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Contact and Lie point symmetries of a certain class of second order differential equations using the Lie symmetry theory are obtained. Generators of these symmetries are used to obtain first integrals and exact solutions of the equations. This class of equations is transformed into the so-called generalized Lane-Emden equations of the second kind y ′ ′ (x) + k x y ′ (x) + g (x) e n y = 0. Then we consider two types of functions g(x) and present first integrals and exact solutions of the Lane-Emden equation for them. One of the considered cases is new. [ABSTRACT FROM AUTHOR]

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]

This work studies the behavior of electrical signals in resonant tunneling diodes through the application of the Lonngren wave equation. Utilizing the method of Lie symmetries, we have identified optimal systems and found symmetry reductions; we have also found soliton wave solutions by applying the tanh technique. The bifurcation and Galilean transformation are found to determine the model implications and convert the system into a planar dynamical system. In this experiment, the equilibrium state, sensitivity, and chaos are investigated and numerical simulations are conducted to show how the frequency and amplitude of alterations affect the system. Furthermore, local conservation rules are demonstrated in more detail to unveil the whole system of movements. [ABSTRACT FROM AUTHOR]

In this paper, the integrable (4 + 1)-dimensional Fokas equation is investigated. Exploiting a set of non-singular local multipliers, we present a set of local conservation laws for the equation. The nonlocally related partial differential equation (PDE) systems are found. Nine nonlocally related systems are discussed reveal thirty five interesting closed form solutions of the equation. These solutions contain different types of wave solutions, double soliton, multi-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. Furthermore, we apply the sine-cosine method to find other traveling wave solutions for this equation. [ABSTRACT FROM AUTHOR]

This work studies the behavior of electrical signals in resonant tunneling diodes through the application of the Lonngren wave equation. Utilizing the method of Lie symmetries, we have identified optimal systems and found symmetry reductions; we have also found soliton wave solutions by applying the tanh technique. The bifurcation and Galilean transformation are found to determine the model implications and convert the system into a planar dynamical system. In this experiment, the equilibrium state, sensitivity, and chaos are investigated and numerical simulations are conducted to show how the frequency and amplitude of alterations affect the system. Furthermore, local conservation rules are demonstrated in more detail to unveil the whole system of movements.

Lie similarity transformations for unsteady flow in a thin film under the influence of internal heating and thermal radiation have been deduced earlier. These transformations have been employed to reduce the partial differential equations of the fluid flow and heat transfer to ordinary differential equations. Fluid velocity and temperature profiles were presented by constructing homotopic solutions for the obtained system of equations. Here we apply a general linear combination of all the Lie point symmetries associated with the equations describing the flow and heat transfer with internal heating and thermal radiation, to construct the general Lie similarity transformations. By applying these transformations to the flow, we map them to ordinary differential equations. The reduced system of differential equations contains the arbitrary constants used to construct the generalised Lie similarity transformations through the linear combination of all the Lie symmetries of the flow equations. Further, we determine analytic solutions using the homotopy analysis method for the reduced system of equations. We show that the presence of arbitrary constants in the reduced system of differential equations enables control of the deduced homotopic analytic solutions, i.e., these constants serve as control parameters. We illustrate the effects of these control parameters on the temperature profiles graphically. [ABSTRACT FROM AUTHOR]

The traditional algorithm for finding Lie point symmetries of ordinary differential equations (ODEs) faces challenges when applied to first-order ODEs. This stems from the fact that for first-order ODEs, unlike higher-order ODEs, the determining equation lacks derivatives, rendering it impossible to decompose into simpler PDEs to be solved for the infinitesimals. Consequently, a common technique for determining Lie point symmetries of first-order ODEs involves making speculative assumptions about the form of the infinitesimal generator. In this study, we propose a novel and more efficient approach for finding Lie point symmetries of first-order ODEs and systems of first-order ODEs. Our method leverages the inherent connection between first-order ODEs and their corresponding second-order counterparts derived through total differentiation. By exploiting this connection, we develop a systematic algorithm for determining Lie point symmetries of a wide range of first-order ODEs. We present the algorithm and provide illustrative examples to demonstrate its effectiveness. [ABSTRACT FROM AUTHOR]

In the present paper Lie theory of differential equations is expanded for finding symmetry geometric vector fields of Poisson equation. Similarity variables extracted from symmetries are applied in order to find reduced forms of the considered equation by using Erdélyi-Kober operator. Conservation laws of the space-space-fractional generalized Poisson equation with the Riemann-Liouville derivative are investigated via Noether's method. By means of the concept of non-linear self-adjointness, Noether's operators, formal Lagrangians and conserved vectors are computed. A collocation technique is also applied to give a numerical simulation of the problem. [ABSTRACT FROM AUTHOR]

In this paper, a special integrable negative-order Calogero–Bogoyavlenskii–Schiff equation (nCBS) in fluid mechanics is studied by means of the symmetry reduction method and consistent tanh expansion method. The Painlevé integrability is investigated to confirm the compatibility conditions. This integrable nCBS equation has been transformed into different reduction equations and the corresponding invariant solutions with arbitrary functions are obtained. The corresponding structures of the invariant solutions for the nCBS equation are also shown graphically. At last, new types of soliton–cnoidal interaction solutions for the nCBS equation are presented through the consistent tanh expansion method on the basis of the truncated Painlevé expansion. [ABSTRACT FROM AUTHOR]

Heat and mass transfer in a viscous film on an unsteady stretching surface in the presence of a variable magnetic field is investigated using Lie symmetry analysis. Combination of translational and scaling Lie point symmetries is used to obtain invariants of the model. The deduced invariants provide a new generalized class of similarity transformations that have not been used before. These transformations restructure the governing problem in a system of eight-parameter nonlinear ordinary differential equations. Analytic series solutions are obtained for the resulting system of equations for different values of these parameters and are illustrated graphically. We found that coefficients of the translational symmetries do not have any effect on the solutions however, coefficients of the scaling symmetries significantly affect the variation of temperature and concentration distribution across the fluid film and help in controlling the heat and mass transfer rates. Also, increasing the value of unsteadiness and magnetic parameter decreases the film thickness and promotes a uniform temperature and concentration across film thickness. Furthermore, we found that at the lower values of Prandtl and Schmidt number, diffusion dominates the heat and mass transfer while at the higher values advection dominates the heat and mass transfer.

For GGKdV equation, solutions of the compatible KdV equation are obtained by using CKdVE method, and Lie point symmetry group of the equation is also obtained. Further, some new exact non-traveling wave solutions are obtained by using the equivalent transformation method and elliptic function method on solving the corresponding symmetric reduction equation, and local excitation modes of three kinds of solutions under three different groups of parameters are presented. Finally, the integrability in the sense of the CkdVE and the Lie Symmetric are proved, which shows the effectiveness of the organic combination of various kinds of nonlinear analytical methods. This CkdVE method communicated the mathematical relations of different nonlinear models. It is a new bridge between the known and unknown solutions of the nonlinear partial differential equations, and it is a new way to explore complex nonlinear complex phenomena.

Symmetry methods are always very useful for discussing the classes of differential equation solutions. This article focuses on traveling wave structures of the generalized Pochhammer–Chree (PHC) equation. First, we will discuss Lie point symmetries of the PHC equation to classify the solutions. Then, we formulate traveling wave structures considering the reduced differential equations (DEs) by using sech method and the new extended direct algebraic (EDA) method. In the end, we will sketch some of the traveling wave structures.

In this work, we study a general form of Burgers’ equation with two variable coefficients depending on space and time. Using symmetry analysis we determine certain coefficient functions for which the corresponding nonlinear partial differential equations have Lie point symmetries. For each such equation we construct its conservation laws by the use of conservation theorem owing to Ibragimov. The importance of conversations laws is also mentioned. Moreover, group invariant and power series solutions are obtained for some special cases of the equation under study. [ABSTRACT FROM AUTHOR]

The unidirectional propagation of long waves in certain nonlinear dispersive waves is explained by the (2 + 1) pKP equation, this equation admits infinite number of infinitesimals. We explored new Lie vectors thorough the commutative product properties. Using the Lie reduction stages and some assistant methods to solve the reduced ODEs, Exploiting a set of new solutions. Exploring a set of non-singular local multipliers; generating a set of local conservation laws for the studied equation. The nonlocally related (PDE) systems are found. Four nonlocally related systems are discussed reveal twenty-one interesting closed form solutions for this equation. We investigate new various solitons solutions as one soliton, many soliton waves move together, two and three Lump soliton solutions. Though three dimensions plots some selected solutions are plotted. [ABSTRACT FROM AUTHOR]

In these analyses, we consider the time-fractional Fisher equation in two-dimensional space. Through the use of the Riemann-Liouville derivative approach, the well-known Lie point symmetries of the utilized equation are derived. Herein, we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries. The diminutive equation's derivative is in the Erdélyi-Kober sense, whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time. The conservation laws for the dominant equation are built using a novel conservation theorem. Several graphical countenances were utilized to award a visual performance of the obtained solutions. Finally, some concluding remarks and future recommendations are utilized.

The non-local symmetries of the coupled Lakshmanan–Porsezian–Daniel (LPD) equations are obtained with the help of the known Lax pair. By introducing an auxiliary variable, the coupled LPD equations are extended to a closed prolonged system and the non-local symmetries are localised to the Lie point symmetries of the prolonged system. Furthermore, based on the Lie point symmetries of the prolonged system, the exact solutions and non-local conservation laws of the coupled LPD equations are derived. [ABSTRACT FROM AUTHOR]

The aim of this work is to explore and find new closed-form nematicon solutions for different nonlinearities which occur in nematic liquid crystals (NLC) along with proposing optical system application that utilizes NLC nonlinearities. In particular, Lie point symmetry method is employed to scrupulously inspect and acquire solutions for some interesting cases of nonlinearities which have not been fully examined in literature such as quadratic, generalized dual-power law, and eighth-order nonlinearities. A variety of different nematicon dynamics are observed, including bright solitons, dark solitons and periodic behaviors. The explicit solution form for each dynamical behavior is obtained and the solution dependence on model parameters is investigated. The proposed optical system enables the flexible realization of different types of NLC nonlinearities. To the best of our knowledge, this is the first time to attain explicit exact nontrivial solutions for the particular cases of generalized dual-power law and eighth-order nonlinearities.

In this work, the consistent tanh expansion (CTE) method is developed for a high-order classical Boussinesq- Burgers (HCBB) equation. Via the CTE method, we obtain many exact significant solutions including soliton-resonant solutions, soliton-periodic wave interactions and soliton-rational wave interactions. The CTE related nonlocal symmetries are also proposed. The nonlocal symmetries can be localized to find finite Backlund transformations by prolonging the model to an enlarged one. [ABSTRACT FROM AUTHOR]

A nonlinear two-dimensional orthotropic filtration equation with the Riemann–Liouville time-fractional derivative is considered. It is proved that this equation can admits only linear autonomous groups of point transformations. The Lie point symmetry group classification problem for the equation in question is solved with respect to coefficients of piezoconductivity. These coefficients are assumed to be functions of the square of the pressure gradient absolute value. It is proved that if the order of fractional differentiation is less than one then the considered equation with arbitrary coefficients admits a four-parameter group of point transformations in orthotropic case, and a five-parameter group in isotropic case. For the power-law piezoconductivity, the group admitted by the equation is five-parametric in orthotropic case, and six-parametric in isotropic case. Also, a special case of power function of piezoconductivity is determined for which there is an additional extension of admitted groups by the projective transformation. There is no an analogue of this case for the integer-order filtration equation. It is also shown that if the order of fractional differentiation $\alpha \in (1,2)$ then dimensions of admitted groups are incremented by one for all cases since an additional translation symmetry exists. This symmetry is corresponded to an additional particular solution of the fractional filtration equation under consideration. Using the group classification results for orthotropic case, the representations of group-invariant solutions are obtained for two-dimensional subalgebras from optimal systems of symmetry subalgebras. Examples of reduced equations obtained by the symmetry reduction technique are given, and some exact solutions of these equations are presented. It is proved that the considered time-fractional filtration equation is nonlinearly self-adjoint and therefore the corresponding conservation laws can be constructed. The components of obtained conserved vectors are given in an explicit form.

We use the method of Lie symmetry analysis to investigate the properties of a (2+1)-dimensional KdV–mKdV equation. Using the Ibragimov method, which relies only on the existence of the commutator table, we construct an optimal system of one-dimensional subalgebras of the Lie algebra and study invariant solutions and similarity reductions by considering representatives of the optimal system. To analyze some nonlocal symmetry properties, we apply the truncated Painlevé expansion method and obtain two Bäcklund transformations that are not autotransformations and one auto-Bäcklund transformation. To localize the nonlocal symmetry and obtain a local Lie point symmetry, we introduce an expanded system. Using solutions of the corresponding Cauchy problems for Lie point symmetries, we prove a theorem on a finite symmetry transformation and find the th Bäcklund transformation in terms of determinants. Based on one of the obtained Bäcklund transformations that are not autotransformations, we derive lump-type solutions. In addition, we prove the integrability of the equation by the consistent Riccati expansion method. We present explicit soliton-cnoidal wave solutions and investigate the dynamical characteristics of the obtained solutions using numerical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a type of time-fractional Fokker-Planck equation (FPE) of the Ornstein-Uhlenbeck process is solved via Riemann-Liouville and Caputo derivatives. An analytical method based on symmetry operators is used for finding reduced form and exact solutions of the equation. A numerical simulation based on the Muntz-Legendre polynomials is applied in order to find some approximated solutions of the equation. [ABSTRACT FROM AUTHOR]

In this paper, the Fujimoto-Watanabe equation is studied with the help of the classical Lie point symmetry analysis method. Infinitesimal generators, the en-tire geometric vector fields and symmetry groups of the Fujimoto-Watanabe equation are given. By using symmetry reduction method, Fujimoto-Watanabe equation is reduced to nonlinear ordinary differential equations (NODEs), which has advantage to provide analytical solutions, and the exact analytical solutions are considered by virtue of the power series method. Finally, the symmetry of the Fujimoto-Watanabe equation with method of undetermined coefficients is obtained. As application, the conservation laws are constructed. It shows the integrability and the existence of soliton solutions of the Fujimoto-Watanabe equation. [ABSTRACT FROM AUTHOR]

The evolution of a two-dimensional, pre-existing, fluid-driven permeable fracture is investigated. Fluid leak-off through the permeable fracture interface into the rock formation is modelled by a non-Darcy model — the Forchheimer model. The injected fluid is a viscous incompressible Newtonian fluid and the flow in the fracture is considered laminar. When lubrication theory and the Perkins–Kern–Nordgren approximation are employed, a system of nonlinear differential equations for the half-width of the fracture and the leak-off depth is obtained. The Lie group method is used to reduce the system of partial differential equations to a boundary value problem for a second order ordinary differential equation and to obtain group invariant and numerical solutions, based on the Forchheimer number. • We model the problem of a two-dimensional fluid-driven fracture in porous media. • Fluid leak-off into the porous rock matrix is modeled using the Forchheimer model. • Group invariant and numerical solutions for a two-dimensional fracture are derived. • Three cases which describe the strength of leak-off at the interface are considered. [ABSTRACT FROM AUTHOR]

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The conservation laws of low-dimensional partial differential equations have been employed by Lou et al. to construct various high-dimensional equations, particularly focusing on high-dimensional integrable equations. However, solving these high-dimensional equations poses significant challenges. In this paper, the (2+1)-dimensional Burgers equation is studied by means of nonlocal symmetry method for the first time. For this high-dimensional equation, we establish a link with the exact solution of the (1+1)-dimensional Burgers equation through nonlocal symmetry. Furthermore, we successfully construct multiple exact solutions for the high-dimensional Burgers equation by leveraging the exact solution of the low-dimensional counterpart. We also give the corresponding images of several solutions to study the dynamic behavior of the equations. [ABSTRACT FROM AUTHOR]

In this paper, Lie point symmetry method is used to obtain the general exact solution of the second order nonlinear ordinary differential equation which governing heat transfer in rectangular fin with variable thermal conductivity. Some new forms of thermal conductivity are introduced and the associated exact solution is obtained in each case. The general relation among the fin efficiency, thermal conductivity and thermo-geometric parameter is obtained.

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions. [ABSTRACT FROM AUTHOR]

We consider the classical Boussinesq-Burgers (BB) equation, which describes the propagation of shallow water waves. Based on the truncated painlevé expansion method and consistent Riccati expansion method, we successfully obtain its nonlocal symmetry and Bäcklund transformation. By introducing auxiliary-dependent variables for the nonlocal symmetry, we find the corresponding Lie point symmetries. By considering the consistent tanh expansion method, the interaction solution of soliton-cnoidal wave for the classical BB equation is studied by using the Jacobi elliptic function. The multi-solitary wave solutions are also obtained by introducing a linear combination of N exponential functions. Moreover, the conservation laws of the equation are successfully obtained with a detailed derivation. [ABSTRACT FROM AUTHOR]

In this article, we investigate a (3+1)-dimensional Schamel–Zakharov–Kuznetsov–Burgers (SZKB) equation, which describes the nonlinear plasma-dust ion acoustic waves (DIAWs) in a magnetised dusty plasma. With the aid of the Kudryashov method and symbolic computation, a set of new exact solutions for the SZKB equation are derived. By introducing two special functions, a variational principle of the SZKB equation is obtained. Conservation laws of the SZKB equation are obtained by two different approaches: Lie point symmetry and the multiplier method. Thus, the conservation laws here can be useful in enhancing the understanding of nonlinear propagation of small amplitude electrostatic structures in the dense, dissipative DIAWs’ magnetoplasmas. The properties of the shock wave solutions structures are analysed numerically with the system parameters. In addition, the electric field of this solution is investigated. Finally, we will study the physical meanings of solutions. [ABSTRACT FROM AUTHOR]

The paper deals with a non-minimally coupled scalar field in the background of homogeneous but anisotropic Kantowski–Sachs space–time model. The form of the coupling function of the scalar field with gravity and the potential function of the scalar field are not assumed phenomenologically, rather they are evaluated by imposing Noether symmetry to the Lagrangian of the present physical system. The physical system gets considerable mathematical simplification by a suitable transformation of the augmented variables ( a , b , ϕ ) → ( u , v , w ) and by the use of the conserved quantities due to the geometrical symmetry. Finally, cosmological solutions are evaluated and analyzed from the point of view of the present evolution of the Universe. [ABSTRACT FROM AUTHOR]

In this paper, we obtained some new exact solutions of some nonlinear KdV-like equations using Lie point symmetry and -symmetry methods. The obtained solutions are in the form of doubly periodic, bright and dark soliton solutions. [ABSTRACT FROM AUTHOR]

In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann‐Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi‐Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations. [ABSTRACT FROM AUTHOR]

We consider the generalised dispersive modified Benjamin-Bona-Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painlevé expansion method, we derive its non-local symmetry and Bäcklund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves. [ABSTRACT FROM AUTHOR]

Group invariant and numerical solutions for the evolution of a two-dimensional fracture with non-zero initial length in permeable rock and driven by a laminar incompressible Newtonian fluid are obtained. The fluid leak-off into the rock mass is modelled using Darcy law. With the aid of lubrication theory and the PKN approximation, a system of nonlinear partial differential equations for the fracture half-width and the extent of leak-off is derived. Since the fluid–rock interface is permeable the nonlinear diffusion equation contains a leak-off velocity sink term. Using the Lie point symmetries the problem is reduced to a boundary value problem for a system of second order ordinary differential equations. [ABSTRACT FROM AUTHOR]

By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov–Pinney equation. In this context, nonlinear superposition rule for second order Kummer–Schwarz equation is rediscovered. Invariance under one-dimensional symmetry group is also used to obtain first integrals (Ermakov–Lewis invariants). Our motivation is a type of equations with singular term that arises in many applications, in particular in the study of general NLS (nonlinear Schrödinger) equations. [ABSTRACT FROM AUTHOR]

The two-dimensional liquid jet of a power-law fluid is considered. The problem is formulated in terms of the components of fluid velocity which satisfy the continuity equation and the momentum boundary layer equation for a power-law fluid. The multiplier method is used to investigate the conservation laws for the system of partial differential equations and a conserved vector and the corresponding conserved quantity for the two-dimensional liquid jet is derived. The Lie point symmetries of the system of partial differential equations are calculated. A linear combination of the Lie point symmetries is associated with the conserved vector for the liquid jet to obtain the associated Lie point symmetry which is used to generate the invariant solution. An analytical solution in parametric form for the liquid jet is derived. It is found that a solution for the liquid jet exists only for 1 / 2 < n < ∞ where n is the power law exponent. The profile of the free surface and the thickness of the liquid jet are compared for a shear thinning fluid with 1 / 2 < n < 1 , a Newtonian fluid with n =1 and a shear thickening fluid with 1 < n < ∞ . [ABSTRACT FROM AUTHOR]

The nth finite transformations of the fifth-order KdV equation are obtained from the Lie point symmetry approach via localisation of nonlocal symmetries to local ones of the enlarged system. Through the obtained transformations, some periodic and soliton solutions are derived. [ABSTRACT FROM AUTHOR]

For GGKdV equation, solutions of the compatible KdV equation are obtained by using CKdVE method, and Lie point symmetry group of the equation is also obtained. Further, some new exact non-traveling wave solutions are obtained by using the equivalent transformation method and elliptic function method on solving the corresponding symmetric reduction equation, and local excitation modes of three kinds of solutions under three different groups of parameters are presented. Finally, the integrability in the sense of the CkdVE and the Lie Symmetric are proved, which shows the effectiveness of the organic combination of various kinds of nonlinear analytical methods. This CkdVE method communicated the mathematical relations of different nonlinear models. It is a new bridge between the known and unknown solutions of the nonlinear partial differential equations, and it is a new way to explore complex nonlinear complex phenomena. • A nonlinear (1+1)-dimensional generalized geophysical KdV equation with three arbitrary constants is considered. • The CKdVE method is successfully applied on the GGKdV equation, and Lie point symmetry group of the equation is obtained. • The equivalent transformation method and the elliptic function method are successfully applied. • Some new and exact non-traveling wave solutions of the GGKdV equation are obtained. [ABSTRACT FROM AUTHOR]