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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.

Το σύγγραμμα επιχειρεί να συνεισφέρει σε μία ομαλή μετάβαση από την ύλη του λυκείου στις μαθηματικές έννοιες που είναι απαραίτητες για την κατανόηση της Γραμμικής Άλγεβρας και της Αναλυτικής Γεωμετρίας. Η ύλη που καλύπτεται περιλαμβάνει τους Μιγαδικούς αριθμούς, την Άλγεβρα πινάκων (πίνακας, πράξεις πινάκων, είδη πινάκων, αντίστροφος), το πραγματικό επίπεδο, ο τρισδιάστατος χώρος, γραμμικές απεικονίσεις (με τη βοήθεια πινάκων), Μέθοδος απαλοιφής Gauss, Ορίζουσες, Επίλυση γραμμικών συστημάτων. Ιδιοτιμές, Ιδιοδιανύσματα Πινάκων. Η έμφαση δίνεται στη γεωμετρική ερμηνεία των παραπάνω εννοιών. Η καινοτομία του προτεινόμενου συγγράμματος έγκειται στο ότι ενσωματώνει τη χρήση του υπολογιστικού προγράμματος Mathematica σε ξεχωριστή ενότητα σε κάθε κεφάλαιο. Έτσι γίνονται δυνατοί οι πειραματισμοί και οι εξερευνήσεις και ο αναγνώστης μπορεί να εμπεδώσει καλύτερα τη γνώση και να εξοικειωθεί με τον ηλεκτρονικό τρόπο επίλυσης αριθμητικών προβλημάτων στη Γραμμική Άλγεβρα.

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

In recent years, substantial attention has been given to the optimal placement and planning of sampling or monitoring based on the Value of Information (VoI). The drawback of the VoI approach is it...

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.

The method of numerical Fourier analysis is used to investigate the fractal properties of 2D objects with micrometer–centimeter sizes. This numerical method simulates the small-angle light scattering experiment. Different geometric 2D-regular fractals, such as the Sierpinski carpet, Sierpinski triangle, Koch snowflake, and Vishek snowflake, are studied. We can divide 2D fractals, by analogy with 3D fractals, into “plane” and “boundary” fractals with fractal dimensions lying in the intervals from 1 to 2 and from 2 to 3, respectively. For an object with a smooth boundary, i.e., a circle, the model small-angle scattering curve decreases according to the power law q–3, where q is the momentum transferred.

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

The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic model in the two dimensional space. Based on Agmon, Douglis, and Nirenberg’s estimates for the stationary Stokes equation and Solonnikov’s theorem on Lp-Lq-estimates for the evolution Stokes equation, it is shown that this coupled magnetohydrodynamic equations possesses a global strong solution. In addition, the uniqueness of the global strong solution is obtained.

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.

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.

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.

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.

Surface hairiness of spun yarns is critical to the yarn post-processing deficiency and resultant fabric quality. Several test methods are available for measuring yarn hairiness but they cannot detect the accurate hair amount and precise hair length. This paper provides a novel method to perform accurate hairiness tests on the spun yarn. A new test apparatus was devised by installing a blowing pipe outside of the hairiness testing area comprising a projection receiver and a corresponding laser. The rectangle end plane of the blowing pipe air inlet is vertical to the laser plane and tangent to the edge line of the projection receiver surface. Pressure generated through this inlet stretches yarn wild hairiness straightly in one direction, and fully maps accurate hairiness amount and length configuration on the projection receiver surface. The subsequent measurement showed that this novel test apparatus detected more hairiness with improved accuracy in both number and length of hairiness. Approximately real gravimetric hairiness weight can be obtained by this method which is impracticable for other conventional photoelectric apparatus.

In this paper, a fourth-order difference scheme (FODS) is proposed for solving the two-dimensional Riesz space-fractional diffusion equations with homogeneous Dirichlet boundary conditions. It is proved that the FODS is uniquely solvable, unconditionally stable, and convergent with order O ( τ 2 + h x 4 + h y 4 ) in the discrete L∞- norm, where τ is the time step size, and hx, hy are the space grid sizes in the x direction and the y direction, respectively. Based on the special structure and symmetric positive definiteness of the coefficient matrix, a fast method is developed for the implementation of the FODS. The fast method reduces the storage requirement of O(N2) and computational cost of O(N3) down to O ( M + J ) and O(Nlog N), where N = M J , M and J are the numbers of the spatial grid points in the x direction and the y direction, respectively. Finally, several numerical results are shown to verify the theoretical results and the efficiency of the fast method.

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.

We extend our previous work on the spatial spread of fox rabies from one dimension to two dimensions. We consider the case when the latent period has fixed length. We use the method of lines to replace the spatial derivatives and the integral equations with algebraic approximations, then we apply the explicit continuous Runge–Kutta method of fourth order and discrete Runge–Kutta method of third order with six stages to numerically integrate the resulting systems of ordinary and delay differential equations. We discuss and confirm some of the major results we obtained in earlier work. The asymptotic speeds of spread observed in the two-dimensional simulations and in earlier work are discussed and compared with those found in nature.