MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonl... more MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonlinear partial differential equations of fluid mechan-ics. The modified equal-width (MEW) equation, which is used in handling the sim-ulation of a single dimensional wave propagation in nonlinear media with dispersion process, is studied first. We compute the optimal system of one-dimensional subalge-bras and then use it to perform symmetry reductions and obtain group-invariant solutions. Also, we derive conservation laws of the MEW equation using the multiplier approach and the Noether theorem. Secondly we study the generalized nonlinear advection-diffusion equation, which describes the movement of a buoyancy-driven plume in an inclined porous medium. We consider three cases of n and in each case, we construct optimal system of one-dimensional subalgebras using the computed Lie point symmetries and then obtain symmetry reductions and group-invariant solutions based on these optimal systems of one-dimensional subalgebra. In addition, we de-termine the conservation laws of the equation by employing the multiplier approach and the new conservation theorem due to Ibragimov.Master
MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonl... more MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonlinear partial differential equations of fluid mechan-ics. The modified equal-width (MEW) equation, which is used in handling the sim-ulation of a single dimensional wave propagation in nonlinear media with dispersion process, is studied first. We compute the optimal system of one-dimensional subalge-bras and then use it to perform symmetry reductions and obtain group-invariant solutions. Also, we derive conservation laws of the MEW equation using the multiplier approach and the Noether theorem. Secondly we study the generalized nonlinear advection-diffusion equation, which describes the movement of a buoyancy-driven plume in an inclined porous medium. We consider three cases of n and in each case, we construct optimal system of one-dimensional subalgebras using the computed Lie point symmetries and then obtain symmetry reductions and group-invariant solutions based on these optimal systems of one-dimensional subalgebra. In addition, we de-termine the conservation laws of the equation by employing the multiplier approach and the new conservation theorem due to Ibragimov.Master
In physics as well as mathematics, nonlinear partial differential equations are known as veritabl... more 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.
Journal of Nonlinear Mathematical Physics, Feb 28, 2023
Higher-dimensional nonlinear integrable partial differential equations are significant as they of... more Higher-dimensional nonlinear integrable partial differential equations are significant as they often describe diverse phenomena in nonlinear systems like laser radiations in a plasma, optical pulses in the glass fibres, fluid mechanics, radio waves in the ion sphere, condensed matter and electromagnetics. This article shows an analytical investigation of a (3?1)-dimensional fifth-order nonlinear model with KdV forming its main part. Lie group analysis of the model is performed through which its infinitesimal generators are obtained. These generators are engaged in the construction of an optimal system of Lie subalgebra in one dimension. Moreover, members of the system secured are utilized in reducing the underlying model to ordinary differential equations (ODEs) for possible exact solutions. In consequence, we achieve various functions, ranging from trigonometric, logarithmic, rational, to hyperbolic. In addition, the results found constitute diverse solitonic solutions such as complex, topological kink and anti-kink, trigonometric and bright. We utilize the power series technique to obtain a series solution of the most complicated ordinary differential equation with forty-four terms. In addition, we reveal the dynamics of these solutions via graphical depictions. In the end, we constructed conserved currents of the underlying equation through the use of the multiplier technique. Further, we utilize the optimal system of the underlying model to derive more conserved vectors using Ibragimov's theorem for conservation laws. Keywords A (3?1)-dimensional fifth-order nonlinear model I • Lie group analysis • Group-invariant and exact solutions • Soliton solutions • Optimal system of Lie subalgebras • Conserved currents Abbreviations ODEs Ordinary differential equations
Communications in Theoretical Physics, Oct 26, 2021
Many physical systems can be successfully modelled using equations that admit the soliton solutio... more Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.
Higher-dimensional nonlinear integrable partial differential equations are significant as they of... more Higher-dimensional nonlinear integrable partial differential equations are significant as they often describe diverse phenomena in nonlinear systems like laser radiations in a plasma, optical pulses in the glass fibres, fluid mechanics, radio waves in the ion sphere, condensed matter and electromagnetics. This article shows an analytical investigation of a (3+1)-dimensional fifth-order nonlinear model with KdV forming its main part. Lie group analysis of the model is performed through which its infinitesimal generators are obtained. These generators are engaged in the construction of an optimal system of Lie subalgebra in one dimension. Moreover, members of the system secured are utilized in reducing the underlying model to ordinary differential equations (ODEs) for possible exact solutions. In consequence, we achieve various functions, ranging from trigonometric, logarithmic, rational, to hyperbolic. In addition, the results found constitute diverse solitonic solutions such as comp...
Nonlinear wave equations emerge naturally in optics as well as fluid dynamics. It is noteworthy t... more Nonlinear wave equations emerge naturally in optics as well as fluid dynamics. It is noteworthy that, an essential class of special solutions that satisfy the underlying equations are observed to be localized waves, which are frequently referred to as solitons or solitary waves. Thus, in the context of small amplitude alongside shallow-water waves, such solutions were discovered in the nineteenth century. Therefore, we present in this paper, the analytical investigations accomplished on a three dimensional generalized nonlinear wave equation in a fluid accommodating gas fizzes with applications. This equation was developed in the field within which liquid in conjunction with gas fizzes exists to describe the proliferation of feebly-nonlinear-waves. The underlying equation is transformed into a nonlinear structured ordinary differential equation (NLODE) by Lie group theory. Direct integration of the resulting NLODE produced periodic, trigonometric bright soliton together with singular soliton solutions. Moreover, some general exact soliton solutions of the equation under study are secured via the simplest equation technique (SET) in the structure of various Jacobi elliptic functions. Thus, we secure diverse periodic solitons of the equation under consideration. In addition, the dynamics of the results are depicted using suitable graphs which were also discussed. Furthermore, We conduct stability analysis on the model under study and outline the significance of our results in fluid dynamics, biomedical sciences and biological cells. Conclusively, we constructed conservation laws of the aforementioned equation by invoking Ibragimov's theorem for conserved quantities via its formal Lagrangian structure.
MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonl... more MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonlinear partial differential equations of fluid mechan-ics. The modified equal-width (MEW) equation, which is used in handling the sim-ulation of a single dimensional wave propagation in nonlinear media with dispersion process, is studied first. We compute the optimal system of one-dimensional subalge-bras and then use it to perform symmetry reductions and obtain group-invariant solutions. Also, we derive conservation laws of the MEW equation using the multiplier approach and the Noether theorem. Secondly we study the generalized nonlinear advection-diffusion equation, which describes the movement of a buoyancy-driven plume in an inclined porous medium. We consider three cases of n and in each case, we construct optimal system of one-dimensional subalgebras using the computed Lie point symmetries and then obtain symmetry reductions and group-invariant solutions based on these optimal systems of one-dimensional subalgebra. In addition, we de-termine the conservation laws of the equation by employing the multiplier approach and the new conservation theorem due to Ibragimov.Master
MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonl... more MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonlinear partial differential equations of fluid mechan-ics. The modified equal-width (MEW) equation, which is used in handling the sim-ulation of a single dimensional wave propagation in nonlinear media with dispersion process, is studied first. We compute the optimal system of one-dimensional subalge-bras and then use it to perform symmetry reductions and obtain group-invariant solutions. Also, we derive conservation laws of the MEW equation using the multiplier approach and the Noether theorem. Secondly we study the generalized nonlinear advection-diffusion equation, which describes the movement of a buoyancy-driven plume in an inclined porous medium. We consider three cases of n and in each case, we construct optimal system of one-dimensional subalgebras using the computed Lie point symmetries and then obtain symmetry reductions and group-invariant solutions based on these optimal systems of one-dimensional subalgebra. In addition, we de-termine the conservation laws of the equation by employing the multiplier approach and the new conservation theorem due to Ibragimov.Master
In physics as well as mathematics, nonlinear partial differential equations are known as veritabl... more 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.
Journal of Nonlinear Mathematical Physics, Feb 28, 2023
Higher-dimensional nonlinear integrable partial differential equations are significant as they of... more Higher-dimensional nonlinear integrable partial differential equations are significant as they often describe diverse phenomena in nonlinear systems like laser radiations in a plasma, optical pulses in the glass fibres, fluid mechanics, radio waves in the ion sphere, condensed matter and electromagnetics. This article shows an analytical investigation of a (3?1)-dimensional fifth-order nonlinear model with KdV forming its main part. Lie group analysis of the model is performed through which its infinitesimal generators are obtained. These generators are engaged in the construction of an optimal system of Lie subalgebra in one dimension. Moreover, members of the system secured are utilized in reducing the underlying model to ordinary differential equations (ODEs) for possible exact solutions. In consequence, we achieve various functions, ranging from trigonometric, logarithmic, rational, to hyperbolic. In addition, the results found constitute diverse solitonic solutions such as complex, topological kink and anti-kink, trigonometric and bright. We utilize the power series technique to obtain a series solution of the most complicated ordinary differential equation with forty-four terms. In addition, we reveal the dynamics of these solutions via graphical depictions. In the end, we constructed conserved currents of the underlying equation through the use of the multiplier technique. Further, we utilize the optimal system of the underlying model to derive more conserved vectors using Ibragimov's theorem for conservation laws. Keywords A (3?1)-dimensional fifth-order nonlinear model I • Lie group analysis • Group-invariant and exact solutions • Soliton solutions • Optimal system of Lie subalgebras • Conserved currents Abbreviations ODEs Ordinary differential equations
Communications in Theoretical Physics, Oct 26, 2021
Many physical systems can be successfully modelled using equations that admit the soliton solutio... more Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.
Higher-dimensional nonlinear integrable partial differential equations are significant as they of... more Higher-dimensional nonlinear integrable partial differential equations are significant as they often describe diverse phenomena in nonlinear systems like laser radiations in a plasma, optical pulses in the glass fibres, fluid mechanics, radio waves in the ion sphere, condensed matter and electromagnetics. This article shows an analytical investigation of a (3+1)-dimensional fifth-order nonlinear model with KdV forming its main part. Lie group analysis of the model is performed through which its infinitesimal generators are obtained. These generators are engaged in the construction of an optimal system of Lie subalgebra in one dimension. Moreover, members of the system secured are utilized in reducing the underlying model to ordinary differential equations (ODEs) for possible exact solutions. In consequence, we achieve various functions, ranging from trigonometric, logarithmic, rational, to hyperbolic. In addition, the results found constitute diverse solitonic solutions such as comp...
Nonlinear wave equations emerge naturally in optics as well as fluid dynamics. It is noteworthy t... more Nonlinear wave equations emerge naturally in optics as well as fluid dynamics. It is noteworthy that, an essential class of special solutions that satisfy the underlying equations are observed to be localized waves, which are frequently referred to as solitons or solitary waves. Thus, in the context of small amplitude alongside shallow-water waves, such solutions were discovered in the nineteenth century. Therefore, we present in this paper, the analytical investigations accomplished on a three dimensional generalized nonlinear wave equation in a fluid accommodating gas fizzes with applications. This equation was developed in the field within which liquid in conjunction with gas fizzes exists to describe the proliferation of feebly-nonlinear-waves. The underlying equation is transformed into a nonlinear structured ordinary differential equation (NLODE) by Lie group theory. Direct integration of the resulting NLODE produced periodic, trigonometric bright soliton together with singular soliton solutions. Moreover, some general exact soliton solutions of the equation under study are secured via the simplest equation technique (SET) in the structure of various Jacobi elliptic functions. Thus, we secure diverse periodic solitons of the equation under consideration. In addition, the dynamics of the results are depicted using suitable graphs which were also discussed. Furthermore, We conduct stability analysis on the model under study and outline the significance of our results in fluid dynamics, biomedical sciences and biological cells. Conclusively, we constructed conservation laws of the aforementioned equation by invoking Ibragimov's theorem for conserved quantities via its formal Lagrangian structure.
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