Search

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 Erdelyi-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.

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetricinequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $n=2$ case of a conjecture of Gir\~ao-Pinheiro., Comment: 17 pages

We develop a fully discrete finite volume element scheme of the two-dimensional space-fractional convection–diffusion equation using the finite volume element method to discretize the space-fractional derivative and Crank–Nicholson scheme for time discretization. We also analyze and prove the stability and convergence of the given scheme. Finally, we validate our theoretical analysis by data from three examples.

Plant morphology modeling can be done mathematically which includes roots, stems, leaves, to flower. Modeling of plant stems using the Lindenmayer System (L-system) method is a writing returns that are repeated to form a visualization of an object. Deterministic L-system method is carried out by predicting the possible shape of a plant stem using its iterative writing rules based on the original object photo. The purpose of this study is to find a model of the plant stem with Deterministic Lindenmayer System method which will later be divided into two dimensional space three. The research was conducted by identifying objects in the form of pine tree trunks measured by the angle, thickness, and length of the stem. Then a deterministic and parametric model is built with L-system components . The stage is continued by visualizing the model in two dimensions and three dimensions. The result of this research is a visualization of a plant stem model that is close to the original. Addition color, thickness of the stem, as well as the parametric writing is done to get the results resembles the original. The iteration is limited to less than 20 iterations so that the simulation runs optimal.

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order $${\mathcal {O}}({\delta \tau }^{2-\alpha }),~ 0

The concise and reliable representation of a given source dataset is available for data based learning and decision-making, which can be obtained through a controllable data fusion process within a formatted space. Based on the density matrix representation of a given source weather dataset, this paper first calculates the Hellinger distances and affinities between different density matrices, and then takes them to construct a two-dimensional metric space using the longest Hellinger metric and affinity paths. Within this space, the given source data units are converted into the corresponding rectangle nodes which are determined by the minimum horizontal and vertical distances between different data units. According to a predefined detection size in this space, the basic blocks centred by different rectangle nodes are classified into different subset blocks for fusion. Each subset block is jointed by the basic blocks with overlapped areas. During the detection process, the detection size keeps increasing until a predefined reference variable, such as the relative density for all subset blocks, reaches a predefined turning point. The fusion of the rectangle nodes in a subset block depends on the distances between the included rectangle nodes’ positions and this subset block’s centre position which is calculated according to the included rectangle nodes’ weights and positions. The experimental analysis shows that the proposed weather data fusion method is controllable and stable and can obtain concise and reliable fusion results for learning inputs and decision-making.

In this article, a fast time two-mesh (TT-M) finite element (FE) method for the two-dimensional space fractional Gray–Scott model is studied and discussed to get the numerical solutions effectively. The method mainly includes three steps: firstly, one uses an iterative method for solving the coupled nonlinear system on the time coarse grid; secondly, by an interpolation formula, one can get any coarse values on the time fine mesh; finally, based on the computed coarser solutions, a linear FE system on time fine mesh can be constructed by using the two-variables Taylor’s formula. Here, some theoretical results, which include stability and a priori error for the fully discrete scheme, are analyzed and proved. Furthermore, the computing data are given to verify the correctness of the theoretical results and to illustrate that the TT-M FE algorithm can reduce the computing time.

Space and time approximations for two-dimensional space fractional complex Ginzburg–Landau equation are examined. The schemes under consideration are discreted by the second-order backward differential formula (BDF2) in time and two classes of the fractional centered finite difference methods in space. A linearized technique is employed by the extrapolation. We prove the unique solvability and stability for both numerical methods. The convergence of both numerical methods is analyzed at length utilizing the energy argument, and the convergence orders under the optimal step size ratio are O ( τ 2 + h 2 ) and O ( τ 2 + h 4 ) in the sense of the discrete L 2 -norm, where τ is the time step size, h = max { h x , h y } , and h x , h y are spatial grid sizes in the x -direction and y -direction, respectively. In addition, we construct a multistep alternating direction implicit (ADI) scheme and a multistep compact ADI scheme based on BDF2 for the efficiently numerical implementation. Finally, numerical examples are carried out to verify our theoretical results.

In this paper, we consider the following system { u t = Δ u − χ ∇ ⋅ ( u ∇ ω ) , v t = d v Δ v − ξ ∇ ⋅ ( v ∇ ω ) , m t = d m Δ m + u − m , ω t = − ( γ 1 u + m ) ω , in two dimensional space with zero-flux boundary conditions. This model was proposed by Franssen et al. [7] to characterize the invasion and metastatic spread of cancer cells. We first establish the global existence of uniformly bounded global strong solutions. Then using the decay of ECM and the positivity of MDE, we further improve the regularity of obtained solutions, and achieve the uniform boundedness of solutions in the classical sense. Subsequently, we also prove the uniqueness of solutions. After that, we turn our attention to the large time behavior of solutions, and show that the global classical solution strongly converges to a semi-trivial steady state in the large time limit.

A simple kinematic approach to the description of the interaction between solitons is developed. It is applicable to both integrable and non-integrable two-dimensional models, including those commonly used for studying the surface and internal oceanic waves. This approach allows obtaining some important characteristics of the interaction between solitary waves propagating at an angle to each other. The developed theory is validated by comparison with the exact solutions of the Kadomtsev-Petviashvili equation and then applied to the observed interaction of solitary internal waves in a two-layer fluid within the two-dimensional Benjamin-Ono model.

We provide and analyze a second order scheme for the model describing the functional distributions of particles performing anomalous motion with exponential Debye pattern and no-time-taking jumps eliminated, and power-law jump length. The model is derived in [M. Chen, J. Shi, W. Deng, arXiv:1809.03263], being called the space fractional diffusion equation with the time Caputo-Fabrizio fractional derivative. The designed schemes are unconditionally stable and have the second order global truncation error with the nonzero initial condition, being theoretically proved and numerically verified by two methods (a prior estimate with $L^2$-norm and mathematical induction with $l_\infty$ norm). Moreover, the optimal estimates are obtained., Comment: 19pages

In this paper, we consider the numerical approximation of the space and time fractional Bloch-Torrey equations. A fully discrete spectral scheme based on a finite difference method in the time direction and a Galerkin-Legendre spectral method in the space direction is developed. In order to reduce the amount of computation, an alternating direction implicit (ADI) spectral scheme is proposed. Then the stability and convergence analysis are rigorously established. Finally, numerical results are presented to support our theoretical analysis.

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions, which is discretized by the space–time finite element method to propagate solutions. We develop a multigrid-reduction-in-time (MGRIT) algorithm with time-dependent time-grid propagators and provide its two-level convergence theory under the assumptions of the stability and simultaneous diagonalizability on time-grid propagators. Numerical results show that the proposed method possesses the saturation error order, theoretical results of the two-level variant deliver good predictions for our model problems, and significant speedups of the MGRIT can be achieved when compared to the two-level variant with F-relaxation (an equivalent version of the parareal algorithm) and the sequential time-stepping approach.

A general ejection theory of polymer is developed in a two- and three-dimensional space.A polymer is confined initially in a cavity and ejects spontaneously to the outer space through a nanopore channel without the help of any external stimulus. A reflective wall boundary is set at the pore entrance to prevent the falling of the head monomer of chain into the cavity. Three stages are distinguished in a process: (1) the entering stage, in which the head monomer enters the pore to search for a way to traverse the pore channel, (2) the main ejection stage, in which the chain body is transported from the cavity to the outer space, (3) the leaving stage, in which the tail monomer passes through and leaves the pore channel. Depending on the number of the monomers remaining in the cavity, the main ejection stage can be divided into the confined and the non-confined stages. The non-confined stage can be further split into the thermal escape and the entropic pulling stages. The Onsager's variational principle is used to derive the kinetics equation of ejection. The escape time is calculated from the corresponding Kramers' escape problem.Extensive molecular dynamics simulations are then performed in a quasi two-dimensional space to verify the theory. The variation of the ejection speed is carefully examined in a process. The decreasing behavior of the number of monomers in the cavity is studied in details. The scaling properties of the spending time at each processing stage are investigated systematically by varying the chain length, the cavity diameter, and the initial volume fraction of chain. The results of simulation support firmly the predictions of the theory, cross-checked in the studies of various topics. Together with the previous investigations in the three-dimensional space, the generalized theory is very robust able to explain the two seemly different phenomena, polymer ejection and polymer translocation, under the same theoretical framework in the two space dimensions.

Space-time image velocimetry (STIV) is a promising technique for river surface flow field measurement with the development of unmanned aerial vehicles (UAVs). STIV can give the magnitude of the velocity along the search line set manually thus the application of the STIV needs to determine the flow direction in advance. However, it is impossible to judge the velocity direction at any points before measurement in most mountainous rivers due to their complex terrain. A two-dimensional STIV is proposed in this study to obtain the magnitude and direction of the velocity automatically. The direction of river flow is independently determined by rotating the search line to find the space-time image which has the most prominent oblique stripes. The performance of the two-dimensional STIV is examined in the simulated images and the field measurements including the Xiasi River measurements and the Kuye River measurements, which prove it is a reliable method for the surface flow field measurement of mountain rivers.

This study investigated the localization ability of an impulse vibration source outside the body in two-dimensional space. We tested whether humans can recognize the direction or distance of an impulse vibration source when using their hand to detect spatiotemporal vibrotactile information provided by the propagated vibrational wave from the source. Specifically, we had users put their hands on a silicone rubber sheet in several postures. We asked users to indicate the position of the vibration source when a location on the sheet was indented. Experimental results suggested that the direction of the impact vibration source can be recognized to some extent, although recognition accuracy depends on hand posture and the position of the vibration source. The best results were achieved when the fingers and palm were grounded and a vibration source was presented around the middle fingertip, and the directional recognition error in this case was 6 degree. In contrast, results suggest it is difficult to accurately recognize the distance of the vibration. The results of this study suggest a new possibility for directional display where vibrotactile actuators are embedded at a distance from the user's hand.

In this study, singular boundary method (SBM) is employed for solving fractional cable problem in two dimensional space with initial and Dirichlet-type boundary conditions. The process is modeled as a two dimensional time-fractional equation in sense of Riemann–Liouville fractional derivatives. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. We present the numerical operation for calculating the particular solution and homogeneous solution. For achieving approximation particular solution and homogeneous solution, we employ Method of Particular solution (MPS) and SBM, respectively. We use θ-weighted and finite difference method as time discretization for time derivatives. A comparison between the present method and other methods is given to show the accuracy of SBM applying on this equation. Consequently, some numerical examples with different domains are tested and compared with the exact analytical solutions to display the validity and accuracy of the numerical method in comparison with other methods.

Recently fabricated two-dimensional (2D) black phosphorene (BP) is considered to be a promising optoelectronic sheet, but its applications are hindered by the poor stability in air. Thus, it is des...

During the design of transducers configuration for an active noise control system, current optimization methods need to predetermine the error sensors configuration, which significantly increases the workload of later optimization of the secondary sources configuration. In this study, a new method free from specific error sensors configuration information is presented that higher order microphones are used to capture the sound field so as to formulate the cost function in wave domain. In addition, according to sparsity characteristics of the primary sound field, sparsity-inducing regularization is introduced to optimize the secondary sources configuration, including the number and positions, by calculating a sparse approximate solution to underdetermined equations. Effects of the number of candidate secondary sources are discussed, and the comparison with the uniform configuration and the optimized configuration using the genetic algorithm is performed. Results show that the proposed method can optimize the secondary sources configuration effectively independent of the error sensors configuration information. The noise reduction of the proposed method is close to that by the genetic algorithm, while other evaluation metrics for the system are much better, which would benefit the stability of active noise control system.

A two-dimensional two-phase flow model for gas-liquid mixture is presented. The model takes into account the relative velocity between the gas and liquid phases and is based on conservation equations for gas-liquid mixtures. The mixture model involves balance equations for the relative velocity and is able to handle it without any physical or artificial stabilization in the source terms. The novel aspect of the mixture model is that it is written in a conservative form and ensures the hyperbolicity of the two-phase flow equations. With this regard, the governing equations are solved with finite volume methods. We extend and apply the framework of Godunov methods of centred-type, namely, the FirstOrder Centered (FORCE) and the Slope Limiter Centered (SLIC) methods to the two-dimensional governing equations without any loss of generality in the numerical solutions. An efficient assessment of both the mixture model and the numerical methods is carried out by simulating physical problems available in the literature. Simulations agree well with those in the literature and include new insights that could be used to explain the relative velocity observations. The favourable results suggest that the two-dimensional mixture model simulations can be employed for practical engineering problems of the non-equilibrium type.

A new (2+1)-dimensional Heisenberg ferromagnetic spin chain equation is investigated, which can be used to describe magnetic soliton excitations in two dimensional space fields and a time field. The Lax pair of the equation is first constructed. Based on the Lax pair, initial seed solution and Darboux transformation, the analytic first-, second- and third-order rogue wave solutions are obtained, and a general expression of the N th-order ( N > 3 ) rogue wave solutions is presented. The impacts of the system parameters on the rogue waves are demonstrated through numerical visualization method.

In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and we showed how this extends the definition for the Euclidean case. The argument is constructive and serves to define the center of mass of a system of particles on the one-dimensional hyperbolic sphere LR1.

The paper presents mesh-connected computer (MCC) algorithms to solve the visibility problem for a set of disjoint simple objects such as line segments, circles, and simple polygons in the plane. Given a set of n such objects, the algorithms compute, on a √n x √n MCC, a view of these objects in O(√n) time. Both parallel and perspective views are considered. Earlier published algorithms for computing the views are sequential and have O(n log n) time complexity. Consider the solutions to solve problems of size n on MCCs with p processors, where p < n. An analysis is presented of the time performance and the limitations imposed by the computational and communication requirements, and the trade-offs between the size of the MCC and the time complexity that can be achieved.

In this paper, we have considered a numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem with source term over a finite domain. The convection and diffusion equation can depend on both spatial and temporal variables. Crank-Nicolson scheme for time combined with weighted and shifted Gr&uuml, nwald-Letnikov difference operator for space are implemented to get second order convergence both in space and time. Unconditional stability and convergence order analysis of the scheme are explained theoretically and experimentally. The numerical tests are indicated that the Crank-Nicolson scheme with weighted shifted Gr&uuml, nwald-Letnikov approximations are effective numerical methods for two dimensional two-sided space fractional convection-diffusion equation.

We prove quasi-optimal $L^\infty$ norm error estimates (up to logarithmic factors) for the solution of Poisson's problem by the standard Hybridizable Discontinuous Galerkin (HDG) method. Although such estimates are available for conforming and mixed finite element methods, this is the first proof for HDG. The method of proof is motivated by known $L^\infty$ norm estimates for mixed finite elements. We show two applications: the first is to prove optimal convergence rates for boundary flux estimates, and the second is to prove that numerically observed convergence rates for the solution of a Dirichlet boundary control problem are to be expected theoretically. Numerical examples show that the predicted rates are seen in practice.

In this paper an expanded mixed formulation is introduced to solve the two dimensional space fractional Darcy flow in porous media. By introducing an auxiliary vector, we derive a new mixed formulation and the well-possedness of the formulation can be established. Then the locally mass-conservative expanded mixed finite element method is applied for the solution. Numerical results are shown to verify the efficiency of the proposed algorithm.

The extrinsic and intrinsic curvature of a two-dimensional waveguide influences wave propagation therein. While this can already be apprehended from a geometric point of view in terms of geodesics generalizing straight lines as the shortest distance between any two points, in wave optics interference phenomena strongly govern the field evolution, too. Radii of curvature in the order of the wavelength of light modify the local effective refractive index by altering the mode profile. Macroscopic radii only influence light propagation for non-vanishing intrinsic (or Gaussian) curvature. A positive Gaussian curvature leads to refocusing and thus an imaging behavior, whereas negative Gaussian curvature forces the field profile to diverge exponentially. These effects can be explained by an effective transverse potential acting on the electromagnetic field distribution’s envelope. This can also be extended to nonlinear beam propagation. In this review paper we give a thorough introduction to differential geometry in two-dimensional manifolds and its incorporation with Maxwell’s equations. We report on first fundamental experiments in this newly emerging field, which may lead to applications in integrated optical circuits. The close conceptual analogy to phenomena in four-dimensional spacetime with constant curvature as well as toy-models of the Schwarzschild metric and a wormhole topology are also discussed.

This work studies a parabolic-parabolic chemotactic PDE's system which describes the evolution of a biological population “U” and a chemical substance “V”, using a Generalized Finite Difference Method, in a two dimensional bounded domain with regular boundary. In a previous paper [12] , the authors asserted global classical solvability and periodic asymptotic behavior for the continuous system in 2D. In this continuous work, a rigorous proof of the global classical solvability to the discretization of the model proposed in [12] is presented in two dimensional space. Numerical experiments concerning the convergence in space and in time, and long-time simulations are presented in order to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.

This paper studies the shape similarity evaluation of free-form surfaces expressed by B-spline with single curvature feature and proposes a similarity evaluation algorithm based on curvature feature. Firstly, we calculate the normal vector direction of the two surfaces compared, and use it as the Z axis, so that the two surfaces are aligned on the Z axis. Then, the two surfaces are cut with planes that all perpendicular to the Z axis, and the intersection sets of two surfaces are obtained respectively. Finally, we design the similarity algorithm of plane curves to realize the similarity comparison of corresponding curves in the two sets of intersection, and which is used as the basis for evaluating the similarity between two surfaces. The algorithm transforms the problem of similarity comparison between 3D surfaces into two dimensional space by plane cutting method, and reduces the complexity of the problem effectively. The algorithm only needs to align one coordinate axis in the process of posture adjustment, so it is easy to implement. In order to test the effect of the algorithm, simulation experiments on different type of single curvature feature B-spline surfaces are carried out. The results show that the proposed similarity comparison algorithm of free-form surfaces is feasible and effective.

The purpose of the present work is to present a direct analogy regarding the formulation and solution of anti-plane problems in the context of couple stress and dipolar gradient elasticity and the von Karman plate framework. We show aspects of boundary conditions in both theories of elasticity and propose a robust Finite Element methodology based on the von Karman plate theory in order to solve complex anti-plane (mode III) crack problems. Furthermore, we establish the equivalence between the anti-plane gradient elasticity J-Integral and the plate I c -Integral (Sanders’ plate integral). In passing, we prove the path independency of the I c -Integral . Finally we examine the near tip fields for both anti-plane and plate problems and establish possible strengthening effects due to the underlying microstructure. The proposed analogy is achieved through the von Karman plate theory where the plate is pre-stressed by a constant biaxial tension. The plate theory involves properties such as the plate thickness h , the Poisson's ratio ν and the bending stiffness D . This information, together with the pre-stress N transforms into properties required by the anti-plane couple stress and dipolar gradient elasticity problems: the shear modulus G , the internal length l and the coefficient η. In both problems the two dimensional space remains the same, including the presence of cracks and other defects. The analogy permits numerical and analytical solutions of demanding anti-plane problems of gradient elasticity (couple stress and dipolar) utilizing the von Karman plate corresponding, and vice versa.

The first order system least squares Legendre and Chebyshev spectral method for two dimensional space linear elasticity is investigated. The drilling rotation is defined as a new variable and the linear elasticity equation is supplemented with an auxiliary equation. The weighted L 2 -norm least squares principle is applied to a stress–displacement–rotation. It is shown that the homogeneous least squares functional is equivalent to weighted H 1 -norm like for stress and weighted H 1 -norm for displacement and rotation. This weighted H 1 -norm equivalence is λ -uniform. Spectral convergence for both Legendre and Chebyshev approaches are given along with some numerical experiments. The generalization for three dimensional spaces is also provided.

Singularities in five-axis machining are a series of deadly positions, which may cause unstable axis movements and deteriorate the performance of machine tools. This paper presents an optimization strategy to locally deform the tool path so that singularities of five-axis machine tools can be avoided. To achieve this purpose, a new concept of “forbidden circle”, which is the projection of the three dimensional “singular cone” associated with tool orientations in a two dimensional space named P-plane, is established for the first time to transform the complicated three dimensional problem into a two dimensional problem. Based on this idea, the projected points in P-plane are interpolated as B-spline first, and then, the B-spline control points are locally optimized by taking the “forbidden circle” as the geometrical constraint to keep the B-spline from crossing it. Especially, in the optimization procedure, the constraints are linearized with the changes of B-spline control points to change the whole optimization problem as a typical positive definite quadratic programming problem, which can achieve one and only global optimal solution. By doing so, singularity is avoided, and at the same time, the machining errors caused by tool path deformation can be minimized. Simulation and experimental results verify the effectiveness of the singularity avoidance method.