In this paper, using the Lie group analysis method, the infinitesimal generators for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained. The new concept of nonlinear self-adjointness ...of differential equations is used for construction of nonlocal conservation laws. The conservation laws for the (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained by using the new conservation theorem method and the formal Lagrangian approach. Transforming this equation into a system of equations involving with two dependent variables, it has been shown that the resultant system of equations is quasi self-adjoint and finally the new nonlocal conservation laws are constructed by using the Lie symmetry operators.
In this paper, new exact solutions of fractional nonlinear acoustic wave equations have been devised. The travelling periodic wave solutions of fractional Burgers–Hopf equation and ...Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation have obtained by first integral method. Nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible. Taking nonlinear effects into account, the ultrasound beam analysis makes more accurate in these applications. The Burgers–Hopf equation is one of the extensively studied models in mathematical physics. In addition, the KZK parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3D diffraction sound beams in dissipative media. In the present analysis, these nonlinear equations have solved by first integral method. As a result, new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations. The obtained results are presented graphically to demonstrate the efficiency of this proposed method.
This paper investigates a reward-driven policy, employed in a closed-loop supply chain (CLSC), for acquiring used products earmarked for remanufacture. Under the examined model, a single manufacturer ...sells products through a retailer as well as directly to end users in a forward supply chain. In the reverse supply chain, three different modes of collection are employed to capture used products for remanufacture: they are through a third party, directly by the manufacturer and from the retailer. Mathematical models for both non-cooperative and centralised scenarios are developed to characterise the pricing decisions and remanufacturing strategies that indicate individual and overall supply chain performance. Optimality of all the proposed models is examined with theory. To coordinate and achieve a win-win outcome for channel members, we proposed a three-way discount mechanism for the manufacturer. Extended numerical investigation provides insights on ways to manage an efficient reward-driven CLSC in a dual-channel environment.
In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics, namely the time fractional KdV–Zakharov–Kuznetsov (KdV–ZK) and ...space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equations by using improved fractional sub equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie’s modified Riemann–Liouville sense.
An efficient design is one of the most critical activities for ensuring structural integrity and operational reliability of an industrial plant. Failure of the piping components is a serious hazard ...leading to loss of assets and endangering environment safety. Typically a hydrocarbon or a power plant is comprised of piping, equipment and structures. The structures are mainly intended to support the piping or equipment mounted on them. In general the composite system including piping, structures and equipment should be analyzed as an integrated assembly. However this is not practical for a variety of reasons and a decoupled analysis (i.e. piping, structure equipment are analyzed as separate independent systems) is carried out as a current practice. Invariably this method is an approximation and generates solution errors which could lead to unrealistic results and unsafe design. The challenge is to estimate the effect of interaction between the different systems and their impact on the overall solution error. In the study a method has been presented for deriving a coupling parameter for static interaction between the primary and secondary systems. The coupling parameter indicates the degree of interaction between the primary and the secondary systems. This is related to the solution error and could give adequate cue to the analyst whether to go for a decoupled or an augmented analysis. Possibly this is a novel approach and not found in the literature till date. The theory has been developed on a mathematical framework and is quite general and not limited to small systems. Numerical simulation has been presented to validate the theory followed by a real life case study.
•Practical constraints entail decoupled analysis (DCA) of primary & secondary systems for strength design.•Different types of models for DCA result in solution errors could lead to unsafe design.•In this work a decoupling parameter (DP) has been developed based on fundamental theory.•DP gives a measure of the possible solution error in the results of DCA which helps in assessing reliability of design.•Results of numerical simulation for the validation and a real life case study have been given.
In this paper, the invariance properties of the time fractional (2+1)-dimensional Zakharov–Kuznetsov modified equal width (ZK-MEW) equation have been investigated using the Lie group analysis method. ...Lie point symmetries of the time fractional (2+1)-dimensional ZK-MEW equation have been derived by using the Lie group analysis method of fractional differential equations. Using the Lie symmetry analysis, the vector fields and the symmetry reduction of this equation are obtained. It is shown that the time fractional (2+1)-dimensional ZK-MEW equation can be transformed to an equation with Erdélyi–Kober fractional derivative. Finally using new conservation theorem with formal Lagrangian, the new conserved vectors are well constructed with a detailed derivation, which constitutes the conservation analysis for the time fractional (2+1)-dimensional ZK-MEW equation.
In this paper, the Lie group analysis is used to carry out the similarity reduction and exact solutions of the
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-dimensional modified KdV–Zakharov–Kuznetsov equation. This research deals ...with the similarity solutions of mKdV–ZK equation. The mKdV–ZK equation has been reduced into a new partial differential equation with less number of independent variables, and again using Lie group symmetry method, the new partial differential equation is reduced into an ordinary differential equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group, and similarity reduction for the mKdV–ZK equation. In addition to that, solitary wave solutions of the mKdV–ZK equation are derived from the reduction equation. Thus, we obtain some new exact explicit solutions of the
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-dimensional mKdV–ZK equation which describes the dynamics of nonlinear waves in plasmas.
In two dimensions, the problem governing a homogeneous phoretic swimmer of circular cross-section is ill-posed because of the logarithmic divergence associated with a purely diffusive solute ...transport. We address here the well-posed problem that is devised by introducing a slight inhomogeneity in the interfacial chemical activity. With the radial symmetry being perturbed, phoretic motion is animated by diffusio-osmosis. Solute advection, associated with that motion, becomes comparable to diffusion at large distances. The singular problem associated with that scale disparity is analysed using matched asymptotic expansions for arbitrary values of the Damköhler number $\textit {Da}$ and the intrinsic Péclet number $\textit {Pe}$. Asymptotic matching provides an implicit equation for the particle velocity in terms of these two parameters. The velocity exhibits a non-trivial dependence upon the sign $M$ of the slip coefficient. For $M=-1$, we observe the appearance of several solutions beyond a $\textit {Da}$-dependent critical value of $\textit {Pe}$. We also address the respective limits of small and large $\textit {Da}$ for fixed $\textit {Pe}$ and arbitrary inhomogeneity, and illuminate their linkage to the limit of weak inhomogeneity.
The recent discovery of graphene has led to many advances in two-dimensional physics and devices. The graphene devices fabricated so far have relied on SiO(2) back gating. Electrochemical top gating ...is widely used for polymer transistors, and has also been successfully applied to carbon nanotubes. Here we demonstrate a top-gated graphene transistor that is able to reach doping levels of up to 5x1013 cm-2, which is much higher than those previously reported. Such high doping levels are possible because the nanometre-thick Debye layer in the solid polymer electrolyte gate provides a much higher gate capacitance than the commonly used SiO(2) back gate, which is usually about 300 nm thick. In situ Raman measurements monitor the doping. The G peak stiffens and sharpens for both electron and hole doping, but the 2D peak shows a different response to holes and electrons. The ratio of the intensities of the G and 2D peaks shows a strong dependence on doping, making it a sensitive parameter to monitor the doping.
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