A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers’ equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers’ equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.     

Modified Burgers’ equation by the local discontinuous Galerkin method

In this paper,we present the local discontinuous Galerkin method for solving Burgers’ equation and the modified Burgers’ equation.We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail.The method is applied to the solution of the one-dimensional viscous Burgers’ equation and two forms of the modified Burgers’ equation.The numerical results indicate that the method is very accurate and efficient.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

Kink-Type Solutions of Disturbed Burgers’ Equation

Russian Physics Journal -     

A new complex line soliton for the two-dimensional KdV–Burgers equation

Making use of a extended tanh method with symbolic computation, we find a new complex line soliton for the two-dimensional (2D) KdV–Burgers equation. Its real part is the sum of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D KdV (KP) equation, and its imaginary part is the product of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D MKdV (MKP) equation.     

Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods

This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable.     

An approximate global solution of Einstein’s equation for a rotating compact source with linear equation of state

We use analytic perturbation theory to present a new approximate metric for a rigidly rotating perfect fluid source with equation of state (EOS) $\epsilon +(1-n)p=\epsilon _0$ . This EOS includes the interesting cases of strange matter, constant density and the fluid of the Wahlquist metric. It is fully matched to its approximate asymptotically flat exterior using Lichnerowicz junction conditions and it is shown to be a totally general matching using Darmois–Israel conditions and properties of the harmonic coordinates. Then we analyse the Petrov type of the interior metric and show first that, in accordance with previous results, in the case corresponding to Wahlquist’s metric it can not be matched to the asymptotically flat exterior. Next, that this kind of interior can only be of Petrov types I, D or (in the static case) O and also that the non-static constant density case can only be of type I. Finally, we check that it can not be a source of Kerr’s metric.     

Exact solutions of nonlinear Schrodinger’s equation with dual power-law nonlinearity by extended trial equation method

In this paper, we acquire the soliton solutions of the nonlinear Schrodinger’s equation with dual power-law nonlinearity. Primarily, we use the extended trial equation method to find exact solutions of this equation. Then, we attain some exact solutions including soliton solutions, rational and elliptic function solutions of this equation using the extended trial equation method.     

Can control banding be useful for the safe handling of nanomaterials? A systematic review

Control banding (CB) is a risk management strategy that has been used to identify and recommend exposure control measures to potentially hazardous substances for which toxicological information is limited. The application of CB and level of expertise required for implementation and management can differ depending on knowledge of the hazard potential, the likelihood of exposure, and the ability to verify the effectiveness of exposure control measures. A number of different strategies have been proposed for using CB in workplaces where exposure to engineered nanomaterials (ENMs) can occur. However, it is unclear if the use of CB can effectively reduce worker exposure to nanomaterials. A systematic review of studies was conducted to answer the question “can control banding be useful to ensure adequate controls for the safe handling of nanomaterials.” A variety of databases were searched to identify relevant studies pertaining to CB. Database search terms included ‘control,’ ‘hazard,’ ‘exposure,’ and ‘risk’ banding as well as the use of these terms in the context of nanotechnology or nanomaterials. Other potentially relevant studies were identified during the review of articles obtained in the systematic review process. Identification of studies and the extraction of data were independently conducted by the reviewers. Quality of the studies was assessed using the methodological index for nonrandomized studies. The quality of the evidence was evaluated using grading of recommendations assessment, development and evaluation (GRADE). A total of 235 records were identified in the database search in which 70 records were determined to be eligible for full-text review. Only two studies were identified that met the inclusion criteria. These studies evaluated the application of the CB Nanotool in workplaces where ENMs were being handled. A total of 32 different nanomaterial handling activities were evaluated in these studies by comparing the recommended exposure controls using CB to existing exposure controls previously recommended by an industrial hygienist. It was determined that the selection of exposure controls using CB were consistent with those recommended by an industrial hygienist for 19 out of 32 (59.4 %) job activities. A higher level of exposure control was recommended for nine out of 32 (28.1 %) job activities using CB, while four out of 32 (12.5 %) job activities had in-place exposure controls that were more stringent than those recommended using CB. After evaluation using GRADE, evidence indicated that the use of CB Nanotool can recommend exposure controls for many ENM job activities that would be consistent with those recommended by an experienced industrial hygienist. The use of CB for reducing exposures to ENMs has the potential to be an effective risk management strategy when information is limited on the health risk to the nanomaterial and/or there is an absence of an occupational exposure limit. However, there remains a lack of evidence to conclude that the use of CB can provide adequate exposure control in all work environments. Additional validation work is needed to provide more data to support the use of CB for the safe handling of ENMs.     

An Explicit Backlund Transformation of Burgers Equation with Applications

In this paper, an explicit Bgcklund transformation （BT） of the Burgers equation is obtained by using the further extended tanh method [Phys. Lett. A 307 （2003） 269; Chaos, Solitons ＆ Fractals 17 （2003） 669]. Based on the BT and some newly obtained seed solutions, infinite sequences of exact solutions for the Burgers equation are generated. Further more, this BT of the Burgers equation is applied to solve the variant Boussinesq equations and the approximate equations of long water wave.     

Bcklund transformations for the Burgers equation via localization of residual symmetries

We obtain the non-local residual symmetry related to truncated Painlev′e expansion of Burgers equation. In order to localize the residual symmetry, we introduce new variables to prolong the original Burgers equation into a new system.By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. More importantly,we also localize the linear superposition of multiple residual symmetries to find the corresponding finite transformations.It is interesting to find that the n-th B¨acklund transformation for Burgers equation can be expressed by determinants in a compact way.     

Wahlquist’s metric versus an approximate solution with the same equation of state

We compare an approximation of the singularity-free Wahlquist exact solution with a stationary and axisymmetric metric for a rigidly rotating perfect fluid with the equation of state $\mu +3p=\mu _0$ , a sub-case of a global approximate metric obtained recently by some of us. We see that to have a fluid with vanishing twist vector everywhere in Wahlquist’s metric the only option is to let its parameter $r_0\rightarrow 0$ and using this in the comparison allows us in particular to determine the approximate relation between the angular velocity of the fluid in a set of harmonic coordinates and $r_0$ . Through some coordinate changes we manage to make every component of both approximate metrics equal. In this situation, the free constants of our metric take values that happen to be those needed for it to be of Petrov type D, the last condition that this fluid must verify to give rise to the Wahlquist solution.     

A review on various aspects of organic synthesis using Comins’ reagent

Molecular Diversity - Recently, researchers have shown great interest in compounds such as triflate and enotriflate that are synthesized by Comins reagent. For the above-mentioned reason, we...     

A stochastic Ginzburg–Landau equation with impulsive effects

In this paper we consider a stochastic Ginzburg–Landau equation with impulsive effects. We first prove the existence and uniqueness of the global solution which can be explicitly represented via the solution of a stochastic equation without impulses. Then, based on our obtained result, we study the qualitative properties of the solution, including the boundedness of moments, almost surely exponential convergence and pathwise estimations. Finally, we give a first attempt to study a fractional version of impulsive stochastic Ginzburg–Landau equations.     

On solitary wave solutions of the compound Burgers–Korteweg–de Vries equation

The goal of this note is to construct a class of traveling solitary wave solutions for the compound Burgers–Korteweg–de Vries equation by means of a hyperbolic ansatz. A computational error in a previous work has been clarified.     

The KdV–Burgers equation in a new continuum model with consideration of driverʼs forecast effect and numerical tests

A new continuum traffic flow model is proposed based on an improved car-following model, which takes the driver?s forecast effect into consideration. The backward travel problem is overcome by our model and the neutral stability condition of the new model is obtained through the linear stability analysis. Nonlinear analysis shows clearly that the density fluctuation in traffic flow leads to a variety of density waves and the Korteweg–de Vries–Burgers (KdV–Burgers) equation is derived to describe the traffic flow near the neutral stability line. The corresponding solution for traffic density wave is also derived. Finally, the numerical results show that our model can not only reproduce the evolution of small perturbation, but also improve the stability of traffic flow.     

A plane-symmetric solution of Einstein's equations with perfect fluid and an equation of statep=γϱ

A new class of plane-symmetric solutions of Einstein's equations with perfect fluid source and an equation of statep= (=const.) is presented. It contains the static vacuum solution, a special Kasner solution and the flat Friedmann-Robertson-Walker spacetime as subclasses. The only class for which the matter distribution is truly inhomogeneous (class D in the sequel) represents matter concentrated around a planar orbit of the symmetry group in an expanding universe.     

Sonoproduction of nanobiomaterials – A critical review

Ultrasound (US) demonstrates remarkable potential in synthesising nanomaterials, particularly nanobiomaterials targeted towards biomedical applications. This review briefly introduces existing top-down and bottom-up approaches for nanomaterials synthesis and their corresponding synthesis mechanisms, followed by the expounding of US-driven nanomaterials synthesis. Subsequently, the pros and cons of sono-nanotechnology and its advances in the synthesis of nanobiomaterials are drawn based on recent works. US-synthesised nanobiomaterials have improved properties and performance over conventional synthesis methods and most essentially eliminate the need for harsh and expensive chemicals. The sonoproduction of different classes and types of nanobiomaterials such as metal and superparamagnetic nanoparticles (NPs), lipid- and carbohydrate-based NPs, protein microspheres, microgels and other nanocomposites are broadly categorised based on the physical and/or chemical effects induced by US. This review ends on a good note and recognises US-driven synthesis as a pragmatic solution to satisfy the growing demand for nanobiomaterials, nonetheless some technical challenges are highlighted.