Two-dimensional Nonlinear Equation Research Articles

A class of monotone and structure-preserving Du Fort-Frankel schemes for nonlinear Allen-Cahn equation

In this study, a class of weighted, monotone and structure-preserving Du Fort-Frankel schemes are constructed for solving one-dimensional (1D) and two-dimensional (2D) nonlinear Allen-Cahn equations, respectively. As parameters θ, β and grid-ratios satisfy suitable conditions, numerical solutions obtained using the current methods satisfy the discrete energy-dissipation laws, monotone increases, and the discrete maximum bound principles. Furthermore, by using the discrete and continuous maximum bound principles, maximum-norm error estimates have been derived, rigorously. Although the current methods are conditionally consistent, they are very easy to be implemented and cost less than some the existent methods because of their explicitness. Finally, numerical results confirm the exactness of theoretical findings, and simulate interesting physical phenomena.

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

Delta-V analysis for impulsive orbital pursuit-evasion based on reachable domain coverage

This paper proposes a delta-V analysis method for the impulsive orbital interception of a non-cooperative maneuvering target from the perspective of reachable domain coverage. In the case of small impulse magnitude and short propagation time, the envelope of the impulsive reachable domain is approximately described by an ellipsoidal equation based on the linearized state transition matrix. Then, the minimum pursuit impulse magnitude under the reachable-domain coverage constraint is obtained by solving a two-dimensional nonlinear equation for given pursuit impulse time and given interception time. Two fuel optimization problems under the reachable-domain coverage constraint are constructed based on whether to optimize the first impulse of the chaser. For the first optimization problem, the optimization variables include the pursuit impulse time, the interception time, and the pursuit impulse magnitude; for the second optimization problem, two additional optimization variables are considered including the time and the aiming point of the first impulse. Numerical examples show that the proposed optimal delta-V approach for the chaser can cover the target's maneuvering reachable domain.

A second-order fitted scheme combined with time two-grid technique for two-dimensional nonlinear time fractional telegraph equations involving initial singularity

In this paper, we derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous L2-1σ formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements.

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

In this article, we use a new technique called conformable fractional reduced differential transform (CFRDT) with Adomian decomposition to estimate the solution of one and two-dimensional time-fractional partial linear and nonlinear differential equations with initial values. We explain the convergence analysis of this technique. The obtained results illustrate that the novel method is efficient and easy to use to find approximate solutions for the time-fractional partial differential equations (PDEs). Thus, the suggested method has a significant impact on how engineering, physics, and other disciplines solve fractional PDEs. Furthermore, we analyze the solution of problems with a 2D or 3D graphical representation by using Mathematica software.

A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

Fourth-order modon in a rotating self-gravitating fluid

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

A Numerical Technique for Solving Optimization Problems

The aim of this paper is to calculate a better approximation value (whether it is maximize or minimize ) for one- and two-dimensional nonlinear equations using the best numerical optimization algorithms, which is Newton's method. The idea of this technique is based on approximating the function by expanding the Taylor series expansion and iteratively updating the estimate of the optimal solution. we have obtained good results in terms of accuracy and speed of approach, as shown in the examples mentioned. We also mentioned the applications of Newton’s method in multiple disciplines, including engineering, physics, economics, finance, computer graphics, machine learning, image processing, and other applications.

Nonlinear internal gravity waves in the atmosphere: Rogue waves, breathers and dark solitons

We study nonlinear internal gravity waves (IGWs) in the atmosphere. The reductive perturbation method is used to derive a system of two-dimensional nonlinear equations for the envelope of velocity stream function and the mean flow. In the one-dimensional case, we obtain a nonlinear Schrödinger (NLS) equation corresponding to both horizontal and vertical propagation of IGWs. Depending on the characteristic wavelengths, the NLS equation is focusing or defocusing. In the focusing case, non-stationary solutions in the form of the Peregrine soliton, the Akhmediev breather and the Kuznetsov–Ma breather are considered as potential candidates for the modeling of rogue waves in the atmosphere. In the defocusing case, stationary nonlinear IGWs are considered in the form of nonlinear periodic waves and dark solitons.

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.

A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

Study of the model of the phase transition envelope taking into account the process of thermal storage under natural draft and by air injection

The study proposed mathematical modeling of the enclosing structure of the building, which includes an energy-active panel accumulating solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear heat conduction equation describing the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel, taking into account the process of heat accumulation under natural and forced traction by air injection. In comparison of the calculation results, it was found. that the efficiency of the use of forced convection relative to natural was 33–44% in the question of the selection of useful heat, depending on the time (second or first day), and the value of heat accumulation showed almost the same results. Taking into account the research carried out by the authors, it is shown by calculation that enclosing structures in which a heat-accumulating material with a phase transition is used to accumulate solar energy are more energy efficient than simple heat accumulation in the carrier layer. The introduction of highly conductive inclusions into the energy-active panel and the use of forced convection instead of natural traction during heat extraction also significantly increase the energy efficiency of the structure, which can subsequently be used as an option for external fencing under appropriate climatic conditions.

Thermo-mechanical buckling analysis of non-uniformly heated rectangular plates with temperature-dependent material properties

This paper examines the thermo-mechanical buckling behavior of a rectangular plate with temperature-dependent (TD) material properties exposed to non-uniform heating. The buckling analysis was performed by using the differential quadrature (DQ) method, which mainly includes three steps: (i) a two-dimensional (2D) non-linear heat transfer equation was proposed to obtain the temperature distribution in the plate considering the convective–radiative conditions on the plate surface; (ii) the in-plane stress distribution of the plate was obtained by solving the thermoelastic equilibrium equation; (iii) the critical load was determined by solving the stability equation based on the stress distribution in the previous step. The comparison study shows that the present results were in good agreement with the finite element (FE) solutions and the results reported in the literature. The effects of the surrounding temperature fields and in-plane load cases on the thermo-mechanical behavior of the plate are analyzed in detail.

A two-grid ADI finite element approximation for a nonlinear distributed-order fractional sub-diffusion equation

<abstract><p>In this paper, a two-grid alternating direction implicit (ADI) finite element (FE) method based on the weighted and shifted Grünwald difference (WSGD) operator is proposed for solving a two-dimensional nonlinear time distributed-order fractional sub-diffusion equation. The stability and optimal error estimates with second-order convergence rate in spatial direction are obtained. The storage space can be reduced and computing efficiency can be improved in this method. Two numerical examples are provided to verify the theoretical results.</p></abstract>

Integrated Chebyshev wavelets for numerical solution of nonlinear one-dimensional and two-dimensional Rosenau equations

In this study, we focus on developing an efficient computational method for one-dimensional and two-dimensional nonlinear Rosenau equations. Suggested computational method in this study uses finite differences for discretization of time variable. For discretization of space variables, firstly unknown function with highest derivative which is appeared in the Rosenau equation is expanded to Chebyshev wavelets and then by successive integrations the unknown function itself is found in terms of truncated Chebyshev wavelets. Also, a linearization technique is applied to handle the nonlinearity. By doing so we obtain a linear system of equations for solving the Rosenau equation. We apply the suggested method to four test problems and compare obtained numerical solutions with exact solutions and with finite element method to assess how accurate are the numerical results. The obtained numerical results endorse the efficiency and practicality of the suggested method for both one-dimensional and two-dimensional nonlinear Rosenau equations.

Construction of a model for an enclosing structure with a heat-accumulating material with phase transition taking into account the process of solar energy accumulation

This paper proposes a mathematical model and a procedure for calculating the thermal state of the enclosing structure of the building, which includes an energy-active panel that accumulates solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear equation of thermal conductivity, which describes the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel. The model also includes equations describing radiant heat transfer between opaque and translucent bodies. To correctly describe solar insolation, the ASHRAE 2009 model was used in conjunction with the daily change in the position of the Sun in the sky. To solve the system of equations that make up the mathematical model, an iterative procedure has been developed, which involves alternating solution at each time step of the two-dimensional equation of thermal conductivity and a set of algebraic equations of convective and radiant heat transfer. The study’s result established that the amount of accumulated energy in the heat-accumulating material of the phase transition during daylight hours increases significantly, from 15 to 35 %. At night, the surface temperature of the heat-accumulating element in structures using a material with a phase transition is greater than in the case of heat accumulation only in the bearing layer. As a result, it is possible to select from 70 to 120 % more accumulated heat while the presence of high-thermal partitions in a heat-accumulating material with a phase transition contributes to an increase in accumulated heat and usable heat

Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation

Over the years, energy-dissipation-preserving (EDP) numerical methods (EDP-NMs) for the nonlinear system of energy dissipation have attracted considerable attention because they are good at long-term computations. More recently, the invariant energy quadratization methods (IEQMs) have been proposed by Xiaofeng Yang and his collaborators to construct linearized EDP-NMs for the nonlinear system of energy dissipation, such as gradient flows. However, the system of the linear algebraic equations corresponding to the obtained EDP-NMs involves the variable coefficient matrices in general, thus resulting in complex computations. Besides, most of efficient alternating direction implicit (ADI) methods are not able to preserve the structures of continuous problem, such as, energy dissipation, energy conservation. To overcome these shortcomings, taking two-dimensional (2D) nonlinear Allen-Cahn equation for example, two energy-dissipation-preserving ADI finite difference methods (FDMs) have been developed by using IEQMs. Numerical analyses of them, such as, the discrete energy-dissipation laws, error estimations and stabilities, have been derived in detail. Numerical results support the theoretical analysis, and confirm the efficiency and accuracy of the proposed algorithms.

A numerical approach to a 2D porous-medium mathematical model: Application to an atherosclerosis problem

This work deals with the study and implementation of a Local Space-Time DG ADER approach, in the finite volume framework with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space, in the context of 2D porous media problems. The particular application performed concerns a biomedical problem dealing with the first stages of atherosclerosis disease, where the artery is considered as a porous medium, although the methodology described can be applied to other situations. The mathematical model on which this research is based is given by a system of two-dimensional nonlinear reaction-diffusion equations with a nonlinear source term in one of the equations, which is a variant of the model proposed originally in N. El Khatib, S. Genieys & V. Volpert (2007) Math. Model. Nat. Phenom., vol 2(2), pp. 126–141 [1]. The 2D model considered in this work has two main features: (i) the artery is taken as a porous medium and (ii) a nonlinear non-homogeneous Neumann boundary condition is incorporated, aimed to account for the recruitment of immune cells through the upper boundary, as a response to the production of cytokines. Certain theoretical properties of the stationary solutions and the evolution solution are stated and proved. Some of them are also verified with the numerical simulation results.

NONLINEAR ELLIPTIC EQUATION WITH NONLOCAL INTEGRAL BOUNDARY CONDITION DEPENDING ON TWO PARAMETERS

In this paper, the two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters is solved by finite difference method. The main aim of this paper is to investigate the conditions under those all eigenvalues of corresponding difference eigenvalue problem are positive. For this purpose, we investigate the spectrum structure of one-dimensional difference eigenvalue problem with integral condition. In particular, conditions of the existence and some properties of negative eigenvalue are analyzed in details.

Maximum error estimates of two linearized compact difference schemes for two-dimensional nonlinear Sobolev equations

In this paper, two classes of high-order numerical schemes on the time discretization for the solutions of two-dimensional nonlinear Sobolev equations are analyzed. The two-level Newton linearized difference scheme had been established in the previous literature, in which the error estimates in L2-norm and H1-norm are obtained, but that in L∞-norm remains unsolved. The second difference scheme is established based on a three-level linearized difference technique in temporal direction. The numerical discretization in spatial dimension for both schemes utilizes the compact difference operator. We prove that the orders of convergence for both difference schemes in L∞-norm are O(τ2+h14+h24) based on the energy argument, where τ denotes the temporal step size, h1 and h2 denote the spatial step sizes in x-direction and y-direction, respectively. The numerical tests further verify the theoretical results and demonstrate the efficiency of both difference schemes.