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The diverse use of heat and mass transport mechanisms on three-dimensional rotating systems has been discovered to improve the efficiency of several physiological and industrial processes. These include, for example, fiber technology, the process of extrusion, and polymer extraction. The purpose of this dissertation is to investigate the transportation of mass and heat in the flow of non-Newtonian fluid processing vortex viscosity and micro-inertial features along two horizontal plates in a rotating scenario. The thermal transport process is influenced by physical parameters such as irregular heat generation/absorption rate, linear radiative heat flow, variable thermal conductivity, and mass diffusion with cubic autolysis chemical reaction. After enforcing conservation laws, a simple constitutive equations are modelled in a rotating system. The mathematical formulation is converted into a dimensionless format by applying comparable variables. The bvp4c process is used to execute numerical experiments to arrive at the numerical solution for the governing problem. Through sketching, the related distribution (velocity, micro rotation, velocity, and concentration) is presented concerning the various parameters. Tabular data provides insights into relevant quantities against different parameters. The credibility of computed results is assumed by agreeing with previous studies.Graphical AbstractThe velocity variation for different values of Reynolds number. It is analysed from the sketch that stronger values of the Reynolds number improve the fluid near the lower plate but as we move towards upper plate flow is reduced.

Across the neuronal membrane, the Korteweg–de Vries equation regulates the spatial and temporal development of the density difference among extracellular and intracellular liquids. In this study, model of the nerve impulse is taken into consideration by means of bifurcation and chaotic theory. The considered model is subjected to an extended direct algebraic method, that revealed abundant analytical results containing hyperbolic, trigonometric and rational functions. Solitons are obtained in terms of pulse stimulation from density difference equation. These results are made accessible together with their fundamental requirements, which ensure the continuance of pulse solitons such as dark, singular and kink that utilize suitable physical parameters in 2D plots. Subsequently, the nonlinear density difference equation is transformed into a planar dynamical system via a Galilean transformation. Bifurcation theory is applied to the dynamical structure of the considered setup to investigate the possible phase portraits of the travelling wave results. Furthermore, we applied the numerical approaches to determine the periodic and quasiperiodic behaviours of the considered equation and the findings are exhibited graphically. Sensitivity and multistability analysis are additionally used to study the periodic and quasiperiodic behaviour for several initial constraints. At the end, the concluding remarks are presented.

Bifurcation analysis is studied for the first time in magneto-hydrodynamic (MHD) Jeffery–Hamel flow nanofluids. The dimensionless ordinary differential equations are formulated by using the similarity transformations. The phase portraits of the traveling wave solutions are examined using bifurcation theory on the dynamical system of the considered equation. Moreover, multistability and sensitivity analysis is used to examine quasi-periodic and periodic behavior for various initial conditions. The (MHD) Jeffery–Hamel flow solutions are derived analytically by applying the extended direct algebraic approach and the influence of the different physical parameters for velocity profiles, have been examined. It is found that the physical parameter i.e, Reynolds number and channel angle rising in convergent flow results in an increase in the velocity profiles, signifying that backflow is excluded, while an inverse behavior can be attributed to a diverging flow.

This research includes the study of the non-linear dynamics of thin-film ferroelectric materials governed by an equation of wave dynamics within the material. This equation plays a key role in both physics and the study of aqueous flow. The work is planned as examining the symmetries group analysis drops, studying the dynamical system features through bifurcation phase portraits, and carrying dynamic phenomena in chaos theory. Diverse techniques are taken, such as Lyapunov exponent, 2D, and 3D phase portraits, Poincaré maps, time series analysis, and sensitivity to multistability under the different conditions of the initial state. In addition, the study involves using the extended hyperbolic function method to obtain the general analytical solutions via which various kinds of solitary wave solutions are produced including trigonometric and hyperbolic functions and periodic, bright, and singular soliton solutions. These solutions are followed by a list of constraint conditions in the form of equations. Visual data of 2D, 3D, and contour plots are presented, with parameters carefully set to reflect various scenarios. Sensitivity analysis is performed using alternative initial conditions, and stability analysis is demonstrated graphically. To fully grasp the dynamic features of these systems and accurately predict outcomes, it is essential to advance new technologies and methodologies that can further enhance our understanding and predictive capabilities in complex systems. [ABSTRACT FROM AUTHOR]

The growing need for effective thermal management systems in engineering applications will improve performance by using nanofluids. Nanofluids, which show enhanced thermal characteristics compared to typical fluids, offer an effective way for heat transmission processes in industries. This study is particularly useful for systems where traditional fluids are insufficient for improving thermal performance. Understanding the overall impacts of Joule heating, magnetic fields, and slip conditions would be beneficial in fields such as aircraft, microelectronics, and biomedical engineering.

This article proposes a mathematical investigation of unsteady flow and heat transfer in the presence of Joule Heating over a radially stretching sheet using a nanofluid of Cu-Ethylene glycol. With an extensive numerical study, we reveal the novel interaction between the shape factors of nanoparticles and surface deformations brought about by stretching. As opposed to earlier studies that have mostly concentrated on traditional nanoparticle forms, our investigation methodically looks at the unique behaviors of Cu-EG nanoparticles on stretching surfaces. The research findings offer great potential for numerous practical applications, in addition to providing insight into basic concepts related to fluid dynamics and heat transfer. The solution to this issue is significant for enhancing thermal management in manufacturing environments, such as cooling systems used in aerospace and electronics. Therefore, our work establishes a foundation for novel methods of creating materials with customized qualities, opening the door for the creation of next-generation technologies that are more sustainable and functional. A numerical solution of the highly non-linear ordinary differential equation is attained with suitable boundary conditions by applying BVP4C in MATLAB. Impact of pertinent parameters on Cu-Ethylene glycol nanofluid Joule Heating concentration, as well as Eckert, Prandtl, and Biot-number on flow and heat transport, are studied. Important results show that the Joule Heating effect raises the total heat transfer rate by roughly 15 %, and the addition of Cu nanoparticles improves thermal conductivity by around 22 %. The findings show that the combined influences of Joule Heating and nanoparticle concentration greatly increase the heat transfer efficiency, offering important new information for the optimization of cooling systems in a range of industrial applications. Finding of the current study is that the shape factor of platelets effectively transfers heat and flow, with sphere forms convey the least amount of heat.

In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

The exploration of the nonlinear dynamics related to the new coupled Konno–Oono equation, which determines the propagation of magnetic fields, is the focus of this work. Through the employing of Lie group analysis, the bifurcation phase portraits, and chaos theory, the project will investigate symmetry reductions in dynamical systems and examine the dynamic behavior of perturbed dynamical systems. The 3D phase portrait, 2D phase portrait, Lyapunov exponent, time series analysis, sensitivity analysis, and an examination of the existence of multistability in the autonomous system under various initial conditions constitute a few of the methods used for recognizing chaotic behavior. Furthermore, the investigation constructs general solutions for solitary wave solutions, such as exponential and hyperbolic function, singular, dark, and bright soliton solutions, by using the new Kudryashov methodology to determine the investigated equation analytically. These solutions are shown graphically as 2D, 3D, and contour plots with specifically selected values. They include as well with the related constraint circumstances. Additionally, a discussion and a visual illustration of the considered equation's sensitivity analysis are presented. The observations demonstrate that the aforementioned approach is an effective procedure for treating a variety of nonlinear PDE systems that arise in nonlinear physics analytically. The plot of the Lyapunov exponents is employed to validate the chaotic dynamics of the studied model. Additionally, the multiplier method is employed to determine the conserved vectors for the analyzed problem. • Coupled Konno–Oono equation within the domain of magnetic fields is considered. • The Lie invariance condition is applied to identify symmetry generators. • Soliton solutions are derived and phase portrait analysis has been done. • Chaotic behaviors are examined with different characteristic settings. • Conserved vectors for the studied equation are calculated. [ABSTRACT FROM AUTHOR]

This paper discusses the extended simplest equation method for analyzing the behavior of a porous fin while taking into account the porous and convection parameters. This method helps in obtaining exact solutions and understanding the temperature distribution and heat transfer characteristics of the fin. Furthermore, the equation is transformed into a planar dynamical system using the Galilean transformation. Under dynamic and quasiperiodic conditions, the behavior of the equation is extensively discussed. The bifurcation of the planar system of dynamics described by the transformed equation is studied using bifurcation theory and phase portrait analysis. Additionally, a perturbed force is introduced to the dynamical system, and the RK4 method is employed to identify periodic or quasiperiodic patterns in the system. Sensitivity and multistability analysis are applied to the porous fin equation, considering different initial conditions to investigate its quasiperiodic and periodic behavior. The chaotic structure of the perturbed dynamical system using Lyapunov exponent analysis has also been discussed. Through a comparison between numerical solutions and graphical representations, the validity and applicability of the exact technique are demonstrated. The reported findings contribute novel insights into understanding the behavior of nonlinear equations encountered in mathematical physics. [ABSTRACT FROM AUTHOR]

A complete group classification of the nonlinear degenerate parabolic equation is presented and symmetry generators are calculated for each f(u). For arbitrary f(u), one dimensional conjugacy classes for symmetry algebras are obtained and similarity reduction of each class is given. Moreover, exact solutions for some particular cases are also presented.

In this manuscript, the perturbed Gerdjikov–Ivanov equation is considered for the investigating model. It is studied to govern the dynamics of soliton propagation through optical fibers, metamaterials or PCF. The observed model is subjected to an extended simple equation technique, which disclosed an abundant of exact solutions including trigonometric function, singular, kink, peakon and compacton solutions. These solutions are made available with their essential conditions, which guarantee the persistence of such optical solitons and portraits using appropriate physical parameters in 3D, 2D and density plots. After that, the traveling wave transformation is used to turn the m t h -order nonlinear perturbed Gerdjikov–Ivanov equation into a planar dynamical system. The dynamical and chaotic behaviors of the considered equation are also discussed. The qualitative analysis of the dynamical system and the chaotic behaviors of the perturbed system are investigated using the theory of plane dynamic systems. In addition, we utilize the RK4 method to discover patterns in the dynamical system that seem to be super nonlinear, periodic, and quasiperiodic. The reported results are new and have not been investigated. They can be used in the explanation of the physical phenomena modeled and will give information about the long-term dynamic behavior. Numerical simulations reveal that by changing the frequency and amplitude parameters have an impact on the dynamic behaviors of the system. It is established that the extended simple equation method and dynamical observations offer further influential mathematical tools for constructing exact solutions and their qualitative analysis in NLEEs in mathematical physics. [ABSTRACT FROM AUTHOR]

The current exploration explains new traveling wave profiles by the Lie symmetry approach Here, we derive the (3 + 1)-dimensional Zakharov–Kuznetsov burgers equation for the shock wave phenomena in plasma with dust-charged particles having quantum effects. The structures of this model are made by Abelian algebra. The considered equation is reduced to an ordinary differential equation by the use of this Abelian algebra. The complete derivation of the classical symmetries of the considered model and reduction corresponding to the first four symmetries has been added. The new auxiliary equation method is implemented for the formation of the new traveling wave structures. The graphical explanation of some obtained results are elaborated in detail. The nonlinear self-adjointness of the considered model is also part of this work. By this concept, we have evaluated the conservation laws of the model. This work is new and does not exist in literature. • The (3+1)-dimensional Zakharov–Kuznetsov burgers equation for the shock wave phenomena in plasma with dust-charged particles having quantum effects is considered. • New traveling wave structures are computed by means of Lie theory. • Nonlinear self-adjointness is reported. • Conservation laws have been computed. [ABSTRACT FROM AUTHOR]

In this paper, the conserved quantities of the non-linear Klein–Gordon equation $$u_{tt}=u_{xx}+f(u)$$are derived. The Noether theorem is applied to the considered equation with an arbitrary function $$f(u)$$. Symmetry operators for all possible cases of the function $$f(u)$$are investigated. Then the conserved vectors are computed for each of these Noether symmetry operators.

The $$(1+2)$$-dimensional heat equation on the plane ($$\mathbb R ^{2}$$) and sphere ($$\mathbb S ^{2}$$) is considered respectively. For each surface a class of functions is presented for which considered equation has nontrivial symmetries. We consider whether the background metric ($$\mathbb S ^{2}$$) or the nonlinearity have the dominant role in the infinitesimal generators of the considered equation. Then the time dependent Ginzburg–Landau equation (TDGL model) is considered on $$\mathbb S ^{2}$$. Lie point symmetry generators are calculated and optimal systems of its subalgebras up to conjugacy classes are obtained. Similarity reductions for each class are performed.

This study is dedicated to the computation and analysis of solitonic structures of a nonlinear Sasa–Satsuma equation that comes in handy to understand the propagation of short light pulses in the monomode fiber optics with the aid of beta derivative and truncated M- fractional derivative. We employ a new direct algebraic technique for the nonlinear Sasa–Satsuma equation to derive novel soliton solutions. A variety of soliton solutions are retrieved in trigonometric, hyperbolic, exponential, rational forms. The vast majority of obtained solutions represent the lead of this method on other techniques. The prime advantage of the considered technique over the other techniques is that it provides more diverse solutions with some free parameters. Moreover, the fractional behavior of the obtained solutions is analyzed thoroughly by using two and three-dimensional graphs. This shows that for lower fractional orders, i.e., β=0.1, the magnitude of truncated M-fractional derivative is greater whereas for increasing fractional orders, i.e., β=0.7 and β=0.99, the magnitude remains the same for both definitions except for a phase shift in some spatial domain that eventually vanishes and two curves coincide.

Abstract: In this paper, relationship between background metric and Noether operators is developed for different surfaces. For this the -dimensional non-linear wave equation on curved surfaces is considered. The Noether approach is applied on the discussed equation and determining equations for the Noether operators are computed in terms of coefficients of the first fundamental form (FFF). Further these determining equations are utilized to compute the Noether operators and conserved vectors of the considered equation on particular surfaces i.e. sphere , torus , flat space and cone . In derivation of conservation laws, two cases of the function are observed. For both cases the conserved vectors of the discussed equation on , and are established. It is noted that on all discussed surfaces Lie point generators coincide with the corresponding Noether operators while the maximal solvable algebra of symmetries is obtained for . [Copyright &y& Elsevier]

Abstract: This paper discusses the nonlinear wave equation on hyperbolic space. We extend some results reported in Nadjafikhah and Zaeim (2011) . Partial Noether theorem is applied to the equation under consideration and partial Noether operators are computed for different cases of . Furthermore, these partial Noether operators are utilized to calculate corresponding conserved quantities. A complete group classification is discussed for the considered equation. Maximal solvable algebra of Lie point symmetries is used to reduce the class of considered equation. Then algebras of translational symmetries are associated with its infinite conservation laws. [Copyright &y& Elsevier]

Abstract: A special type of (1+ n)-dimensional linear evolution equation is considered. A class of the equations generated by the Fokker–Planck equation becomes the subcase of the considered equation. Conserved vectors using the partial Lagrangian approach is derived in terms of the coefficients of the discussed equation. Derived results are used for the different models from different sciences. We also discuss the conservation laws of the heat equation on curved manifolds and in different coordinate systems. Potential systems are also obtained for some models. At last conclusion is given. [Copyright &y& Elsevier]

Abstract: The problem of symmetry classification for the heat equation on torus is studied by means of classical Lie group theory. The Lie point symmetries are constructed and Lie algebra is formed for equation under consideration. Then these algebras are used to classify its subalgebras up to conjugacy classes. In general the heat equation on torus admits one-, two-, three- and four-dimensional algebras. For one-dimensional algebra £1 and £2 the heat equation on torus is reduced in independent variables whereas in two-dimensional algebras £3 and £4 the considered heat equation is investigated by quadrature. While three- and four-dimensional algebras lead to a trivial solution. [Copyright &y& Elsevier]

• Nonlinear Modified Gardner Equation • Lie group analysis • Power series solution • Conservation laws • quantum electron-positron-ion magneto plasmas • Ion-acoustic waves In this paper, the nonlinear modified Gardner (mG) equation is under consideration which represents the super nonlinear proliferation of the ion-acoustic waves and quantum electron-positronion magneto plasmas. The considered model is investigated with the help of Lie group analysis. Lie point symmetries are computed under the invariance criteria of Lie groups and symmetry group for each generator is reported. Furthermore, the one-dimensional optimal system of subalgebras is developed by adjoint technique and then we compute the similarity reductions corresponding to each vector field present in the optimal system, with the help of similarity reduction method we have to convert the PDE into the ODE. Some exact explicit solutions of obtained ordinary differential equations were constructed by the power series technique. With the aid of the Galilean transformation, the model is transformed into a planer dynamical system and the bifurcation behaviour is recorded. All practicable types of phase portraits with regard to the parameters of the problem considered are plotted. Meantime, sensitivity is observed by utilizing sensitivity analysis. In addition, the influence of physical parameters is studied by the application of an extrinsic periodic power. With additional perturbed term, quasi-periodic and quasi-periodic-chaotic behaviours is reported. [ABSTRACT FROM AUTHOR]

• Solitonic structures of Zakharov-Kuznetsov (mZK) equation. • Transformation of the considered model in the planer dynamical system. • Quasiperiodic and chaotic behavior of the mZk equation. • Conservation laws of mZK equation via Lie symmetry. In this paper, modified Zakharov-Kuznetsov (mZK) equation is taken into consideration. Diverse variety of solitonic structures are computed by using extended algebraic method. Sufficient conditions for the existence of the computed solutions are presented. Then, Galilean transformation is utilized to transform the considered model in the planer dynamical system. All possible forms of phase portraits with respect to the parameters are plotted. Moreover, we utilized the numerical techniques to find out the nonlinear periodic structures of the discussed model and results are shown graphically. Moreover, after deploying an extrinsic periodic force the effect of frequency is observed then sensitive analysis is applied for different initial value problems to analyze the quasiperiodic and chaotic behaviour. In addition to this, Lie point symmetries and their corresponding conservation laws are reported at the end. [ABSTRACT FROM AUTHOR]

The objective of this manuscript is to examine the non-linear characteristics of the modified equal width-Burgers equation, known as the generalized Kadomtsive–Petviashvili equation, and its ability to generate a long-wave with dispersion and dissipation in a nonlinear medium. We employ the Lie symmetry approach to reduce the dimension of the equation, resulting in an ordinary differential equation. Utilizing the newly developed generalized logistic equation method, we are able to derive solitary wave solutions for the aforementioned ordinary differential equation. In order to gain a deeper understanding of the physical implications of these solutions, we present them using various visual representations, such as 3D, 2D, density, and polar plots. Following this, we conduct a qualitative analysis of the dynamical systems and explore their chaotic behavior using bifurcation and chaos theory. To identify chaos within the systems, we utilize various chaos detection tools available in the existing literature. The results obtained from this study are novel and valuable for further investigation of the equation, providing guidance for future researchers. • Application of the generalized logistic equation method to address the dynamics of solitary wave solutions for the generalized KP–MEW-Burgers equation. • Computation of the invariant transformations and symmetry reductions using Lie symmetry analysis. • Bifurcation and phase portraits of the unperturbed dynamical system using the idea of bifurcation theory of dynamical systems. • Chaos behavior of the perturbed dynamical system is identified through different chaos detecting tools using chaos theory of dynamical systems. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate the deeper characteristics of the system of shallow water wave equations that is used to model the turbulence in the atmosphere and oceans from different standpoints. The multi-soliton structures such as 1-soliton, 2-soliton, 3-soliton solutions are successfully generated with the help of simplified Hirota's method. For the sake of physical demonstration and visual presentation, we graphically illustrate the identified solutions in 3D, 2D, and contour plots using Mathematica software. The Lie symmetry technique is used to identify the Lie group invariant transformations and symmetry reductions of the examined system, which aid in reducing the dimension by one or into an ordinary differential equation. Further, the qualitative behavior of the time-dependent dynamical system is observed using the bifurcation and chaos theory. The phase portraits of bifurcation are observed at the equilibrium point of a planar dynamical system. We discuss various tools to identify chaos (random and unpredictable behavior) in autonomous dynamic systems, such as 3D phase portraits, 2D phase portraits, time series, and Poincaré maps. At various initial conditions, the sensitivity and modulation instability analyses are also presented, and it is discovered that the investigated system is stable, as a small change in the initial conditions does not cause an abrupt change in solutions. The findings of this investigation will contribute in the overall depiction of soliton theory and nonlinear dynamical systems. • Simplified Hirota's method used to find soliton for the proposed model. • Computation of the invariant transformations and symmetry reductions done. • Bifurcation and phase portraits of the unperturbed dynamical system is done. • Chaos behavior of the perturbed dynamical system is identified. • Sensitivity and modulation instability analyses for the considered system done. [ABSTRACT FROM AUTHOR]