Special Session 7: Lie Symmetries, Conservation Laws, and Other Approaches in Solving Nonlinear Differential Equations

Solitons, abundant analytical solutions and conserved quantities for a three-dimensional soliton model in plasma physics

Oke OD ADEYEMO North-West University So Africa Co-Author(s):    Prof C.M. Khalique

Abstract: In this talk, we consider the analytical investigation carried out on a three-dimensional soliton model, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. In the real sense of it, solitons as well as solitary waves have been discovered in numerous situations and often dominate long-term behaviour. Thus, the Lie group method is applied to obtain the symmetries of the equation. These are consequently used to obtain various solutions of interests. Moreover, conservation laws of the soliton equation are constructed.

Analysis of the Caputo Space-Time Fractional Phase Separation Cahn-Hilliard Model Using the Shehu Homotopy Transform Method

Shelly Arora Punjabi University, Patiala India Co-Author(s):    Wen Xiu Ma; SS Dhaliwal; Atul Pasrija

Abstract: The Shehu homotopy transform method is applied to examine the space-time fractional phase separation Cahn-Hilliard model in the Caputo sense. The Shehu transform is used to resolve the time fractional operator, while homotopy perturbation method is employed to decompose the non-linear term. Results obtained through the Shehu homotopy perturbation method have been compared with existing schemes in the literature for specific cases. For time fractional equations, fractional homotopy transform method needs to determine the time fractional derivative for each successive component of the series solution. This constrains the application of fractional homotopy transform method for time fractional equations. The Shehu transform with homotopy transform method is an admirable attempt to get over the limitations of fractional homotopy transform method.

Distributed Position and Velocity Delay Effects in a Van der Pol System with Time-periodic Feedback

Sudipto Roy Choudhury University of Central Florida USA Co-Author(s):    Ryan Roopnarain

Abstract: The effects of a distributed delay on a parametrically forced Van der Pol limit cycle oscillator are considered. Delays modeling time lags due to a variety of factors in self-excited systems, have been considered earlier in the context of modification and control of limit cycle and quasiperiodic responses. Those studies are extended here to include the effects of periodically amplitude modulated {\it distributed} delays in both the position and velocity. A normal form or `slow flow` is employed to search for various bifurcations and transitions between regimes of different dynamics, including amplitude death and quasiperiodicity. The existence of quasiperiodic solutions then motivates the derivation of a second slow flow. A detailed comparison of the results and predictions from the second slow flow to numerical solutions is made. The second slow flow is also employed to approximate the amplitudes of the quasiperiodic solutions, yielding close agreement with the numerical results on the original system. Finally, the effect of varying the delay parameter is briefly considered, and the results and conclusions are summarized.

A study of a generalized nonlinear (3+1)-D breaking soliton equation

Chaudry Masood Khalique North-West University, Mafikeng Campus So Africa Co-Author(s):

Abstract: Higher-order nonlinear wave models have recently attracted significant interest from researchers due to their importance in mathematical physics, various nonlinear sciences, and engineering applications. In this talk, we present analytical studies focused on a generalized form of a nonlinear breaking soliton equation with higher-order nonlinearity, highlighting its relevance to both science and engineering. We employ Lie group theory and derive a Lie algebra associated with the equation. This approach also facilitates reductions of various subalgebras related to the model. Additionally, we use direct integration techniques to obtain an analytic solution. Furthermore, we apply the simplest equation technique to uncover additional general solutions. Finally, we compute the conserved quantities associated with the equation using the well-known Noether theorem.

Symmetry reductions and dynamic formation with distinct solitons of exact invariant solutions of (3+1)-dimensional nonlinear evolution equations

SACHIN KUMAR University of Delhi India Co-Author(s):    Sachin Kumar

Abstract: In this talk, we will explore how to find a variety of generalized invariant solutions and show the dynamics of exact solutions to highly nonlinear evolution equations such as the (3+1)-dimensional evolution equation. Infinitesimals and a few similarity reductions are produced from the governing equations by using the Lie symmetry technique. The equation is transformed into a set of nonlinear ordinary differential equations (NLODEs) using the required stages of Lie symmetry reductions. Then, by solving the number of resulting ODEs, we obtain an extensive variety of invariant solutions in terms of the arbitrary functional parameters. Closed-form invariant solutions are successfully presented in the form of solitons and other combo-form solitons. Furthermore, dynamic and graphic representations of the resulting solutions are produced via computerized symbolic work.

Nonlocal integrability and solitons

Wen-Xiu Ma University of South Florida USA Co-Author(s):

Abstract: We will explore integrable models involving involution points. Classical Lax pairs are employed to derive nonlocal integrable models and their associated Riemann-Hilbert problems. Soliton solutions are obtained through reflectionless Riemann-Hilbert problems. Novel solution phenomena will also be discussed within the nonlocal framework.

Group classification of a system of reaction-diffusion and shallow water equations

Motlatsi Molati National University of Lesotho Lesotho Co-Author(s):    Lebohang Khatebe

Abstract: \begin{document} \begin{abstract} The main purpose of this work is to perform Lie group classification of a system of reaction-diffusion and shallow water equations. The forms that the arbitrary function takes are obtained via Lie symmetry analysis, for each functional form, the corresponding symmetry Lie algebra is utilized for the derivation of invariant solutions. \end{abstract} \end{document}

Preliminary group classification of nonlinear reaction-diffusion equation

Mpho DM Nkwanazana Sefako Makgatho Health Science University So Africa Co-Author(s):    Daniel Mpho Nkwanazana and Raseelo Joel Moitsheki

Abstract: \abstractformat{Abstract.} In this article we consider heat transfer models prescribed by reaction-diffusion equations. The diffusivity term and internal heat generator are given by the power law. The effects of parameters appearing in our model are presented and explained.

Study of the 2D generalized Calogero-Bogoyavlenskii-Schiff equation

Karabo KP Plaatjie University of Johannesburg So Africa Co-Author(s):    Chaudry Masood Khalique

Abstract: In our talk, we investigate the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff (2DgCBS) equation by computing its various wave solutions with the aid of Lie symmetry method and other techniques. These solutions are significant milestones in the study of water waves in mathematical physics and engineering phenomena. Furthermore, we construct conserved vectors for this equation by means of the two various techniques. This conserved quantities may be used to determine the integrability of this nonlinear model.

Lie Optimal Solutions of Heat Transfer in a Liquid Film over an Unsteady Stretching Surface with Viscous Dissipation and External Magnetic Field

Muhammad Safdar National University of Sciences and Technology Pakistan Co-Author(s):    Harris Ahmad, Bisma Jamil, S Taj

Abstract: This research aims at analyzing heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation and external magnetic field by incorporating Lie symmetry approach and corresponding one dimensional optimal system. This facilitates reduction of partial differential equations of the flow model to equations namely ordinary differential equations with one independent variable only. Further, analytic schemes such as the homotopy analysis and the finite difference methods are employed to solve the obtained system of equations. These solutions lead to the primary objective of this study which is to unravel temperature, velocity, and concentration distributions in the liquid film and provide a comprehensive analysis of the heat transfer characteristics. The outcomes of this research are expected to contribute significantly to the design and optimization of heat transfer in various industrial and biomedical applications. The results are presented with the help of graphs and tables, that are also compared and validated with existing literature to ensure accuracy and reliability.

Capillary rise in partially saturated rigid porous media

Javed I Siddique Penn State York USA Co-Author(s):    Javed I Siddique and Daniel M. Anderson

Abstract: \begin{abstract} We explore predictions of two models of one-dimensional capillary rise in rigid and partially saturated porous media. One is an existing one from the literature and the second is a free-boundary model based on Richards` equation with two moving boundaries of the evolving partially saturated region. Both models involve the specification of saturation-dependent functions for local capillary pressure and permeability and connect to classical models for saturated porous media. Existing capillary-rise experiments show two notable regimes: (1) an early time regime typically well-described by classical capillary-rise theory in a fully saturated porous media, and (2) a long-time regime that has anomalous dynamics in which the capillary-rise height may scale with a non-classical power law in time or have more complicated dynamics. We demonstrate that the predictions of both models compare well with experimental capillary-rise data over early- and long-time regimes gathered from three independent studies in the literature. The model predictions also shed light on recent scaling laws that relate the capillary pressure and permeability of the partially saturated media to the capillary-rise height. We use these models to probe computationally observed permeability relationships to capillary-rise height. We demonstrate that a recently proposed permeability scaling for the anomalous capillary-rise regime is indeed realized and is particularly apparent in the lower portion of the partially saturated media. For our free-boundary model we also compute capillary pressure measures and show that these reveal the linear relation between the capillary pressure and capillary rise height expected for a capillarity/gravity balance in the upper portion of the partially saturated porous media. \end{abstract}

Analytic Homotopy Analysis and Perturbation Solutions for MHD Casson Fluid Flow and Heat Transfer Near A Stagnation Point on A Stretching Sheet

Safia Taj National University of sciences and Technology, College of Electrical and Mechanical Engineering Pakistan Co-Author(s):    Muhammad Safdar and Muhammad Khurram Rashid

Abstract: Similarity transformations for a magnetohydrodynamic Casson fluid flow over a stretching surface have been derived earlier. These transformations map the flow partial differential equations to ordinary differential equations as there exist efficient analytic and numerical solutions schemes for the later as compared to the former. On the obtained system of equations the Runge-Kutta of order four has been engaged. Here we employ two analytic solution procedures namely Homotopy analysis and perturbation methods, on the reduced systems of ordinary differential equations corresponding to the considered flow and heat transfer model. We consider constant as well as variable viscosity and thermal conductivity to conduct this comparison of the homotopy analytic solution procedures. Further, we use finite boundary conditions instead of those at infinity that were employed to conduct this sort of study earlier. Both the analytic solution procedures are employed through codes developed on MAPLE. The effects of various parameters on velocity and temperature are presented through graphs and tables.