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We study the long-time behavior of scalar viscous conservation laws via the structure of $ \omega $-limit sets. We show that $ \omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ \omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics. [ABSTRACT FROM AUTHOR]

We give a complete description of inequivalent nontrivial local conservation laws of all orders for a natural generalization of the dissipative Westervelt equation and, in particular, show that the equation under study admits an infinite number of inequivalent nontrivial local conservation laws for the case of more than two independent variables. [ABSTRACT FROM AUTHOR]

The Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation is often used in dealing with long-wave propagation interactions, and is widely used in mathematics, physics, and engineering. This paper proposes a new extended (3+1)-dimensional KdV-CBS equation, and it’s never been studied. Additionally, we verify the integrability of the equation based on the Painlevé test. By employing Hirota’s method, a bilinear auto-Bäcklund transformation, the multiple-soliton solutions, and the soliton molecules of the equation are derived. New exact solutions of the equation are constructed utilizing the power series expansion method and (G ′ / G) -expansion method. These exact solutions are also presented graphically. Finally, the conservation laws of the equation are obtained. Our results are helpful for understanding nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018. https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983. https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023. https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023. https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022. https://doi.org/10.1007/s00205-021-01736-2). For L t q L x r suitable Leray–Hopf solutions of the d - dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure P s , which gives s = d - 2 as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity. [ABSTRACT FROM AUTHOR]

This paper explores the evolutionary behavior of some specific waves for a Generalized Hirota Bilinear (GHB) equation with applications in modern sciences. More precisely, the linear superposition principle is first adopted to construct the resonant multi-wave of the GHB equation. Through the resonant N-wave and some particular computations, positive multi-complexiton to the governing model is then extracted with the use of computational packages. Furthermore, after deriving nonclassical Lie symmetries and their corresponding invariant solutions, Conservation Laws (CLs) for the GHB equation are formally constructed based on a general method developed by Ibragimov. The propagation dynamics of resonant double- and triple-waves as well as positive single- and double-complexitons are examined in detail by considering several case studies. The present results demonstrate how adjusting the nonlinear parameter can be used to control specific waves in nonlinear systems. [ABSTRACT FROM AUTHOR]

Wave propagation in the form of fronts or kinks, a common occurrence in a wide range of physical phenomena, is studied in the context of models defined by their Hamiltonian structure. Motivated, for dispersive wave evolution equations such as a strongly nonlinear model of two‐layer internal waves in the Boussinesq limit, by the symmetric properties of a class of front‐propagating solutions, known as conjugate states or solibores, a generalized formulation based purely on the dispersionless reduction of a system is introduced, and a class of undercompressive shock solutions, here referred to as "Hamiltonian shocks," is defined. This analysis determines whether a Hamiltonian shock, representing locally a kink for the parent dispersive equations, will interact with a sufficiently smooth background wave without inducing loss of regularity, which would take the form of a classical dispersive shock for the parent equations. This property is also related to an infinitude of conservation laws, drawing a parallel to the case of completely integrable systems. [ABSTRACT FROM AUTHOR]

This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators. [ABSTRACT FROM AUTHOR]

Optical fibers are used in the communications, biological sensors and chemical sensors. We investigate a variable-coefficient modified Hirota equation for the amplification or absorption of pulses propagating in an inhomogeneous optical fiber. With respect to the complex envelope of the optical field, we construct the infinitely-many conservation laws based on the existing Lax pair. According to the existing Darboux transformation, we derive the three-soliton solutions, the higher-order breather solutions and breather-to-soliton transition condition. Amplitudes of the two solitons change after the interaction, while velocities of them are unchanged via asymptotic analysis. When $ P(z)=0 $ P (z) = 0 , interactions among the three parabolic or wavy solitons, interaction between the two parabolic or wavy or crooked breathers, and interactions among the three parabolic and wavy breathers are presented, where $ P(z) $ P (z) is related to the nonlinear focus length. Velocities of three solitons or two crooked breathers with $ P(z)\neq 0 $ P (z) ≠ 0 are different from those with $ P(z)=0 $ P (z) = 0. Based on the breather-to-soliton transition condition, when $ P(z)=0 $ P (z) = 0 , parabolic or wavy multi-peak and M-shaped solitons are presented; when $ P(z)\neq 0 $ P (z) ≠ 0 , the crooked periodic wave and anti-dark soliton are shown. [ABSTRACT FROM AUTHOR]

In this paper, we study the Camassa–Holm type equation, which has applications in mathematical physics and engineering. Its applications extend across disciplines, contributing to our understanding of complex systems and helping to develop innovative solutions in diverse areas of research. Our main aim is to construct closed-form solutions of the equation using a powerful technique, namely the Lie group analysis method. Firstly, we derive the Lie point symmetries of the equation. Thereafter, the equation is reduced to non-linear ordinary differential equations using symmetry reductions. Furthermore, the solutions of the equation are derived using the extended Jacobi elliptic function technique, the simplest equation method, and the power series method. In conclusion, we construct conservation laws for the equation using Noether's theorem and the multiplier approach, which plays a crucial role in understanding the behavior of non-linear equations, especially in physics and engineering, and these laws are derived from fundamental principles such as the conservation of mass, energy, momentum, and angular momentum. [ABSTRACT FROM AUTHOR]

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of α which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail. [ABSTRACT FROM AUTHOR]

We construct the classical interaction Lagrangian for an electric charge q and a magnetic dipole m in relative motion. In the rest frame of m the resulting force acting on q is f q = q E + c - 1 v × B + c - 1 q (v · ∇) A . Application to the Aharonov-Bohm (AB), and the equivalent Spavieri effect, indicates that the observed AB phase shift is due to the classical lag effect between interfering particles caused by the local force c - 1 q (v · ∇) A = (v · ∇) Q em with nonvanishing longitudinal component in the direction of motion and with Q em representing the gauge-invariant electromagnetic momentum. Our results confirm the validity of the same expression for f q derived in literature with an approach based on the stress-energy tensor T μ ν , Maxwell’s equations, and the momentum conservation law. Similar results apply to the force f m = - f q acting on m, indicating conservation of the action and reaction principle in the effects of AB type, which can be interpreted classically in terms of the lag effect caused by a local force. [ABSTRACT FROM AUTHOR]

The interaction between a Riemann wave propagating along the y-axis and a long wave along the x-axis results in a generalised breaking soliton (gBS) equation. Lie symmetries of the equation are generated in this article to derive some rarely available classes of invariant solutions. The presence of arbitrary functions in each solution opens up a broad class of solution profiles. 3D profiles are used to explore more properties of the solutions to the gBS equation. The profiles describe doubly solitons, annihilation of parabolic, periodic solitons, line solitons and solitons on curved surface types. Solution profiles are useful in optical fibre, acoustic waves in a crystal lattice, long waves in stratified oceans, long-distance transmission and shallow water waves. The Lie symmetry approach has future scope to provide more variety in solutions due to the capability of solutions to include functions and arbitrary constants. This research effectively demonstrates the uniqueness of the solutions when compared with the previously published result. Moreover, the adjoint equation and conserved vectors are determined using Noether’s theorem. [ABSTRACT FROM AUTHOR]

We consider the energy-minimizing configuration of the Sadowsky-type functional for narrow rectifying strips. We show that the functional is proportional to the p -Willmore functional using classical analysis techniques and the geometry of developable surfaces. We introduce hyperelastic strips (or p-elastic strips) as rectifying strips whose base curves are the critical points of the Sadowsky-type functional and find the Euler-Lagrange equations for hyperelastic strips using a variational approach. We show a naturally expected relationship between the planar stationary points of the Sadowsky-type functional and the hyperelastic curves. We derive two conservation vector fields, the internal force and torque, using Euclidean motions and obtain the first and second conservation laws for hyperelastic strips. [ABSTRACT FROM AUTHOR]

This study deals with exact solutions and soliton dynamics of Burgers equation. The Lie symmetry method under one parameter transformation is employed to establish symmetry condition, infinitesimals and their commutative relations. In consequence, the similarity variables are derived and cause to first symmetry reduction. A twice employment of method allows further symmetry reductions of test equations and results to systems of ODEs. Then, the ODEs have been integrated under parametric constraints and evolve desired exact solutions. These solutions consist of all the functions f 1 (t) , f 2 (t) , g 1 (y) , g 2 (y) existed in infinitesimals, more arbitrary functions F(X), H(X) and several arbitrary constants that make obtained results generalize than previous findings. To analyze the nature of physical phenomena associated with Burgers equation, these solutions have been supplemented with numerical simulation. Consequently, single soliton, doubly soliton, multisoliton, asymptotic nature, soliton fusion and fission nature are discussed systematically. Moreover, the Lagrangian formulation and conserved vectors associated with Lie symmetries are obtained. [ABSTRACT FROM AUTHOR]

We consider the energy-minimizing configuration of the Sadowsky-type functional for narrow rectifying strips. We show that the functional is proportional to the $ p $-Willmore functional using classical analysis techniques and the geometry of developable surfaces. We introduce hyperelastic strips (or p-elastic strips) as rectifying strips whose base curves are the critical points of the Sadowsky-type functional and find the Euler-Lagrange equations for hyperelastic strips using a variational approach. We show a naturally expected relationship between the planar stationary points of the Sadowsky-type functional and the hyperelastic curves. We derive two conservation vector fields, the internal force and torque, using Euclidean motions and obtain the first and second conservation laws for hyperelastic strips.

A preliminary group classification based on symmetry operators is applied to study invariance properties of the time-fractional $(\alpha +2)$(α+2)-dimensional fractional evolution equation. The concepts of Riemann–Liouville and Caputo fractional derivatives are used in this study. The similarity variables obtained from symmetries and one-dimensional optimal systems of constructed Lie algebras are used in order to find the group-invariant solutions of the equation. Finally conservation laws of the equation are derived via a modified version of Noether’s theorem. [ABSTRACT FROM AUTHOR]

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation Dtαu = aupu xx + buqu x2 is studied by the symmetry analysis method. Liu et al. obtained two group generators X1 = ∂ ∂x, X2 = 2t α ∂ ∂t + x ∂ ∂x and only one trivial solution u(t,x) = f(t) = C0 Γ(α)tα−1 for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case p = q + 1, we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically. [ABSTRACT FROM AUTHOR]

We are concerned with the Riemann problem for the isentropic Euler equations of mixed type in the dark energy fluid. This system is non-strictly hyperbolic on the boundary curve of elliptic and hyperbolic regions. We obtain the unique admissible shock waves by utilizing the viscosity criterion. Assuming fixed left states are in the elliptic and hyperbolic regions, respectively, we construct the unique Riemann solution for the mixed-type models with the initial right state in some feasible regions. Finally, we present numerical simulations which are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

Summary: The paper introduces a computational framework using a novel Arbitrary Lagrangian Eulerian (ALE) formalism in the form of a system of first‐order conservation laws. In addition to the usual material and spatial configurations, an additional referential (intrinsic) configuration is introduced in order to disassociate material particles from mesh positions. Using isothermal hyperelasticity as a starting point, mass, linear momentum and total energy conservation equations are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first‐order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex‐based Finite Volume method is employed and suitably adapted. To guarantee stability from both the continuum and the semi‐discretisation standpoints, an appropriate numerical interface flux (by means of the Rankine–Hugoniot jump conditions) is carefully designed and presented. Stability is demonstrated via the use of the time variation of the Hamiltonian of the system, seeking to ensure the positive production of numerical entropy. A range of three dimensional benchmark problems will be presented in order to demonstrate the robustness and reliability of the framework. Examples will be restricted to the case of isothermal reversible elasticity to demonstrate the potential of the new formulation. [ABSTRACT FROM AUTHOR]

· Demonstrated theoretical inconsistency in recent three-dimensional magnetohydrodynamical lattice Boltzmann models. · Verified Lorentz force model requirements to simultaneously recover the Maxwell stress tensor and mass conservation. · Continuity violation impacts density, velocity, and field variations in pressure-driven and closed channel flows. · Incorrect Maxwell stress tensor impacts velocities and induced fields in flows with magnetic pressure gradients. · A theoretically consistent and easily implementable model eliminates spurious errors in a broad range of flows. The lattice Boltzmann method (LBM) for magnetohydrodynamics (MHD) was first developed for two-dimensional (2D) flows. The magnetic component of the Lorentz force is obtained from the divergence of the Maxwell stress tensor, which is recovered by an appropriate modification to the LBM equilibrium distribution function for the solution of the velocity field. Although this method has subsequently been applied to three-dimensional (3D) flows, we show that several common modifications of the 3D equilibrium distribution function cannot simultaneously recover the Maxwell stress tensor and conserve mass exactly. Our study brings attention to this theoretical issue since such inconsistent models have been used in recent 3D LBM MHD simulations. Both requirements can be simultaneously satisfied by incorporating an additional model term to the 3D equilibrium distribution function. Three variants of Hunt's flow in a rectangular channel are considered to illustrate the undesirable consequences of using inconsistent models: 1) flow driven by a pressure difference, 2) flow driven by an external body force, and 3) inclusion of a streamwise gradient in magnetic pressure. While spurious errors were observed using inconsistent models due to the violation of mass conservation or inaccuracy in the Maxwell stress tensor, the consistent model always achieves the correct solutions in these cases and is recommended for 3D MHD simulations. [ABSTRACT FROM AUTHOR]

This article utilizes the Lie symmetry method to analyze the (n + 1) -dimensional modified Zakharov–Kuznetsov (mZK) equation, which characterizes weakly nonlinear traveling waves in plasma with a constant magnetic field, comprising cold ions and hot isothermal electrons. The model is also applicable to dusty and magnetized plasma. Lie point symmetries and associated group invariant solutions are computed using Lie group theory, with the underlying equation. The improved tan ( Φ (ξ) 2) -expansion method is then employed to derive soliton solutions, including hyperbolic, trigonometric, rational, and exponential forms. Graphic interpretations of specific solutions are provided. Nonlinear self-adjointness is used to compute non-local conservation laws in lower dimensions, and conservation laws are developed based on the equation's formal Lagrangian structure, applying Ibragimov's theorem. These findings highlight the novelty and reliability of the methodology employed. [ABSTRACT FROM AUTHOR]

Considered herein is the orbital stability of the periodic peaked solitons for the higher-order modified Camassa–Holm equation. This equation can be viewed as a natural higher-order generalization of the modified Camassa–Holm equation, and admits a single peaked soliton and multi-peakons. We first show that the equation possesses the periodic peakons. Furthermore, it is proved that the periodic peakons are dynamically stable under small perturbations in the energy space by utilizing the inequalities with the maximum and minimum of the solutions related to the first two conservation laws. [ABSTRACT FROM AUTHOR]

For the disciplinary training of the undergraduate student in Physics at the Francisco José de Caldas District University, it is essential to understand and analyze the concepts of collision and disintegration of relativistic particles and the conservation laws that govern them, these are immersed within the area of modern physics and quantum mechanics, mandatory core subjects belonging to the 7th and 8th semester curriculum, the aforementioned contents are challenging due to their abstract and complex nature, therefore the need emerges to provide tools that energize and strengthen the interpretation, analysis and therefore the memory fixation of said concepts, which with visual and interactive aids such as Virtual Learning Objects (OVA) provide greater flexibility in teaching and learning. For this, a virtual learning object is designed using the ADDIE instructional model: (Analysis, Design, Development, Implementation and Evaluation) which has a constructivist model that shows the interactions of collisions and decay between particles constituted by the first generations of the model. standard and that give rise to the SU(3) symmetries, the hadrons; Taking bubble chamber experiments as a starting point as a tool that helps solve this abstract nature, this work contemplates the first two stages of the ADDIE model. [ABSTRACT FROM AUTHOR]

We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ > 0 . This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities. [ABSTRACT FROM AUTHOR]

In this research, the (2+1)‐dimensional normal biological population model, incorporating the Verhulst law for population growth, is employed to explore species population dynamics. Employing Lie symmetry analysis, we address a nonlinear degenerate parabolic partial differential equation, yielding much‐improved results. This analysis includes computing one‐dimensional optimal subalgebras, reduced ordinary differential equations, and obtaining invariant solutions with a visual depiction of the physical behavior of the Verhulst biological population model through symmetry group transformations. Additionally, the multiplier method leads to novel conservation laws and potential systems not locally connected to the governing partial differential equation (PDE). These findings have significant implications for understanding and controlling biological populations, offering insights for applications in ecology and the environment. [ABSTRACT FROM AUTHOR]

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modeled using non-linear partial differential equations whose main ingredients are the adsorption isotherms , which are non-linear functions modeling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (2018) [3] to other adsorption isotherms. In particular, we propose a family of generalized Langmuir -type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with Tóth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We prove that if the middle components of the non-stationary solutions of the Maxwell equations verify a Morawetz-type estimate supported on a compact region in space around the trapped surface, then the components of the Maxwell fields decay uniformly in the entire exterior of the Schwarzschild black hole, including the event horizon. This is shown by making only use of Sobolev inequalities combined with energy estimates using the Maxwell equations directly. The proof does not pass through the scalar wave equation on the Schwarzschild black hole, does not need to decouple the middle components for the Maxwell fields, and would be in particular useful for the non-abelian case of the Yang–Mills equations where the decoupling of the middle components cannot occur. In fact, the estimates for the hereby argument are still valid for the Yang–Mills fields except for the Lie derivatives of the fields that are involved in the proof. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with anti-cubic and cubic-quintic-septic nonlinearities is considered. A series of direct transformations is performed for the traveling wave reduction of the original equation. The implicit function method is used for nonlinear ordinary differential equation with some constraints on the parameters. Some analytical solutions are found in the form of periodic and solitary waves. Three conservation laws, corresponding to the generalized nonlinear Schrödinger equation are obtained and conserved quantities corresponding to the solution in implicit form are found [ABSTRACT FROM AUTHOR]

A set of arbitrarily high-order WENO schemes for reconstructions on nonuniform grids is presented. These non-linear interpolation methods use simple smoothness indicators with a linear cost with respect to the order, making them easy to implement and computationally efficient. The theoretical analysis to verify the accuracy and the essentially non-oscillatory properties are presented together with some numerical experiments involving algebraic problems in order to validate them. Also, these general schemes are applied for the solution of conservation laws and hyperbolic systems in the context of finite volume methods. [ABSTRACT FROM AUTHOR]

The reaction diffusion equation arises in physical situations in problems from population growth, genetics and physical sciences. In many practical situations, the physical domain of the problem is adequately described in cylindrical Coordinates. Therefore, we consider the Fisher equation in cylindrical coordinates. We consider the generalised Fisher equation in cylindrical coordinates from Lie theory stand point. An invariance method is performed and the optimal set of nonequivalent symmetries is obtained. Finally, the conservation laws are constructed using 'multiplier method'. We determine multipliers as functions of the dependent and independent variables only. The conservation laws are computed and presented in terms of conserved vector corresponding to each multiplier. [ABSTRACT FROM AUTHOR]

We apply the basic Lie symmetry method to investigate the time-dependent negative-order Calogero-Bogoyavlenskii-Schiff (vnCBS) equation. In this case, the symmetry classification problem is answered. We obtain symmetry algebra and create the optimal system of Lie sub- algebras. We obtain the symmetry reductions and invariant solutions of the considered equation using these vector fields. Finally, we determine the conservation laws of the vnCBS equation via the Bluman-Anco ho- motopy formula [ABSTRACT FROM AUTHOR]

Conservation laws with two-parameter hysteretic fluxes are introduced. The total variation of the flux of a viscous solution does not increase as time progresses. The maximum principle for the flux is also established. Viscous shock waves are identified, and solutions to the Riemann problem are obtained. The one with the least total variation of flux is preferred. A definition of weak solutions for the hysteretic flow model, a non-conservative hyperbolic system, is provided. Approximate solutions of the system are constructed using the front tracking method with randomly adjusted discretization locations. These approximate solutions have uniformly bounded total variation, demonstrating their compactness. [ABSTRACT FROM AUTHOR]

Carbonated water flooding (CWI) increases oil production due to favorable dissolution effects and viscosity reduction. Accurate modeling of CWI performance requires a simulator with the ability to capture the true physics of such process. In this study, compositional modeling coupled with surface complexation modeling (SCM) are done, allowing a unified study of the influence in oil recovery of reduction of salt concentration in water. The compositional model consists of the conservation equations of total carbon, hydrogen, oxygen, chloride and decane. The coefficients of such equations are obtained from the equilibrium partition of chemical species that are soluble both in oleic and the aqueous phases. SCM is done by using the PHREEQC program, which determines concentration of the master species. Estimation of the wettability as a function of the Total Bound Product (TBP) that takes into account the concentration of the complexes in the aqueous, oleic phases and in the rock walls is performed. We solve analytically and numerically these equations in 1 - D in order to elucidate the effects of the injection of low salinity carbonated water into a reservoir containing oil equilibrated with high salinity carbonated water. [ABSTRACT FROM AUTHOR]

We investigate variational, Lie and Killing symmetries of the Lagrangian of an essential class of four-dimensional (pseudo-)Riemannian manifolds, i.e., non-reductive homogeneous spaces. Moreover, a Lie algebra analysis is shown. With the help of a result of the Noether's theorem, we have presented the expressions for conservation laws corresponding to all variational symmetries. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

This paper aims to study a generalized extended (2 + 1) -dimensional Kadomstev–Petviashvili (KP) equation. The KP equation models several physical phenomena such as shallow water waves with weakly nonlinear restoring forces. We will use a variety of wave ansatz methods so as to extract bright, singular, shock waves also referred to as dark or topological or kink soliton solutions. In addition to soliton solutions, we will also derive periodic wave solutions and other analytical solutions based on the invariance surface condition. Moreover, we will establish the multiplier method to derive low-order conservation laws. In order to have a better understanding of the results, graphical structures of the derived solutions will be discussed in detail based on some selected appropriate parametric values in 2-dimensions, 3-dimensions and contour plots. The findings can well mimic complex waves and their underlying properties in fluids. [ABSTRACT FROM AUTHOR]

In this study, we consider model equations K(m,n) with fractional Caputo time derivatives. By applying the Lie group symmetry method, we determine all symmetries for these equations and present the reduced symmetric equations for the equation K(m,n) with fractional Caputo time derivatives. Furthermore, we obtain the exact solution for K(1,1) with the fractional Caputo time derivative and provide graphs depicting the behavior at different orders of the fractional time derivative. Additionally, by considering the symmetries of the equation, we establish the conservation laws for K(m,m) with the fractional Caputo time derivative. [ABSTRACT FROM AUTHOR]

For a certain natural generalization of the Infeld—Rowlands equation we prove nonexistence of nontrivial local Hamiltonian structures and nontrivial local symplectic structures of any order, as well as of nontrivial local Noether and nontrivial local inverse Noether operators of any order, and exhaustively characterize all cases when the equation in question admits nontrivial local conservation laws of any order; the method of establishing the above nonexistence results can be readily applied to many other PDEs. [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional time-fractional chiral non-linear Schrödinger equation in physical media is considered in this paper. At the outset, the Lie group analysis is applied to build a set of infinitesimal generators for this equation with the aid of the Riemann–Liouville fractional derivatives. Consequently, the reduction for the considered equation into an ordinary differential equation of fractional order is obtained by using these generators and the ErdLélyi–Kober fractional operator. As a result of this reduction, we use power series analysis to get an analytical solution provided by a convergence analysis of the obtained solution. Furthermore, we construct a numerical solution based on hyperbolic functions using the fractional reduced differential transform method in the sense of Caputo fractional derivatives. Also, we detect absolute errors by performing a comparison between the exact and numerical solutions of the equation under study, while investigating the effect of fractional order α on the numerical solution. Finally, conservation laws are derived using the the formal Lagrangian and new conservation theorem. [ABSTRACT FROM AUTHOR]

This study covers the traveling wave analysis of the two-mode Cahn–Allen (TMCA) equation. The TMCA model plays the role of transmitting information into two different locations and preserving its physical properties. A wave transformation converts the considered model into an ordinary differential equation (ODE). By applying the two analytical techniques, the extended tanh - coth technique and the extended G ′ G 2 -expansion scheme, we have obtained a new type of the wave structures like singular, kink and periodic. To present the significance of the acquired outcomes, we have shown the graphical behaviors of the results by taking some suitable values of the involved parameters. The acquired ODE can be transformed into a dynamical system by applying the Galilean transformation. To examine the sensitive behavior, the dynamical system is expressed in a non-autonomous form, and the effects are analyzed by considering various initial conditions. The considered model admits the two-dimensional Lie algebra. The nonlinear self-adjointness of the considered model is checked using a new conservation theorem. The results show that the considered model is not self-adjoint and is made self-adjoint by computing a new dependent variable. Finally, the conserved quantities corresponding to each symmetry generator are computed. [ABSTRACT FROM AUTHOR]

Following the solution formula method given in Dong et al. (High order discontinuities decomposition entropy condition schemes for Euler equations. CFD J. 2002;10(4): 448–457), this article studies a type of one‐step fully‐discrete scheme, and constructs a third‐order scheme which is written into a compact form via a new limiter. The highlights of this study and advantages of new third‐order scheme are as follows: ① We proposed a very simple new methodology of constructing one‐step, consistent high‐order and non‐oscillation schemes that do not rely on Runge–Kutta method; ② We systematically studied new scheme's theoretical problems about entropy conditions, error analysis, and non‐oscillation conditions; ③ The new scheme achieves exact solution in linear cases and performing better in nonlinear cases when CFL → 1; ④ The new scheme is third order but high resolution with excellent shock‐capturing capacity which is comparable to fifth order WENO scheme; ⑤ CPU time of new scheme is only a quarter of WENO5 + RK3 under same computing condition; ⑥ For engineering applications, the new scheme is extended to multi‐dimensional Euler equations under curvilinear coordinates. Numerical experiments contain 1D scalar equation, 1D,2D,3D Euler equations. Accuracy tests are carried out using 1D linear scalar equation, 1D Burgers equation and 2D Euler equations and two sonic point tests are carried out to show the effect of entropy condition linearization. All tests are compared with results of WENO5 and finally indicate EC3 is cheaper in computational expense. [ABSTRACT FROM AUTHOR]

If a system of differential equations admits a continuous transformation group, then, in some cases, the system can be represented as a combination of two systems of differential equations. These systems, as a rule, are of smaller order than the original one. This information pertains to the linear equations of elasticity theory. The first system is automorphic and is characterized by the fact that all of its solutions are obtained from a single solution using transformations in this group. The second system is resolving, with its solutions passing into themselves under the group action. The resolving system carries basic information about the original system. The present paper studies the automorphic and resolving systems of two- and three-dimensional time-invariant elasticity equations, which are systems of first-order differential equations. We have constructed infinite series of conservation laws for the resolving systems and automorphic systems. There exist infinitely many such laws, since the systems of elasticity equations under consideration are linear. Infinite series of linear conservation laws with respect to the first derivatives are constructed in this article. It is these laws that permit solving the first boundary value problem for the equations of elasticity theory in the two- and three-dimensional cases. The solutions are constructed by quadratures, which are calculated along the boundary of the studied domains. [ABSTRACT FROM AUTHOR]

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

This paper presents a Lie group analysis of the Schamel Burger’s equation, notable for producing shock-type traveling waves in distinctive physical contexts. We determine the infinitesimal generators for this equation using the Lie group theory of differential equations. By applying Lie point symmetries, we establish commutation relations, the adjoint representation, and identify the optimal system of sub-algebras. Using elements from this optimal system, we perform symmetry reductions, resulting in various nonlinear ordinary differential equations (ODEs). Some of these reductions yield exact explicit solutions, while others necessitate the use of the new auxiliary equation method to obtain optical soliton solutions. We illustrate the dynamics of these soliton solutions graphically through both two and three-dimensional representations of wave structures. Additionally, we compute the conservation laws for the Schamel Burger’s equation by applying Ibragimov’s theorem, deriving conserved quantities corresponding to its point Lie symmetries. This analysis underscores our novel contribution, offering insights not previously explored in the literature.

This paper examines a one-dimensional (1D) model that appears in arterial blood flow. The mathematical model for blood flow via arteries is similar to that of unstable incompressible flows in thin-walled collapsible tubes. We present the Riemann invariants of the suggested model, which is one of the fundamental components of this work. For numerical modeling of blood flow model, we present a nonhomogeneous Riemann solver (NHRS) technique. Next, we demonstrate the simulation of how pressure, velocity, and cross section area waveforms propagate through arteries. Specifically, we present numerical test cases with various initial data sets. In addition, we compare the NHRS scheme to the classic Rusanov, Lax–Friedrichs, and Roe schemes. Theoretical models for thin-walled collapsible tubes are applicable to a wide range of physiological events and may be used to build clinical devices for actual biomedical science. The NHRS method’s accuracy and efficiency are demonstrated by the numerical tests.

This paper aims to derive analytical solutions of an extended (2+1)-dimensional constant coefficients new coupled Korteweg–de Vries system. This will be achieved by implementing the classical symmetry method in conjunction with some simplest equation methods. The simplest equations that will be utilised includes among others the Bernoulli and Riccati equations. Furthermore, the conservation laws will be constructed through the multipliers approach, which subsequently reveals the conserved quantities. Moreover, a brief presentation of results obtained consisting of a variety of profile structures which include the kink type, bell and inverted bell shaped and singular wave solutions will be discussed.

We study various classes of the nonlinear dynamics of some high order parabolic equations like the Oskolkov–Benjamin–Bona–Mahony–Burgers and Benjamin–Bona–Mahony–Peregrine–Burger equations that arise in the study of some wave phenomena. Also, a broader class of partial differential equations are used in modelling ocean waves that originate from the Ostrovsky equation. We study the invariance properties via the Lie invariance method for the nonlinear systems and further establish classes of conservation laws for models arises in this study. We show how the relationship leads to double reductions of the systems. This relationship is determined by a recent result involving multipliers that lead to a total divergence or closed form of the differential equation under investigation.