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In this manuscript, the Korteweg-de Vries-Burgers (KdV-Burgers) partial differential equation (PDE) is investigated under nonlocal operators with the Mittag-Leffler kernel and the exponential decay kernel. For both fractional operators, the existence of the solution of the KdVBurgers PDE is demonstrated through fixed point theorems of ff-type z contraction. The modified double Laplace transform is utilized to compute a series solution that leads to the exact values when fractional order equals unity. The effectiveness and reliability of the suggested approach are verified and confirmed by comparing the series outcomes to the exact values. Moreover, the series solution is demonstrated through graphs for a few fractional orders. Lastly, a comparison between the results of the two fractional operators is studied through numerical data and diagrams. The results show how consistently accurate the method is and how broadly applicable it is to fractional nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams from parameterized nonlinear PDEs. Additionally, a neural network approach is also presented for solving eigenvalue problems to analyze solution linear stability, focusing on identifying the largest eigenvalue. The effectiveness of the proposed neural network is examined through experiments on the Bratu equation and the Burgers equation. Results from a finite difference method are also presented as comparison. Varying numbers of grid points are employed in each case to assess the behavior and accuracy of both the neural network and the finite difference method. The experimental results demonstrate that the proposed neural network produces better solutions, generates more accurate bifurcation diagrams, has reasonable computational times, and proves effective for linear stability analysis. • A neural network is combined with a pseudo-arclength continuation to construct bifurcation diagrams from nonlinear PDEs. • An additional neural network for solving eigenvalue problems to analyze solution linear stability is presented. • Experiments are conducted on the Bratu equation and the Burgers equation. • The proposed neural network is compared to a finite difference method. • The neural network generates more accurate bifurcation diagrams and proves effective for linear stability analysis. [ABSTRACT FROM AUTHOR]

Nonlinear Burgers equation with variable coefficients (BEVC) which involves mathematical physics in plasma and fluid dynamics and so on. Investigating the exact solutions of BEVC to show the different dynamics of wave phenomena becomes a hot topic both on many mathematicians and physicists. In this paper, by (G′G2)-expansion and the Jacobian elliptic functions (JEFs) two different methods, we found that various forms for exact wave solutions of BEVC. To our best knowledge, we found a variety of new solutions that have not been studied in previous articles such as u13,u14,u511,u512,u513,u514. The most important thing is that there are double Jacobian elliptic functions ideas in finding solution process, which has not been seen before in seeking for nonlinear BEVC. These new exact soliton solutions contain variable coefficients derived in the form of trigonometric function, rational function, and Jacobian elliptic function, hyperbolic function. The obtained results showed many different types such as annihilation, parabolic kink, curved shaped kink, tine shaped, shock solitons and so on. Furthermore, the above obtained solutions for Burgers equation with variable coefficients are different from Mohanty et al. (2022), Zayed and Abdelaziz (2010).

The global random walk on grid method (GRWG) is developed for solving two-dimensional nonlinear systems of equations, the Navier–Stokes and Burgers equations. This study extends the GRWG which we have earlier developed for solving the nonlinear drift-diffusion-Poisson equation of semiconductors (Physica A556 (2020), Article ID 124800). The Burgers equation is solved by a direct iteration of a system of linear drift-diffusion equations, while the Navier–Stokes equation is solved in the stream function-vorticity formulation. [ABSTRACT FROM AUTHOR]

This article deals with the regularity of a solution for a non-homogeneous heat equation in Sobolev spaces. The problem is subject to Cauchy–Dirichlet boundary conditions considered in a rectangle or domain that can be transformed into a rectangle. [ABSTRACT FROM AUTHOR]

In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation F (u , ε) = u t ′ - u xx ′ ′ + u u x ′ + ε u 2 = f (x , t) , where F : Ω → C ([ 0 , π ] × [ 0 , ∞)) , Ω = C 2 ([ 0 , π ] × [ 0 , ∞)) × R , u (0 , t) = u (π , t) = 0 , u (x , 0) = g (x) , and F, f(x, t), g(x) will be describe in the text. The first derivative of operator F at the solution point is degenerate. By virtue of p-regularity theory and Michael selection theorem, we prove the existence of continuous solution for this nonlinear problem. [ABSTRACT FROM AUTHOR]

Changes in atmospheric pressure continuously ventilate soils and snowpacks. This physical process, known as pressure pumping, is a major factor in the exchange fluxes of H2O, CO2, and other trace gases between the soil and atmosphere. Thus models of pressure pumping are relevant to many areas of critical importance. This study compares the three principal models used to describe pressure pumping. Beginning with the fundamental physical principles and whether the flow field is compressible or incompressible, these models are categorized as linear parabolic (one model—compressible) or nonlinear hyperbolic (two models—incompressible). Using observed soil surface pressure data, measured vertical profiles of soil permeability and standard linear analysis and numerical methods, this study shows that nonlinear models produce advective velocities that are one to two orders of magnitude greater than those associated with the linear model. Incorporating soil temperature and moisture dynamics made very little difference to the linear model, but a significant difference in the nonlinear models suggesting that advective velocities induced by pressure changes associated with soil heating and moisture dynamics may not always be small enough to ignore. All numerical results are sensitive to the frequency of the pressure forcing, which was band‐pass filtered into low, mid and high frequencies with the greatest model differences at low frequencies. Partitioning the pressure forcing and model responses helped to establish that mid‐frequency weather‐related phenomena (empirically identified as inertia gravity waves and solitons) are important drivers of gas exchange between the soil and the atmosphere. Plain Language Summary: Atmospheric pressure is constantly moving air in, out and through the surface of the soil or a snowpack. This pressure pumping (or ventilating) mechanism influences soil water evaporation and snow sublimation, the fluxes of key climate‐warming greenhouse gases, and rate at which contaminants can be removed from soils. Thus the ability to model the movement of these gases through these two media is relevant to many current environmental concerns. The present study discusses the differences between the two broad categories of pressure pumping models and points out that these models predict very different ventilation rates. So different that it may be possible to conclude (incorrectly) that the daily cycle for pressure is not significant to soil or snowpack gas exchange. These two model types also respond differently when pressure changes are influenced by changes in soil temperature and moisture, with one type suggesting that dynamic soil temperature and moisture effects are small, in agreement with expectations, and the other suggesting that they can cause surprisingly large effects. Present results also identify (for the first time) specific mesoscale atmospheric waves, often associated with frontal systems, convective activity and rain, can be significant drivers of pressure induced gas exchange. Key Points: The efficiency of atmospheric pressure pumping to the transport of trace gases out of soils and snowpacks is strongly model dependentSome models suggest that the daily temperature cycle and rapid changes in soil moisture have an unexpectedly large impact on such transportMesoscale atmospheric phenomena often associated with severe weather and the daily cycle of turbulence cause significant advective transport [ABSTRACT FROM AUTHOR]

We propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks. [ABSTRACT FROM AUTHOR]

Particular solutions of the Burgers equations (BE) with zero boundary conditions are investigated in an analytical form. For values of the shape parameter greater than 1, but approximately equal to 1, the amplitude of the initial periodic perturbations depends nonmonotonically on the spatial coordinate, i.e. the initial perturbation can be considered as a shock wave. Particular BE solutions with zero boundary conditions describe a time decrease of the amplitude of initial nonmonotonic perturbations, which indicates the decay of the initial shock wave. At large values of the shape parameter , the amplitude of the initial periodic perturbations depends harmoniously on the spatial coordinate. It is shown that over time, the amplitude and the spatial derivative of the profile of such a perturbation decrease and tend to zero. Emphasis was put on the fact that particular BE solutions can be used to control numerical calculations related to the BE-based description of shock waves in the region of large spatial gradients, that is, under conditions of a manifold increase in spatial derivatives. These solutions are employed to describe the profile of a one-dimensional train of elementary steps with an orientation near , formed during the growth of a NaCl single crystal from the vapor phase at the base of a macroscopic cleavage step. It is shown that the distribution of the step concentration with distance from the initial position of the macrostep adequately reflects the shock wave profile at the decay stage. The dimensionless parameters of the wave are determined, on the basis of which the estimates of the characteristic time of the shock wave decay are made.

As a class of nonlinear partial differential equations, the Burgers equations are widely used in various fields. A new space-time polynomial collocation method was presented for particular solutions to 3D Burgers equations. The basic process was divided into 2 steps. The 1st step is to find the polynomial particular solutions of the linear differential operator terms (including the time differential term) in the governing equation. The 2nd step is to solve the nonlinear term of the 3D Burgers equation iteratively. The proposed method is simple and easy to program. The approximate solution has high accuracy. Especially, the stability of the method is excellent, which improves the programming simplicity and deepens the understanding of high-dimensional Burgers equations and the practical application. [ABSTRACT FROM AUTHOR]

In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation. [ABSTRACT FROM AUTHOR]

We contribute an answer to a quantitative variant of the question raised in Coron (in: Perspectives in nonlinear partial differential equations. Contemporary mathematics, vol 446, American Mathematical Society, Providence, pp 215–243, 2007) concerning the controllability of the viscous Burgers equation u t + (u 2 / 2) x = u xx for initial and terminal data prescribed for x ∈ (0 , 1) . We investigate the (non)-controllability under the additional a priori bound imposed on the (nonlinear) operator that associates the solution to the terminal state. In contrast to typical techniques on the controllability of the viscous Burgers equation invoking the heat equation, we combine scaling and compensated compactness arguments along with observations on the non-controllability of the inviscid Burgers equation to point out wide sets of terminal states non-attainable from zero initial data by solutions of restricted size. We prove in particular that, given L ≥ 1 , for sufficiently large |C| and T < (1 + Δ) / | C | (where Δ > 0 depends on L), the constant terminal state u (· , T) : = C is not attainable at time T, starting from zero data, by weak solutions of the viscous Burgers equation satisfying a bounded amplification restriction of the form ‖ u ‖ ∞ ≤ L | C | . Our focus on L ∞ solutions is due to the fact that we rely upon the classical theory of Kruzhkov entropy solutions to the inviscid equation. In Part II of this paper, we will extend the non-controllability results to solutions of the viscous Burgers equation in the L 2 setting, upon extending the Kruzhkov theory appropriately. [ABSTRACT FROM AUTHOR]

In the present paper, a modified simple equation method is used to obtain exact solutions of the equal width wave Burgers and modified equal width wave Burgers equations. By giving specific values to the parameters, particular solutions are obtained and the plots of solutions are drawn. It shows that the proposed method can be easily generalized to solve a variety of non-linear equations by implementing a robust and straightforward algorithm without the need for any tools. [ABSTRACT FROM AUTHOR]

A theoretical investigation has been carried out to examine the ion-acoustic shock waves (IASHWs) in a magnetized degenerate quantum plasma system containing inertialess ultra-relativistically degenerate electrons, and inertial non-relativistic positively charged heavy and light ions. The Burgers equation is derived by employing the reductive perturbation method. It can be seen that under the consideration of non-relativistic positively charged heavy and light ions, the plasma model only supports the positive electrostatic shock structure. It is also observed that the charge state and number density of the non-relativistic heavy and light ions enhance the amplitude of IASHWs, and the steepness of the shock profile is decreased with ion kinematic viscosity. The findings of our present investigation will be helpful in understanding the nonlinear propagation of IASHWs in white dwarfs and neutron stars.

The Smoothed Particle Hydrodynamics (SPH) is a meshless particle-based method mainly used to solve dynamical problems for partial differential equations (PDE). By means of dispersion analysis we investigated four classical SPH-discretizations of parabolic PDE differing by the approximation of Laplacian. We derived approximate dispersion relations (ADR) for considered SPH-approximations of the Burgers equation. We demonstrated how the analysis of the ADR allows both studying the approximation and stability of numerical scheme and explaining the features of the method that are known from practice, but are counter-intuitive from the theoretical point of view. By means of the mathematical analysis of ADR, the phenomenon of conditional approximation of some schemes under consideration is shown. Moreover, we pioneered in obtaining the necessary condition for the stability of the SPH-approximation of parabolic equations in terms of the Fredholm integral operator applied to the function defined by the kernel of the SPH method. Using this condition, we revealed that passing from the classical second-order kernels to high-order kernels for some schemes leads to the appearance of tensile (short-wave) instability. Among the schemes under consideration, we found the one, for which the necessary condition for the stability of short waves is satisfied both for classical and high-order kernels. The fourth order of approximation in space of this scheme is shown theoretically and confirmed in practice. • Approximate dispersion relation for different SPH-approximation of parabolic equation is derived. • Asymptotic approximate dispersion relation reveals tensile instability. • Fourier transforms of SPH high-order kernels are non-positive. • Fredholm transforms of SPH high-order kernels are often positive. [ABSTRACT FROM AUTHOR]

Abstract Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

First, we discuss the non-Gaussian type of self-similar solutions to the Navier–Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne et al. (1996 Commun. Partial. Differ. Equ.21, 179–193). In order to shed some light on it, we study self-similar solutions to the one-dimensional Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution, we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer's function M. For the two-dimensional Navier–Stokes equations, on top of the celebrated Burgers vortex, we derive yet another solution to the associated Fokker–Planck equation. This can be regarded as a 'conjugate' to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the three-dimensional (3D) Navier–Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier–Stokes equations, we consider what we could tell about more general solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Polytechnic is the property of Journal of Polytechnic and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The present work investigates the single and overtaking collision of multi-shock wave excitations having space fractional evolution in a thin viscoelastic tube filled with incompressible inviscid fluid. The previously proposed model equations are considered to study such physical scenarios. The spatial fractional Burgers equation is formulated by implementing the reductive perturbation method form the considered model equations. The new analytical solutions for single and overtaking collision of multi-shocks are constructed by implementing the rational exponential functions directly. With the changes of physical parameters, the behaviors of single and overtaking collision of multi-shocks are displayed graphically and described physically. It is found that the overtaking collisions of multi–shocks are produced with the presence of beta nonlocal operator. The single and interactions of multi-shocks are also significantly changed with the change of physical parameters. The obtained results are very useful in describing the nature of overtaking collision of multi-shocks in various environments, particularly in large blood vessels and further laboratory studies.

This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

In this paper I give a short review on the continuous, semi-discrete and discrete Burgers equations, which are featured as integrable equations that are linearisable. The review focuses more on connections of these three kinds of equations and connections with (2+1)-dimensional systems. The continuous Burgers hierarchy is a reduction of the derivative nonlinear Schrödinger hierarchy. The later is a result of the squared eigenfunction symmetry constraint of the modified Kadomtsev–Petviashvili system. The semi-discrete Burgers hierarchy can be considered as Bäcklund transformations of the continuous Burgers hierarchy. They are also reductions of the relativistic Toda hierarchy, which is a consequence of the squared eigenfunction symmetry constraint of the semi-discrete modified Kadomtsev–Petviashvili system. The fully discrete Burgers equation is the Bianchi identity of the Bäcklund transformation for the continuous Burgers hierarchy. It is 3-point lattice equation, three-dimensionally consistent, and linearisable.

In this research, we have used a technique known as the homotopy perturbation method (HPM) to discover a solution to the one-dimensional convection diffusion problem. Equations of convection and diffusion hold a unique place in the domains of engineering and science, and they serve as a useful model for a great deal of different kinds of systems that are found in a variety of disciplines. The homotopy perturbation technique is one of the most recent analytic methods that can be utilized to determine the exact solution of a number of different classes of PDE. Through the use of the HPM, a complex issue is reduced to a manageable and straightforward issue that may be resolved with relative ease. When contrasted with the results of numerical solution, the results generated by HPM are analyzed. [ABSTRACT FROM AUTHOR]

This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity of solutions and their statistical quantities in this stochastic dynamical system. The quantities include moment estimate, structure function and energy spectrum of the turbulent velocity field. Furthermore, we provide qualitative and quantitative properties of the stochastic Burgers equation when the kinematic viscosity ν tends towards zero. The inviscid limit describes the strong stochastic turbulence. [ABSTRACT FROM AUTHOR]

In this paper, Reduced Differential Transformmethod in the framework of (p, q)-calculus, denoted by Rp,qDT, has been introduced and applied in solving a variety of differential equations such as diffusion equation, 2Dwave equation, K-dV equation, Burgers equations and Ito system. While the diffusion equation has been studied for the special case p = 1, i.e., in the framework of q-calculus, the other equations have not been studied even in q-calculus. [ABSTRACT FROM AUTHOR]

We solve the inviscid Burgers equation involving a logistic reaction. The goal is to investigate the effects of the logistic reaction to numerical solutions of the inviscid Burgers equation. We use standard and nonstandard finite difference methods. Based on our research results, the logistic reaction in the inviscid Burgers equation makes the shock wave propagates faster.

In this article, the well-posedness of the boundary value problem for the Burgers equation with dynamic boundary conditions is studied in Sobolev spaces with a degenerating domain. [ABSTRACT FROM AUTHOR]

Stochastic subgrid-scale parametrizations aim to incorporate effects of unresolved processes in an effective model by sampling from a distribution usually described in terms of resolved modes. This is an active research area in fluid dynamics where processes evolve on a wide range of spatial and temporal scales. We propose a data-driven framework where resolved modes are defined as local spatial averages and deviations from these averages are the unresolved degrees of freedom. The proposed approach is applicable to a wide range of finite volume and finite difference numerical schemes commonly used to discretize many realistic problems in fluid dynamics. In this study, we evaluate the performance of conditional generative adversarial network (GAN) in parametrizing subgrid-scale effects in a finite difference discretization of stochastically forced Burgers equation. We train a Wasserstein GAN (WGAN) conditioned on the resolved variables to learn the distribution of the subgrid flux and, thus, represent the effect of unresolved scales. The resulting WGAN is then used in an effective model to reproduce the statistical features of resolved modes. We demonstrate that various stationary statistical quantities such as spectrum, moments and autocorrelation are well approximated by this effective model. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a numerical method to approximate the solution of non-linear fractal-fractional Burgers equation. In this model, differential operators are defined in the Atangana-RiemannLiouville sense with Mittag-Leffler kernel. We first expand the spatial derivatives using barycentric interpolation method, and then we derive an operational matrix (OM) of the fractal-fractional derivative for the Legendre polynomials. To be more precise, two approximation tools are coupled to convert the fractal-fractional Burgers equation into a system of algebraic equations which is technically uncomplicated and can be solved using available mathematical software such as MATLAB. To investigate the agreement between exact and approximate solutions, several examples are examined. [ABSTRACT FROM AUTHOR]

In this paper, we propose the barycentric interpolation collocation method for the Burgers equation. Firstly, Burgers equation is transformed into the heat conduction equation by the Hopf-Cole transformation. Then we use the barycentric interpolation collocation method to discretize the equation in the time and space directions. The collocation scheme of the Burgers equation is constructed and the corresponding linear algebraic equations are derived. Moreover, the consistency analysis of barycentric Lagrange interpolation method is given. Numerical examples validate the efficiency of our scheme. [ABSTRACT FROM AUTHOR]

Copyright of Latin-American Journal of Physics Education is the property of Latin-American Physics Education Network and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider some initial boundary value problems for the Burgers equation in a rectangular domain, which in a sense can be taken as a model one. The fact is that such a problem often arises when studying the Burgers equation in domains with moving boundaries. Using the methods of functional analysis, priori estimates, and Faedo-Galerkin in Sobolev spaces and in a rectangular domain, we show the correctness of the initial boundary value problem for the Burgers equation with nonlinear boundary conditions of the Neumann type.

The dusty plasma system (containing nonthermally distributed inertialess electron species, warm inertial ion species, and positively charged stationary dust species) is considered. The basic features of subsonic and supersonic modified-ion-acoustic solitary and shock waves formed in such a dusty plasma system have been investigated by the reductive perturbation method. It has been shown that positively charged dust species play a new role in favor of the formation of subsonic solitary and shock waves. On the other hand, the ion fluid temperature (represented by the parameter σ) and the electron nonthermal parameter (represented by α) play new significant roles in against the formation of subsonic solitary and shock waves, and give rise to the formation of the supersonic solitary and shock waves after their (σ’s and α’s) certain values. It is also shown that after a certain value of the nonthermal parameter α, the subsonic as well as supersonic solitary and shock waves are formed with negative potential. The important applications of the results of this theoretical investigation in space and laboratory dusty plasma systems are pinpointed.

In recent years, artificial neural networks developed rapidly. Application of this method to partial differential equations became a new idea for exploring numerical solutions to differential equations. Compared with the traditional methods, it has some advantages, such as a wide range of applications (i.e. the same model can be used to solve multiple types of equations) and low meshing requirements. In addition, the trained model can be directly used to calculate the numerical solution at any point in the computation domain. The weight coefficients in the traditional finite volume method were optimized based on the convolutional neural network model to get a new numerical scheme with high-resolution results on the coarse grid. The proposed model helps solve the Burgers and level set equations efficiently and stably with high accuracy. [ABSTRACT FROM AUTHOR]

In this study, an indirect method is proposed based on the Chebyshev pseudo-spectral method for solving optimal control problems governed by Burgers’ equation. Pseudo-spectral methods are one of the most accurate methods for solving nonlinear continuous-time problems, specially optimal control problems. By using optimality conditions, the original optimal control problem is first reduced to a system of partial differential equations with boundary conditions. Control and state functions are then approximated by interpolating polynomials. The convergence is analyzed, and some numerical examples are solved to show the efficiency and capability of the method.

In this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; vt+vvx-vxx=0, x>0, t>0,v(x,0)=u+, x>0,v(0,t)=ub, t>0,$\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

Abstract This article deals with the fractional multi-dimensional Burgers equation in the sense of the Caputo fractional derivative. An approximate analytical solution of the problem is established by the homotopy perturbation method (HPM). Furthermore, the convergence analysis and the error estimation derived by the HPM are shown.

This paper introduces new fully implicit numerical schemes for solving 1D and 2D unsteady Burgers' equation. The non-linear Burgers' equation is discretized in the spatial direction by using second order Finite difference method which converts the Burgers' equation to non-linear system of ODEs. Then, the backward differentiation formula of order two (BDF-2) is employed to march the solution in the time direction. The non-linear term in the obtained system is linearized without any transformation; and it forms a system of linear algebraic equations that is solved by using Thomas's algorithm. Accuracy and performance of the proposed schemes are studied by using four test problems with Dirichlet and Neumann boundary conditions. Comparing the numerical results with exact solutions and the solutions of other schemes shows that the proposed schemes are simple, efficient and accurate even for the cases with high Reynolds numbers.

In this paper we consider a class of Burgers equations. We propose a new method for investigation for existence of classical solutions.