General Symmetry Multi-Reduction Method for Partial Differential Equations with Conservation Laws 

Speaker: Prof. Stephen Anco 

Affiliation: Professor, Department of Mathematics & Statistics, Brock University, Canada 

Abstract: A powerful application of symmetries is finding symmetry-invariant solutions of nonlinear partial differential equations (PDEs). These solutions satisfy a reduced differential equation with one fewer independent variable. It is well known that a double reduction occurs whenever the starting nonlinear PDE possesses a conservation law that is invariant with respect to the symmetry.

Recent work has developed a broad generalization of the double-reduction method by considering the space of invariant conservation laws with respect to a given symmetry. In its simplest formulation, the generalization is able to reduce a nonlinear PDE in 2 variables to an ODE with m first integrals where m is the dimension of the space of invariant conservation laws. Nonlinear PDEs in 3 or more variables can be reduced to an ODE similarly by using an algebra of given symmetries. Importantly, the algebra does not need to be solvable.

The general method employs multipliers and is fully algorithmic. In particular, no a priori knowledge of conservation laws of the nonlinear PDE is necessary, and the multi-reduction is carried out in one step.

In this talk, a summary of the general multi-reduction method will be presented for obtaining invariant solutions of physically interesting PDEs. Examples will be shown for quadruple reduction from a single symmetry; complete integration from a solvable algebra in one step; reduction via a non-solvable algebra.nsion of the space of invariant conservation laws. Nonlinear PDEs in 3 or more variables can be reduced to an ODE similarly by using an algebra of given symmetries. Importantly, the algebra does not need to be solvable.

The general method employs multipliers and is fully algorithmic. In particular, no a priori knowledge of conservation laws of the nonlinear PDE is necessary, and the multi-reduction is carried out in one step.

In this talk, a summary of the general multi-reduction method will be presented for obtaining invariant solutions of physically interesting PDEs. Examples will be shown for quadruple reduction from a single symmetry; complete integration from a solvable algebra in one step; reduction via a non-solvable algebra.

Selected Case Studies from Mathematical Biology Using Differential/Difference Equations

Speaker: Prof. Cang Hui 

Affiliation: Stellenbosch University, South Africa

Symmetries of Differential Equations and Their Applications 

Speaker: Prof. Sibusiso Moyo

Affiliation: DVC Research, Innovation and Postgraduate Studies, Stellenbosch University, South Africa

Abstract: Most natural phenomena can be described in the form of Differential Equations. Understanding properties of differential equations and the types of transformations that can reduce them to more amenable and integrable forms is part of a bigger set of recent research results that have emanated from research on Lie groups, symmetry analysis and the integrability of differential equations. In this abstract, a review of a set of results pertaining to the treatment of differential equations and their applications using symmetry analysis is given with selected examples and illustrations. The aim is to delineate these methods with a view of applying them to emerging areas in the mathematical biology, mathematical physics and in the modelling of physical systems in general.

Keywords: Symmetries, Lie Symmetries, Noether Symmetries, Lie-groups, Infinitesimal Transformations, Group Theory, Integrable Systems, Matrices

On First Integrals and Reduction of Classes of Emden and Lienard Equations 

Speaker: Prof. Abdul Hamid Kara

Affiliation: Head of Department, School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

Abstract: We present a general method to construct first integrals for some classes of well known second-order ordinary differential equations, viz., the Emden and Lienard classes of equations. The approach does not require a knowledge of a Lagrangian, which is the usual, cumbersome route. A study of the invariance properties  are used to `twice' reduce the equations to solutions. 

On the Method of Differential Constraints 

Speaker: Prof. Sergey Meleshko

Affiliation: School of Mathematics, Institute of Science, Suranaree University of Technology, 30000,Thailand

Abstract: Almost all methods for finding exact particular solutions of partial differential equations require compatibility analysis of overdetermined systems. One method differs from the other in the way the overdetermined system is defined. One of such methods is the method of differential constraints, proposed by N.N. Yanenko [1]. The presentation gives a short review of the method [2]. The main part of the presentation is devoted to a particular class of solutions of the one-dimensional equations of continuum mechanics: solutions of this class are called the generalized simple waves, and they are obtained by the method of differential constraints. The class of generalized simple waves reduces to integrating a system of ordinary differential equations. Magneto- and gas- dynamics equations, and the equations describing the behavior of a nonlinear elastic material are considered in the talk. The general forms of differential constraints for the existence of the generalised simple waves of the models studied are obtained . In particular, solutions of the one-dimensional equations of the gasdynamics and the equations describing the behavior of a nonlinear elastic material, which are reduced to solving a system of homogeneous differential equations written in Riemann invariants, are considered. It is shown that the solution of any Cauchy problem for such systems admitting a differential constraint reduces to solving a system of ordinary differential equations. Examples of solutions for certain initial data are given. This presentation is dedicated to the memory of Russian academician Nikolay N. Yanenko (1921-1984).
References
[1] N. N. Yanenko. Compatibility theory and methods of integrating systems of nonlinear partial differential equations. In Proceedings of the Fourth All-Union Congress on Mathematics, page 613. Nauka, Leningrad, 1964.
[2] S. V. Meleshko. Methods for Constructing Exact Solutions of Partial Differential Equations. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2005.
[3] S. V. Meleshko, S. Moyo, and G. M. Webb. Solutions of generalized simple wave type of magnetic fluid. Communications in Nonlinear Science and Numerical Simulation, 103:105991, 2021.
[4] S. V. Meleshko and E. Schulz. Application of the method of differential constraints to systems of equations written in riemann invariants. Journal of Applied Mechanics and Technical Physics, 62(3):351-360, 2021.

Exact Solutions of One of Kinetic Models of Inelastic Processes

Speaker: Prof. Yu. N. Grigoriev 

Affiliation: Institute of Computational Technologies, 630090, Novosibirsk

Abstract: The talk is devoted to the construction of exact solutions to the system of two nonhomogeneous Boltzmann kinetic equations. The source functions in them model the integrals of double and triple inelastic collisions. An extension of the Lie group admitted by a system of homogeneous equations  is constructed, which is considered as an equivalence group for inhomogeneous equations. Conditions are found under which a transformation from the extended group vanishes the sources in the transformed equations. A class of sources linear in the distribution function is given, for which the generalized Bobylev-Krook-Wu [3, 4] solutions allowing physical interpretation are explicitly obtained . This work was supported financially by the Russian Science Foundation (project code 2311-00027).
References
[1] S. V. Meleshko. Methods for Constructing Exact Solutions of Partial Differential Equations. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2005.
[2] Yu. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev, and S. V. Meleshko. Symmetries of integro-differential equations and their applications in mechanics and plasma physics. Lecture Notes in Physics, Vol. 806. Springer, Berlin / Heidelberg, 2010.
[3] A. V. Bobylev. On exact solutions of the Boltzmann equation. Dokl. AS USSR, 225(6):1296.
[4] M. Krook and T. T. Wu. Formation of Maxwellian tails. Phys. Rev. Lett., 36(19):1107-1109, 1976.
[5] S. V. Meleshko, Yu. N. Grigoriev, A. Karnbanjong, and A. Suriyawichitseranee. Invariant solutions in explicit form of the Boltzmann equation with a source term. Journal of Physics: Conf. Series, (012063), 2017.
[6] Yu. N. Grigoriev and S. V. Meleshko. Equivalence group and exact solutions of the system of nonhomogeneous Boltzmann equations. Continuum Mech. Thermodyn., 35:2117-2124, 2023

Minisuperspace Description in Gravitational Physics and the Role of Symmetries as a Selection Rule for Dark Energy Models

Speaker: Prof. A. Paliathanasis

Affiliation: DUT, South Africa

Abstract: I will discuss the Lagrangian formulation of gravitational models and how the symmetries can be used for the description of dark energy. 

On a Family of Higher Order Difference Equations: Symmetries, Formula Solutions, Periodicity and Stability Analysis
Speaker: Prof. Mensah Folly-Gbetoula
Affiliation:  School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Abstract: We present formula solutions of a family of difference equations of higher order. We discuss the periodic nature of the solutions and we investigate the stability character of the equilibrium points. We utilize Lie symmetry analysis as part of our approach together with some number theoretic functions.
References
[1] Hydon, P.E., Difference Equations by Differential Equation Methods, Cambridge University Press: Cambridge, UK (2014).
[2] Aljoufi, L. S.; Almatrafi, M. B.; Seadawy, A. R., Dynamical analysis of discrete time equations with a generalized order, Alexandria Engineering Journal 64, 937-945, (2023).
[3] Grove, E. A.; Ladas, G., Periodicities in Nonlinear Difference Equations, Vol. 4; Chapman & Hall/CRC: Boca Raton, USA (2005).

On some Classical Partial Differential Equations

Speaker: Prof. Sameerah Jamal
Affiliation:  School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Abstract: We present some recent advances of the applications of Lie group transformations of classical partial differential equations. For example, we discuss the heat and Burgers' equation, which are often benchmarks in the study of differential equations. In the former case, the equation admits a fundamental solution and the latter equation may be linked to a recursion operator to gain further insights.

FLUID FRACTURE

Speaker: Prof. David Paul Mason
Affiliation:  School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Abstract:  The fracture of rock by high pressure fluids has many applications, for example in breaking rocks in mines, in geothermal reservoirs, in the formation of dykes and sills in geophysics and in the extraction of oil and gas trapped underground in shale rock. The bulk flow model for laminar and turbulent fluid flow in fractures is first described. Two models for the net normal stress which keeps the fracture open will be considered. In the PKN model the net normal stress is proportional to the half-width of the fracture which leads to a nonlinear diffusion equation. The Lie point symmetry and the conservation laws for the nonlinear diffusion equation are derived. The Lie point symmetries associated with the conserved vectors are determined and the invariant solutions that follow from the double reduction theorem are investigated. In the second model the net normal stress is expressed as a Cauchy principal value integral which leads to an integro-differential system of equations. The scaling transformation method of Dresner is used to write the system in similarity form. Further progress with this model uses asymptotic methods and contour integration which will not be considered in this talk.

A New Method for Discovering Lie Point Symmetries of First-Order ODEs Through Total Differentiation 

Speaker: Prof. Winter Sinkala
Affiliation: Department of Mathematical Sciences and Computing, Faculty of Natural Sciences, Walter Sisulu University, Private Bag X1, Mthatha 5117, Republic of South Africa

Abstract: Order reduction is a common approach to solving ordinary differential equations (ODEs). When an ODE admits a Lie point symmetry, the equation can be reduced to a simpler form using routines of Lie symmetry analysis. In the case of first-order ODEs, the reduction amounts to the complete integration of the equation. However, the main algorithm employed for finding Lie point symmetries of ODEs does not apply to first-order ODEs. A common approach to finding Lie point symmetries in this case is to guess the form of the infinitesimals of the infinitesimal generator of the symmetry. With some luck, this may lead to the discovery of some of the admitted Lie point symmetries. However, this approach can be time-consuming and not always successful. In this talk, we present a new approach to finding Lie point symmetries of first-order ODEs. Our approach exploits the connection between first-order and secondorder ODEs through total differentiation. We show that this connection can be used to derive a systematic algorithm for finding Lie point symmetries of firstorder ODEs. We also provide a number of illustrative examples to demonstrate the effectiveness of our approach.
References
[1] Cheb-Terrab E.S., Kolokolnikov T., First-order ordinary differential equations, symmetries and linear transformations, Eur. J. Appl. Math. 14(2), 231-246, (2013).

Toward an Extended Unified Method Solutions of Partial Differential Equations and Nonlinear Wave configurations

Speaker: Dr. Mohamed Tantawy
Affiliation: October 6 University, Egypt
Abstract: We present a summary of the extended unified method EUM for finding exact solutions to Partial Differential Equations PDEs with time or space dependent [1]. The solutions are classified as polynomial and rational function solutions with multi-auxiliary functions that satisfy appropriate ordinary differential equations ODEs. Direct and indirect nonlinear interactions are suggested via nonlinear combinations and bilinear transformations with nonlinear combinations of basic traveling wave solutions TWS. The EUM technique was adopted providing examples of PDEs with time dependent coefficients (Nonautonomies). They led self-similar-traveling wave of the  Nizhnik Novikov Veselov equation (NNVE) revealed that they exhibit symmetrical profiles about  and . Nonlinear waves were obtained by varying nonlinear and dispersion terms [2]. (3+1)-dimensional Yu-Toda-Sasa-Fukuyama 3D-YTSFE equation in space-time-dependent coefficients in two-layer heterogeneous liquid is constructed [3]. Solutions of the equations obtained, which contain arbitrary functions and their space and time derivatives. The solutions are evaluated explicitly and are represented in graphs. It is shown that they reveal abundant novel waves geometric structures. Also, the wave configurations are classified.

[1] H.I. Gawad, N.S. Elazab, M. Osman, Exact solutions of space dependent Korteweg-de Vries equation by the extended unified method, J. Phys. Soc. Japan 82, 044004, (2013)
[2] H. I Abdel-Gawad, M. Tantawy, J.A. Tenreiro Machado,Abundant structures of waves in plasma transitional layer sheath. Chin. J. Phys. 67, 147-154, (2020)
[3] M. Tantawy, H. I. Abdel-Gawad, Complex Physical Phenomena of a Generalized (3+1)- Dimensional Yu-Toda-Sasa-Fukuyama equation in a Two-Layer Heterogeneous Liquid. Eur. Phys. J. Plus. 137, 1001 (2022)

Lie Symmetries with Generalized Invariant Solutions and Dynamics Structures and Patterns of the (2+1)- and (3+1)-Dimensional Nonlinear Evolution Equations

Speaker: Dr. Sachin Kumar
Affiliation: Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi-110007, India

Abstract: In this talk, we will discuss obtaining a variety of generalized invariant solutions and demonstrate the dynamics of exact solutions to the highly nonlinear (2+1) and (3+1)-dimensional evolution equations. By taking advantage of the Lie symmetry technique, infinitesimals and a few similarity reductions are obtained from the governing equations. Using the required stages of Lie symmetry reductions, the equation is transformed into several nonlinear ordinary differential equations (NLODEs). Then, by solving the different resulting ODEs, we obtain plenteous invariant solutions in terms of the arbitrary functional parameters. Closed-form invariant solutions are successfully provided in the form of various dynamical waveforms of solutions, such as solitons and other combo-form solitons. Furthermore, utilizing computerized symbolic work, the dynamic and graphical representations of the derived solutions are displayed in 3D graphics. Therefore, this talk will be more advantageous and extremely helpful for new researchers in obtaining more generalized invariant solutions and their dynamical wave patterns for understanding the propagation of excitation waves in shallow water wave models and diverse nonlinear phenomena.

Keywords: Lie symmetry analysis; highly nonlinear evolution equation; Invariant solutions; wave patterns

Analysis of Flow and Pressure Variation Inside Horizontal Filter Chamber: Lie symmetry

Speaker: Dr. Tanki Motsepa
Affiliation: School of Computing and Mathematical Sciences, University of Mpumalanga, Mbombela Campus, South Africa
Abstract: Closed-form solutions improve various industrial processes by giving operators a greater grasp of how industrial processes work and operate. In pursuit of understanding the dynamics of flow and pressure during the filtration process, this study seeks to obtain closed-form solutions of momentum and pressure during the unsteady regime filtration process. Lie point symmetries are used to transform a system of PDEs equation representing the momentum and pressure variation inside the filter chamber into solvable ODEs. The ODEs representing the case study under investigation are then solved to obtain velocity and pressure solutions. Effects of physical parameters arising from the current flow dynamics are analysed to identify the parameters that yield the best permeates outflow.
References
[1] S. Das, Effect of constant suction and injection on MHD three dimensional couette flow and heat transfer through a porous medium, Journal of Naval Architecture and Marine Engineering 6, 41-51, (2010).

Linearisation of a System of Two Second-Order Stochastic Ordinary Differential Equations and Their Applications

Speaker: Dr. Thembisile Gloria Mkhize
Affiliation: DUT, South Africa
Abstract:  We give a new treatment for the linearization of two second-order stochastic ordinary differential equations. We provide the necessary and sufficient conditions for the linearization of these differential equations. The linearization criteria are given in terms of coefficients of the system. We further, consider the underlying group theoretic properties of a system of two linear second-order stochastic ordinary differential equations for selected cases. Some applications and occurrence of these types of equations are given.

Keywords: Linearisation, Stochastic Differential Equation, Ordinary Differential Equations

A Similarity Solution for a Pre-Existing Fluid Driven Fracture in a Permeable Rock

Speaker: Dr. M.W Nchabeleng
Affiliation: University of Pretoria, Pretoria, South Africa
Abstract: The problem of a two-dimensional, pre-existing, fluid-driven fracture propagating in a permeable rock is investigated. The fracturing fluid is a viscous, incompressible Newtonian fluid and the flow of fluid inside the fracture is laminar. The elasticity of the rock is modelled using the Cauchy principal value integral derived from linear elastic fracture mechanics. With the aid of lubrication theory, a nonlinear partial integrodifferential equation relating the fracture half-width to the pressure and leak-off velocity is derived. Similarity solutions are derived for the fracture half-width, pressure and leak-off velocity and are used to reduce the partial integrodifferential equation to an ordinary integro-differential equation. In order to close the problem, a model in which the leak-off velocity is proportional to the fracture half-width and the gradient of the fluid-rock interface was used. Numerical results are obtained for the fracture length, fracture half-width and the net fluid pressure.

[1] M. Nchabeleng, and A. Fareo, A similarity solution for a pre-existing fluid driven fracture in a permeable rock, International Journal of Non-Linear Mechanics, Volume 154, 104446, (2023).

Solutions of the Linear and Nonlinear Differential Equations and Applications

Speaker: Dr. M.O. Aibinu
Affiliation: Institute for Systems Science, Durban University of Technology, Durban, South Africa

Abstract: A powerful tool to test hypotheses, confirm experiments and simulate the dynamics of complex systems is mathematical modelling. Modelling of real-life phenomena often results in nonlinear differential equations whose exact solutions are unknown or difficult to obtain. A large class of dynamical systems are associated with delays as their natural components. The scenario of find a solution of a differential equation is often more complicated if it involves a delay. This study presents a method that is efficient in accuracy and computational time for solving linear and nonlinear differential equations that involve a delay. We applied the results of study to a problem in nuclear physics. We obtained the solutions of mathematical models that describe the decay of radioactive elements.
References
[1] First Author, First Author, Title of the paper, Journal Name Volume(Issue), Pages, (Year).
[2] E.K. Akgül, A. Akgül, M. Yavuz, New illustrative applications of integral transforms to financial models with different fractional derivatives, Chaos, Solitons & Fractals, 146, (2021), 110877.
[3] D. C. Tayal, Nuclear Physics, Himalaya Publishing House, 581 (2009).

 Invariance, Conservation Laws and Reductions of Some Classes of "High" Order Partial Differential Equations

Speaker: Tebogo Malatsi
Affiliation: School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Abstract: Using underlying invariance/symmetry properties and related/associated conservation laws, we investigate some 'high' order nonlinear equations. The multiplier method is mainly used to construct conserved vectors for these equations. When the partial differential equations are reduced to the nonlinear ordinary differential equation (NLODE), exact solutions for the ODEs are constructed and graphical representations of the resulting solutions are provided. In some cases, the solutions obtained are the Jacobi elliptic cosine function and the solitary wave solutions. We study the third-order 'equal width equation' followed by a new fourth-order nonlinear partial differential equation (NLPDE), which was recently established in the literature and, finally, the Korteweg-de Vries (KdV) equation having three dispersion sources.
References
[1] T. D. Malatsi and A. H. Kara, "Invariance, conservation laws and reductions of some classes of 'high' order partial differential equations," Transactions of the Royal Society of South Africa, vol. 77, no. 3, pp. 255-270, 2022.

Adaptive Strategies Destabilise the Rock-Paper-Scissors Game but Increase the Eco-Evolutionary Performance 

Speaker: Mmatlou Kubyana  
Affiliation: Stellenbosch University, South Africa
Abstract: The rock-paper-scissors (RPS) game is a classic model used in evolutionary game theory to explore the dynamics of how multiple strategies interact and evolve over time. The classic RPS game assumes a fixed benefit and cost for each strategy when its player interacts with another player of a specific strategy, while an evolutionary RPS game considers the dynamics of three populations whose fitnesses are influenced by the net payoff of each particular strategy. This may not accurately reflect the complexity of real-world scenarios. In this study, we introduce an adaptive evolutionary game (AEG) that captures simultaneously both the trait-mediated population dynamics, and the adaptive dynamics of traits, with the traits determining the benefits and costs of the payoff matrix in the RPS game through particular kernel functions. We investigate how strategies change through the trait evolution and how such adaptive changes can affect the population performance of each strategy and the presence and stability of Nash equilibrium. Our study reveals several key findings regarding the adaptive evolutionary RPS game. Firstly, the AEG reaches a steady state (asymptotic phase) more quickly than the evolutionary RPS game, in terms of population dynamics. Additionally, the stable coexistence of all strategies in the evolutionary RPS game is easily destabilised if allowed even small mutation rates in the AEG. We also show that adaptive strategies can enhance the population performance of the RPS game, as measured by average population densities at the steady state. Adaptive dynamics of gaming strategies exhibit diverse attractors in the trait space that depend both on the initial population densities and initial trait values, as well as parameters. Finally, the evolutionary attractors (i.e., the type of games) emerged in the AEG typically experience greater benefits than costs. These findings, thus, highlight the effect of adaptive strategies in the RPS game that destabilises the eco-evolutionary dynamics while increases the performance of each strategy.

Keywords: Evolutionary games, rock-paper-scissors, adaptive dynamics, traits, interactions 

Compactification of Gravitational Field Equations in Higher Dimensions

Speaker: Dr Robert S Bogadi
Affiliation: Institute for Systems Science, Durban University of Technology
Abstract: The Einstein field equations of general relativity, a coupled system of ten non-linear partial differential equations, are well suited to describing gravitational phenomena. On a cosmological scale, general relativity is less successful in accounting for the observed accelerated expansion of the universe and this has lead to investigations of higher dimensional systems. A popular choice is five-dimensional Einstein-Gauss-Bonnet gravity. Interpretation of the matter quantities in higher dimensions and comparison with quantities computed in standard four dimensional spacetime suggest the use of compactification. Although higher dimensions are presently undetectable, they assist in modified gravity work and are favourable from a mathematical perspective.

Diophantine sets

Speaker: Prof. Florian Luca
Affiliation: School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Abstract: A Diophantine m-tuple is a set of m-positive integers {a1, a2, . . . , am} such as the product of any two of them plus 1 is a square. For example, Fermat pointed out that {1, 3, 8, 120} is a Diophantine quadruple. Euler added the number

777480/8288641

as a fifth element of Fermat’s quadruple. As of 2019, it is known that this is the only rational number that can be added to the Fermat quadruple. There is no Diophantine qunituple of integers. In my talk, I will survey what is known about this subject along with variations of it with rational contents, or polynomial contents, or replacing the squares by larger powers, or by Fibonacci numbers, etc, or on how symmetries come into the picture.

Symmetries and Conservation Laws of Boundary Value Problems 

Speaker: Dr. Alexander Davison  
Affiliation: School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Abstract: This talk is about finding symmetries and associated conservation laws of boundary value problems. Symmetry methods are not always conducive to solving boundary value problems because boundary conditions may not always be compatible with symmetry generators. Conditions for symmetries to be admitted by a BVP are reviewed; such symmetries may then be used to find solutions and conservation laws. Examples are given.