MTDM has been proposed for electromagnetic wave propagation simulations in complex shaped domains. In this method, FDTD is employed in rectangle domains, and MTDM is used in other kinds of domains. Numerical experiments show that the electric field is smoothly propagated, containing connection parts of domains calculated by FDTD and MTDM. In addition, the convergence value of amplification/damping rate of a hybrid method is almost the same as that of FDTD, and the accuracy of the hybrid method is better than that of FDTD for the case where the number of nodes is relatively small. Furthermore, the computation time of the hybrid method is less than that of MTDM.

Convergence of an Anisotropic Perfectly Matched Layer Method for Helmholtz Scattering Problems

AbstractThe anisotropic perfectly matched layer (APML) defines a continuous vector field outside a rectangle domain and performs the complex coordinate stretching along the vector field. Inspired by [Z. Chen et al., Inverse Probl. Imag., 7, (2013):663–678] and based on the idea of the shortest distance, we propose a new approach to construct the vector field which still allows us to prove the exponential decay of the stretched Green function without the constraint on the thickness of the PML layer. Moreover, by using the reflection argument, we prove the stability of the PML problem in the PML layer and the convergence of the PML method. Numerical experiments are also included.