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This paper aims to study the modified Gardner (mG) equation that was proposed in the literature a short while ago. We first construct conserved vectors of the mG equation by invoking three different techniques; namely the method of multiplier, Noether’s theorem, and the conservation theorem owing to Ibragimov. Thereafter, we present exact solutions to the mG equation by invoking a complete discrimination system for the fifth degree polynomial. Finally, we simulate collisions of solitons for the mG equation.

Higher-order nonlinear wave models have been a source of attraction to a huge number of researchers in recent times as a result of their significance in mathematical physics, other nonlinear sciences as well as engineering. In consequence, we outline in this paper the analytical studies entrenched on a generalized structure of a nonlinear breaking soliton equation with higher-order nonlinearity in four variables which have applications in science as well as engineering. Lie group theory is utilized to generate an 11-dimensional Lie algebra associated with the equation under consideration and in addition one parametric group of transformations related to the algebra is calculated. Besides, the technique is further invoked in performing reductions of the various subalgebras of the understudy model. Moreover, in conjunction with the theory, the direct integration technique is engaged to secure an analytic solution of the equation and as a result, a general analytic solution with regard to the second-kind elliptic-integral function is furnished. Moreover, we engage the novel simplest equation technique to gain more general solutions to the equation. In consequence, solitonic solutions comprising periodic, dark-bright, topological kink as well as singular solutions are achieved. We complemented that by exhibiting the dynamics of the secured solutions with the aid of graphical representations. In conclusion, we calculate conserved quantities associated with the aforementioned equation by invoking the well-celebrated classical Noether theorem technique.

In this study, we investigate the travelling wave solutions of the (2+1)-dimensional new generalized Korteweg–de Vries equation by employing Lie group analysis along with various techniques which include direct integration, simplest equation method and Kudryashov’s method. The results obtained consists of periodic, kink, soliton and hyperbolic solutions. The symbolic computation software Maple is used to check the accuracy of all solutions obtained. Finally, 3D, density, and 2D plots of the derived solution are displayed to show the physical appearance of the model. Furthermore, we utilize the general multiplier technique and Ibragimov’s method to derive its conserved vectors. Conservation of energy and momentum, amongst others were found. Conservation laws have many significant uses with regards to integrability, linearization and analysis of solutions.

Solow Growth Model (SGM) is an economic model that is exogenous in nature and examines the relationship between the output and input levels in an economy over a period of time. It projects long-term economic growth in relation to labour (population growth), savings rate, and technological development. However, traditional approaches to solving the Solow growth model may rely on complex mathematical techniques that might not give an accurate representation of real-world economic dynamics. Thus, this paper applies the Natural Decomposition Method (NDM) to the Solow growth financial model. The NDM is a numerical technique that combines the Natural Transform (NT) and the ADM-Adomian decomposition method. The NDM simplifies problem-solving by converting the original differential equations into algebraic equations as regards limitations associated with nonlinear models. From the results obtained by applying the NDM to the Solow growth financial model, researchers and policymakers can better understand the interplay between financial variables, such as savings rates, investment, and capital allocation, and their impact on economic growth dynamics, as a systematic approach to capturing the complex relationship between finance and economic development within the Solow framework is ensured. Further research and application of the NDM can contribute to advancing the knowledge of economic dynamics and support evidence-based decision-making in economic policy.

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

This work delves into the analysis of the Gilson–Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behaviour of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

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.

This article investigates a new generalized two-dimensional nonlinear wave equation of engineering physics with various applications in the fields of sciences and engineering. In this study, shock wave and solitary wave solutions were secured via the sine–Gordon technique. Moreover, various new group invariants along-side exact classical results of the equation were achieved through the utilization of Lie group theoretic techniques. Some of the solutions are gained with regards to Weierstrass functions, complex soliton, topological kink soliton as well as singular soliton. Besides, several algebraic and other solitary-wave-type solutions are obtained. Wave dynamics of the solutions are plotted to give more physical meanings to the obtained results and have a better knowledge of what the nonlinear wave equation represents in terms of physical phenomena. Application of the secured results in engineering (signal processing), physics (electromagnetism) and complex media are presented.

Deposit insurance is a mechanism by which financial institutions are stabilized. The danger of a bank’s inability to meet its consumer commitments due to its suspended license is insured through deposit insurance practices. A flat-rate insurance scheme would contribute to moral hazard and a financial panic when banks indulge in dangerous practices. Hence, a reliable model with an explicit solution is required. This paper considers a risk rate model for deposit insurance engendered by the classical Black Scholes Option Pricing Model. The solutions are obtained via the application of Banach Contraction Mapping or Method. The procedures involved are straightforward, easy, and flexible, even without giving up accuracy. The desired explicit solutions are obtained with less computational time.

This paper presents analytical studies carried out explicitly on a higher-dimensional generalized Zakharov–Kuznetsov equation with dual power-law nonlinearity arising in engineering and nonlinear science. We obtain analytic solutions for the underlying equation via Lie group approach as well as direct integration method. Moreover, we engage the extended Jacobi elliptic cosine and sine amplitude functions expansion technique to seek more exact travelling wave solutions of the equation for some particular cases. Consequently, we secure, singular and nonsingular (periodic) soliton solutions, cnoidal, snoidal as well as dnoidal wave solutions. Besides, we depict the dynamics of the solutions using suitable graphs. The application of obtained results in various fields of sciences and engineering are presented. In conclusion, we construct conserved currents of the aforementioned equation via Noether’s theorem (with Helmholtz criteria) and standard multiplier technique through the homotopy formula.

Nonlinear partial differential equations emerge in an extensive variants of physical problems inclusive of fluid dynamics, solid mechanics, plasma physics, quantum field theory as well as mathematics and engineering. It has also been noticed that systems of nonlinear partial differential equations arise in biological and chemical applications. This article presents the analytical investigation of a completely generalized (3 + 1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation which has applications in the fields of engineering and physics. The generalized version of the potential Yu-Toda-Sasa-Fukuyama equation is more comprehensively studied in this paper compared to other research work previously done on the equation, with various new solutions of interests achieved. The theory of Lie group is applied to the nonlinear partial differential equation to basically reduce the equation to an integrable form which consequently allows for direct integration of the result. The rigorous process involved in performing a comprehensive reduction of the model under consideration using its Lie algebra makes it possible to achieve various nontrivial solutions. Besides, more general solutions are found via a well-known standard technique. In consequence, we secured diverse solitons and solutions of great interest including topological kink solitons, singular solitons, algebraic functions, exponential function, rational function, Weierstrass function, Jacobi elliptic function as well as series solutions of the underlying equation. Moreover, the completeness of the result was ascertained by presenting the solutions graphically. In addition, discussions of the pictorial representations of the results are done. Conclusively, we constructed conserved quantities of the underlying equation via both the variational and non-variational approaches using the classical Noether’s theorem as well as the standard multiplier technique respectively. In addition, some pertinent observations made from the secured results via both techniques are analyzed.

This work presents a generalized hyper-elastic rod wave (gHRW) equation from the Lie symmetry method’s standpoint. The equation illustrates dispersive waves generating in hyper-elastic rods. Using multiplier approach we find conserved vectors of the underlying equation. We subsequently obtain first integrals of the conserved vectors under the time–space group invariant u(t,x)=H(x−νt). Finally, by analysing various attainable instances of the arbitrary coefficient function g(u), we perform symmetry reductions of gHRW equation to lower order ordinary differential equations and in some instances obtain analytic solutions for special values of arbitrary constants.

In physics as well as mathematics, nonlinear partial differential equations are known as veritable tools in describing many diverse physical systems, ranging from gravitation, mechanics, fluid dynamics to plasma physics. In consequence, we analytically examine a two-dimensional generalized Bogoyavlensky-Konopelchenko equation in plasma physics in this paper. Firstly, the technique of Lie symmetry analysis of differential equations is used to find its symmetries and perform symmetry reductions to obtain ordinary differential equations which are solved to secure possible analytic solutions of the underlying equation. Then we use Kudryashov's and (G′/G)-expansion methods to acquire analytic solutions of the equation. As a result, solutions found in the process include exponential, elliptic, algebraic, hyperbolic and trigonometric functions which are highly important due to their various applications in mathematic and theoretical physics. Moreover, the obtained solutions are represented in diagrams. Conclusively, we construct conservation laws of the underlying equation through the use of multiplier approach. We state here that the results secured for the equation understudy are new and highly useful.

Eddy current damper (ECD), a contactless energy-dissipating device, is applying to control the vibration induced by earthquake and strong wind in civil structures. Combining with motion magnification mechanisms improves the damping effect of the ECD while significantly strengthens its nonlinearity. The response of single degree of freedom and multi-degree of freedom system with ECDs under a stationary stochastic earthquake characterized by the power spectral density function is evaluated using the stochastic linearization technique and expressions of the equivalent linear damping coefficient based on force criterion and energy criterion have been found, respectively. Comparisons with results obtained by Monte Carlo simulations confirm that for the nonlinearity of eddy current dampers the force-based criterion stochastic linearization technique gives accurate estimation.

In this work, we examine a nonlinear partial differential equation of fluid mechanics, namely, the generalized nonlinear advection–diffusion equation, which portrays the motion of buoyancy driven plume in a bent-on porous medium. Firstly, we classify all (point) symmetries of the equation, which prompt three cases of n. Next, for each case, we construct an optimal system of one-dimensional subalgebras and use them to perform symmetry reductions and symmetry invariant solutions. In a bid to explain the physical significance of some invariant solutions secured, we present a graphic display of some solutions in 3D, 2D as well as density plots via the exploitation of numerical simulations. Besides, we categorically state here that the results obtained in this study are new when compared with the outcomes previously achieved by Loubens et al., 2011 Quart. Appl. Math. 69 389–401. Interestingly, kink shape soliton, dark soliton, singular soliton together with exponential function solution wave profiles are displayed to make this work more valuable. Furthermore, we determine the conserved vectors in two different ways: engaging the general multiplier approach and Ibragimov’s conservation law theorem. Finally, we provide the physical meaning of these conservation laws.

This article studies a (2+1)–dimensional first extended Calogero-Bogoyavlenskii-Schiff equation, which was recently introduced in the literature. We derive Lie symmetries of this equation and then use them to perform symmetry reductions. Using translation symmetries, a fourth-order ordinary differential equation is obtained which is then solved with the aid of Kudryashov and (G′/G)−expansion techniques to construct closed-form solutions. Besides, we depict the solutions with the appropriate graphical representations. Moreover, conserved vectors of this equation are computed by engaging the multiplier approach as well as Noether's theorem.

Introduction: The purpose of this paper is to study, a (1 + 1)-dimensional generalised coupled modified Korteweg-de Vries-type system from Lie group analysis point of view. This system is studied in the literature for the first time. The authors found this system to be interesting since it is non-decouplable and possesses higher generalised symmetries. Objectives: We look for the closed-form solutions and conservation laws of the system. Methods: Optimal system of one-dimensional subalgebras for the system was obtained and then used to perform symmetry reductions and construct group invariant solutions. Power series solutions for the system were also obtained. The system has no variational principle and as such, we employed the multiplier method and used a homotopy integral formula to derive the conserved quantities. Results: Group invariant solutions and power series solutions were constructed and three conserved vectors for the system were derived. Conclusion: The paper studies the (1 + 1)-dimensional generalised coupled modified Korteweg-de Vries-type system for the first time and constructs its exact solutions and conservation laws.

We study the derivative nonlinear Schrödinger equation which has several applications, such as the propagation of circular polarized nonlinear Alfvén waves in plasmas. We present general and special solutions of this equation using first integrals. Classical Lie group theory along with power series method is also applied to obtain exact analytical solutions of this equation. Finally, conservation laws of the underlying equation are constructed through the use of the multiplier method.

This paper studies the third-order nonlinear deformed equation with variable coefficients, which is a generalization of CH, DP and DGH equations. Firstly, we investigate its conservation laws for different values of the three variable coefficients in the equation. Thereafter, we find principal Lie algebra of the equation and then find the coefficients that extends the principal Lie algebra. Finally, we present one symmetry reduction case.

In a suspension of tangent hyperbolic bionanofluid keeping both nanoparticles and motile microorganisms, the thermobioconvective boundary layer flow was studied through an exponentially stretching surface utilizing response surface methodology (RSM). The constructed model of a tangent hyperbolic nanofluid in boundary layer flow is studied with implications of thermophoresis and Brownian motion. Condition of zero normal flux of nanomaterials is added at the surface to scatter the nanomaterials from the plate surface. The rate of heat transfer is analyzed using convective boundary condition. Numerical shooting strategy with Runge-Kutta scheme is to follow intently behind the similarity transformation to solve the system of governing equations. It is assumed that the output variables of interest are dependent on the governing input parameters. The sensitivity analysis is additionally introduced. It is discovered that the sensitivity of local Nusselt number increments by expanding Lewis and thermophoresis number while the highest non-dimensional Nusselt number appears close to the significant level for the thermophoresis and low level for the Brownian motion variable. Additionally, it is demonstrated that the average maximum mean thickness of motile microorganism appears at the highest level of Brownian motion and thermophoresis number and thermophoresis and Lewis numbers. The results would provide initial guidance for potential manufacture of devices.

In this article, significance of inclined magnetohydrodynamic flow of an upper-convected Maxwell (UCM) liquid along a penetrable stretched plate is explored. Heat transfer phenomenon is studied with heat generation and absorption effect. We utilize Lie group methodology for adaptation of non-linear differential equations and calculate its absolute invariants explicitly. A numerical method, namely the fourth-order Runge Kutta algorithm with shooting technique is used to solve non-linear ordinary differential equations along with boundary conditions. Velocity profiles for flow of Newtonian and UCM liquid are compared. Plots reflecting effect of inclined and non-inclined MHD for the variations of various parameters are explored and examined. It is observed that oposite behaviour occurs corresponding to heat absorption and generation parameter for both cases of inclination and non-inclination MHD.

In this work three recently introduced (3 + 1)-dimensional nonlinear modified Benjamin-Bona-Mahony equations are studied from the modern group-theoretical analysis standpoint. The (3 + 1)-dimensional nonlinear differential equations are considered to be more realistic equations compared to the (1 + 1) and (2 + 1)-dimensional equations. Here we construct soliton and Jacobi elliptic function solutions of these three underlying equations and compute their conservation laws by employing Noether’s approach. The obtained solutions are presented graphically.

This paper analytically investigates a new (2+1)-dimensional extended shallow water wave equation. Lie group theory along with direct integration is used to achieve some solutions of the equation. The solutions obtained are in terms of Jacobi elliptic function as well as Weierstrass elliptic function. Besides, we apply the He’s variational technique to secure some non-topological soliton solutions of the equation. Series solution of the equation is also achieved by employing power series technique and we show the convergence of the series. Furthermore, graphical exhibitions of the dynamical character of the gained result are presented and discussed in a bid to have a sound understanding of the physical phenomena of the underlying model. In addition, we examine the stability analysis of the equation. Conclusively, we give the conserved currents of the aforementioned equation by employing the homotopy formula together with the Noether theorem.

In this article, we study two-dimensional generalized breaking soliton equation, which describes two-dimensional interchange of Riemann wave disseminating alongside y-axis with a long wave disseminating alongside x-axis. We derive Lie symmetry generators of this nonlinear partial differential equation and then utilize them to perform symmetry reductions. Travelling wave variables are used to obtain most general closed-form solutions of this equation by using two procedures. In addition, we compute the conserved vectors of this equation by engaging the classical Noether’s theorem.

In this work, we investigate the two-dimensional Yu–Toda–Sasa–Fukuyama equation, which has many applications in plasma physics and other fields of study in natural sciences. Exact analytical solutions of this model are presented by invoking Lie symmetry analysis technique. Moreover, the power series expansion method is also used to construct exact analytical solutions. We present the 2D and 3D graphical representations of the obtained solutions for some parametric values so as to demonstrate the dynamic behaviour of the solutions. Furthermore, conserved vectors are derived by making use of two methods, namely the multiplier method and Noether’s theorem.

A two-dimensional laminar flow driven by fluid injection through porous surface which represent an incompressible fluid inside a filtration chamber during extraction of particles from the fluid is investigated. The study constructs a mathematical model that represents internal flow field during filtration process proficiently by using basic conservation laws of mass, momentum and energy. For better understanding of dynamics of the case study, solutions that lead to stable filtration process (balanced dynamical system) are obtained and consequently provide an insight of the important dynamics that lead to an optimal filtration process. The solution of partial differential equations representing the internal flow field similarity transformation based on Lie group method is utilized to reduce the system to ordinary differential equations. Thereafter, double perturbation method is employed to determine semi-analytical solutions of reduced system. Effects of various parameters that arise from the configuration (design) of the filter are presented graphically and analysed to show the connection between the case study and findings. Keywords: Two-dimensional viscous laminar flow, Stable filtration process, Lie group analysis, Perturbation method

This paper studies the second extended Calogero-Bogoyavlenskii-Schiff (eCBS) equation in (2+1)–dimensions, which was proposed in the literature a short time ago. Firstly, Lie symmetries of the equation are derived and thereafter we use them to perform symmetry reductions. Using its translation symmetries, the eCBS equation is reduced to a fourth-order ordinary differential equation, which is then solved with the aid of three techniques to construct closed-form solutions. In addition, we portray the solutions with the appropriate graphical representations. Furthermore, conserved vectors of eCBS equation are computed by invoking multiplier procedure as well as Noether’s theorem.

The nonlinear phenomena in numbers are modelled in a wide range of fields such as chemical physics, ocean physics, optical fibres, plasma physics, fluid dynamics, solid-state physics, biological physics and marine engineering. This research article systematically investigates a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. We achieve a five-dimensional Lie algebra of the equation through Lie group analysis. This, in turn, affords us the opportunity to compute an optimal system of fourteen-dimensional Lie subalgebras related to the underlying equation. As a consequence, the various subalgebras are engaged in performing symmetry reductions of the equation leading to many solvable nonlinear ordinary differential equations. Thus, we secure different types of solitary wave solutions including periodic (Weierstrass and elliptic integral), topological kink and anti-kink, complex, trigonometry and hyperbolic functions. Moreover, we utilize the bifurcation theory of dynamical systems to obtain diverse nontrivial travelling wave solutions consisting of both bounded as well as unbounded solution-types to the equation under consideration. Consequently, we generate solutions that are algebraic, periodic, constant and trigonometric in nature. The various results gained in the study are further analyzed through numerical simulation. Finally, we achieve conservation laws of the equation under study by engaging the standard multiplier method with the inclusion of the homotopy integral formula related to the obtained multipliers. In addition, more conserved currents of the equation are secured through Noether’s theorem.

In this work, we analytically examine a (3+1)-dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation (gKdV-ZKe). Solutions of this equation, including a non-topological soliton, are obtained by Lie symmetry reductions and direct integration. Moreover, Kudryashov’s method is utilized to generate some closed-form solutions of the equation. Furthermore, cnoidal and snoidal periodic wave solutions are displayed for a special case of the gKdV-ZKe. The obtained solutions are presented graphically. Conclusively, we provide conservation laws of gKdV-ZKe by engaging Noether’s theorem.

We study coupled Burgers equations that model polydispersive sedimentation from Lie symmetry standpoint. We perform Lie group analysis technique on the system and obtain symmetry reductions. Travelling wave solutions are constructed using the translation symmetries in time and space. Furthermore, we compute conservation laws of the system in two ways; firstly, by using the multiplier approach and secondly, by the application of theorem on conservation laws due to Ibragimov.

The aim of this paper is to study three space-time (3 + 1)-dimensional modified Korteweg-de Vries equations. Nonlinear space-time (3 + 1)-dimensional partial differential equations model many realistic problems in the fields of engineering, wave propagation, fluids, etc. Firstly we construct exact closed-form solutions for the three (3 + 1)-dimensional modified Korteweg-de Vries equations using Lie symmetry method together with the extended Jacobi elliptic expansion method. The solutions obtained are soliton, cnoidal and snoidal waves. Secondly we determine conservation laws for the underlying equations using the multiplier method. Keywords: (3 + 1)-dimensional modified KdV equation, Lie point symmetries, Exact closed-form solutions, Conservation laws

Exact solutions for velocity field and tangential stress for rotational flow of a generalized Burgers’ fluid within an infinite circular pipe are derived by using the methods of Laplace and finite Hankel transformations. Firstly we take the position of fluid at rest and then the fluid flow due to the rotation of the pipe around the axis of flow having time dependant angular velocity. The exact solutions are presented in terms of the generalized Ga,b,c(.,t)-functions. The corresponding results can be freely specified for the same results of Burgers’, Oldroyd B, Maxwell, second grade and Newtonian fluids (performing the same motion) as particular cases of the results obtained earlier. The impact of the different parameters, individually and in comparison, are represented by graphical demonstrations. Secondly the numerical solutions for velocity and stress are also obtained with the help of Laplace transformation, Gaver Stehfest’s algorithm and MATHCAD. Finally a comparison of both methods for the same problem is done and shows the consistency of results. Keywords: Generalized Burgers’ fluid, Velocity, Shear stress, Integral transform, Stehfest’s algorithm, MATHCAD

In this work we study the Korteweg-de Vries-Benjamin-Bona-Mahony (KdV-BBM) equation, which describes the two-way propagation of waves. Using Lie symmetry method together with Jacobi elliptic function expansion and Kudryashov methods we construct its travelling wave solutions. Also, we derive conservation laws of the KdV-BBM equation using the variational derivative approach. In this method, we begin by computing second-order multipliers for the KdV-BBM equation followed by a derivation of the respective conservation laws for each multiplier. Keywords: Korteweg-de Vries-Benjamin-Bona-Mahony equation, Lie point symmetries, Variational derivative, Conservation laws

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

A (3+1)-D generalized B-type KP-Boussinesq equation, which was recently formulated in the literature, is investigated here from Lie group standpoint. A solution is obtained by Lie symmetry reductions and direct integration in terms of incomplete elliptic integral. Furthermore, hyperbolic and trigonometric functions solutions are derived by invoking the (G′/G)- expansion method. Finally, we construct conservation laws of the aforementioned equation by utilizing the multiplier method and conservation theorem due to Ibragimov. Keywords: A (3+1)-dimensional generalized BKP-Boussinesq equation, Lie point symmetries, Exact solutions, Incomplete elliptic integral, (G′/G)-expansion method, Conservation laws

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the (G′/G)-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov’s theorem.

In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

In this paper, we study the diversity of interaction solutions of a shallow water wave equation, the generalized Hirota–Satsuma–Ito (gHSI) equation. Using the Hirota direct method, we establish a general theory for the diversity of interaction solutions, which can be applied to generate many important solutions, such as lumps and lump-soliton solutions. This is an interesting feature of this research. In addition, we prove this new model is integrable in Painlevé sense. Finally, the diversity of interactive wave solutions of the gHSI is graphically displayed by selecting specific parameters. All the obtained results can be applied to the research of fluid dynamics.

In this paper Lie symmetry analysis of the seventh-order time fractional Sawada–Kotera–Ito (FSKI) equation with Riemann–Liouville derivative is performed. Using the Lie point symmetries of FSKI equation, it is shown that it can be transformed into a nonlinear ordinary differential equation of fractional order with a new dependent variable. In the reduced equation the derivative is in Erdelyi–Kober sense. Furthermore, adapting the Ibragimov’s nonlocal conservation method to time fractional partial differential equations, we obtain conservation laws of the underlying equation. In addition, we construct some exact travelling wave solutions for the FSKI equation using the sub-equation method. Keywords: Fractional Sawada–Kotera–Ito equation, Lie symmetry, Riemann–Liouville fractional derivative, Conservation laws, Exact solutions

The Hirota-Satsuma-Ito equation in (2+1)-dimensions passes the three-soliton test. This paper aims to generalize this equation to a new one which still has abundant interesting solution structures. Based on the Hirota bilinear formulation, a symbolic computation with a new class of Hirota-Satsuma-Ito type equations involving general second-order derivative terms is conducted to require having lump solutions. Explicit expressions for lump solutions are successfully presented in terms of coefficients in a generalized Hirota-Satsuma-Ito equation. Three-dimensional plots and contour plots of a special presented lump solution are made to shed light on the characteristic of the resulting lump solutions.

Mixture theory provides a continuum framework to model a multi-phase system. The basic assumption is, at any instant of time all phases are present at every material point and momentum and mass balance equations are postulated. This paper reviews the recent developments in mixture theory and focuses on the applications of the theory in particular areas of biomechanics, composite manufacturing and infiltration into deformable porous materials. The complexity based upon different permeability and stress functions is also addressed. The review covers the literature presented in the past fifty years and summarizes applications of mixture theory in specific areas of interest, for the sake of brevity, only necessary details are provided rather than complete modeling and simulation.

An analytical approach based on the compatibility concept is employed to solve a nonlinear reaction–diffusion model arising in mathematical biology. The solution process makes it extremely easy to obtain a relatively accurate closed-form solution of the model. The pencil-and-paper solution procedure can be extended to other class of nonlinear problems of similar kind. Keywords: Compatibility approach, Reaction–diffusion equation, Exact solvability

We study two nonlinear partial differential equations, namely, the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations. The Lie symmetry approach along with the simplest equation and exp-function methods are used to obtain solutions of the symmetric regularized long wave equation, while the travelling wave hypothesis approach along with the simplest equation method is utilized to obtain new exact solutions of the Klein-Gordon-Zakharov equations.

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

We study the Drinfel'd-Sokolov-Wilson system, which was introduced as a model of water waves. Firstly we obtain exact solutions of this system using the (G′/G)-expansion method. In addition to exact solutions we also construct conservation laws for the underlying system using Noether's approach.

We study a generalized two-dimensional nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation, which is in fact Benjamin-Bona-Mahony equation formulated in the ZK sense. Conservation laws for this equation are constructed by using the new conservation theorem due to Ibragimov and the multiplier method. Furthermore, traveling wave solutions are obtained by employing the (G'/G)-expansion method.