Special Session 20: Lie Symmetries, Conservation Laws, and Other Approaches in Solving Nonlinear Differential Equations

Totally non-negative Pfaffian and its constructions

Jen-Hsu Chang National Defense University Taiwan Co-Author(s):    Jen-Hsu Chang

Abstract: The totally non-negative pfaffian (TNNP) is define for a skew-symmetric matrix such that all the sub-pfaffians are non-negative.It appears in the pfaffian structure of tau-function for the non-singular web solitons of the BKP equation . One constructs TNNP using the Perfect matching, chord diagram and the Dyck paths enumeration. Its tridiagonal form is also investigated.

A study of a fourth-order nonlinear generalized Ablowitz-Kaup-Newell-Segur water wave equation in fluid dynamics

Chaudry M Khalique North-West University So Africa Co-Author(s):

Abstract: In this talk we study a fourth-order nonlinear generalized Ablowitz-Kaup-Newell-Segur water wave equation in fluid dynamics. Utilizing the Lie theoretic approach, symmetries of the equation are obtained and then used to construct invariant solutions. Conclusively, we derive conserved quantities of the equation by employing both the general multiplier and Noether techniques.

Reduced integable equations and Darboux transformations

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

Abstract: We will explore new integrable equations via similarity transformations. Lax pairs will be used to derive reduced integrable equations, their binary Darboux transformations, and soliton solutions generated from the zero seed. Representative examples include reduced matrix nonlinear Schr\"odinger equations and modified Korteweg-de Vries equations.

Solutions and conservation laws of a (3+1)-dimensional Zakharov-Kuznetsov equation

Letlhogonolo LD Moleleki North-West University So Africa Co-Author(s):    B. Muatjetjeja, A.R. Adem

Abstract: In this talk, we study a nonlinear evolution partial differential equation, namely the (3+1) dimensional Zakharov Kuznetsov equation. Kudryashov method together with Jacobi elliptic function method is used to obtain the exact solutions of the (3+1) dimensional Zakharov Kuznetsov equation. Furthermore, the conservation laws of the (3+1) dimensional Zakharov Kuznetsov equation are obtained by using the multiplier method.

Role of OpenFOAM in Computatonal Fog Dynamics

Noor Muhammad King Fahd University of Petroleum and Minerals Saudi Arabia Co-Author(s):    Noor Muhammad, Taha Aziz, Haitham M.S. Bahaidarah

Abstract: Condensation occurs when the air temperature drops to its dew point, causing water vapors in the air to cool, lose kinetic energy, and transform into liquid droplets. This process is influenced by the relative humidity and temperature fluctuations, where lower temperatures reduce the air's capacity to hold water vapor, resulting in the formation of liquid water droplets on cooler surfaces. The condensation process is fundamental to the operation of fog harvesting systems, where specialized meshes capture the transformed water droplets. However, the behavior of droplets attached to meshes under background airflow is not well understood. Consequently, controlling the motion and merging of these droplets with neighboring ones poses a significant challenge. In this study, for fog airflow, it is demonstrated that droplets on parallel meshes can aerodynamically interact with both downstream and upstream neighbors at different temperatures. These interactions lead to diverse behaviors, including alignment, coalescence, and repulsion. This study explores the key factors influencing the efficiency of material and design used for fog harvesting systems, and environmental conditions such as fog density and wind speed. The dynamical model includes the single-phase transport equation along with the $k-\Omega$ SST (Shear Stress Transport (SST) k-omega) a subclass of RAS (Reynolds-Averaged Simulation) model. The computational analysis of fog harvesting mesh for water collection is performed in OpenFOAM (Open-source Field Operation And Manipulation). The Finite Volume Method (FVM) is employed for solution of the model to check the efficiency and effectiveness of fog harvesting computational designs. Using OpenFOAM, condensation, alignment and merging behaviors based on the interactions between wakes and droplets are visualized. The computational design enhances the surface area available for fog capture, thereby increasing the droplets collection efficiency and resulting in a higher yield of liquid water. The computational results obtained in this study can lead to more sustainable and efficient fog water collectors.

Approximate Symmetry Methods with Applications

Mehmet Pakdemirli Manisa Celal Bayar University Turkey Co-Author(s):

Abstract: Three common approximate symmetry methods with the newly proposed additional three methods are briefly discussed. Exact and approximate symmetry generators are calculated for sample ordinary and partial differential equations. The first three methods and the new three methods are contrasted within each other as well as with the exact symmetries. The characteristic features of the methods are exploited using sample problems. Approximate solutions are found using the various approximate symmetry theories and contrasted with each other and with the exact solutions.

The Invariance, Conserwation Laws and Integeration of Differential Equations

Ali Raza Stellenbosch University So Africa Co-Author(s):    Sibusiso Moyo, F.M. Mahomed, F.D. Zaman, A.H. Kara

Abstract: In this talk, we begin with symmetry classification for a class of nonlinear PDEs using the Lie symmetry method and discuss the fundamental simplifications of variables that arise from the Lie invariance criterion. Following this, we discuss the construction of the optimal system of subalgebras to identify unique group-invariant solutions via reductions under optimal system.