Applications of the homotopy perturbation method for some linear and non-linear partial differential equations

In this study, some linear PDEs and nonlinear PDEs are investigated using the homotopy perturbation method (HPM). The primary objective of this research is to employ the HPM as a tool for investigating a range of PDEs and extracting their analytical solutions. To clarify the practicality and efficacy of this method, we present illustrative examples of linear PDEs encompassing the classical heat, wave, and Laplace equations. Subsequently, a comparative analysis is performed, contrasting the outcomes derived from the HPM with established accurate solutions. Through this comparative approach, we aim to provide a comprehensive understanding of the HPM's applicability, robustness, and precision in solving a spectrum of PDEs. Our study contributes to the broader exploration of innovative mathematical techniques for tackling complex PDEs, while also shedding light on the potential advantages and limitations of the homotopy perturbation method in practical applications

Intuitionistic regular subspaces in intuitionistic topological spaces

This paper explores the concept of intuitionistic topological spaces, delving into their definitions and essential properties. It also examines intuitionistic topological subspaces, providing insights into their characteristics. Additionally, the paper investigates intuitionistic regular spaces and demonstrates their hereditary nature, specifically focusing on  and . To illustrate these concepts in practical terms, the paper presents two real-world examples of intuitionistic sets. Through a comprehensive analysis of intuitionistic topological spaces and their subsets, the study sheds light on the inheritability of regularity in these spaces. Furthermore, the work emphasizes the significance of intuitionistic topological spaces within the realm of mathematical research, showcasing their applicability through concrete instances of intuitionistic sets

A review of some properties of persistent homology

Every day, enormous complex geometric data are accumulating rapidly, and qualitative analysis is needed, which cannot be done properly without studying the shapes of those data. Persistent homology describes the homology of data sets of arbitrary size, producing state-of-the art results in data analysis across a significant number of fields and sparking a rigorous study of persistence in homology theory. In this study, persistent homology has been demonstrated as a homology theory by satisfying the Eilenberg-Steenrod axioms. A brief background on persistent homology groups has been written to understand their construction. Then other definitions of persistent homology based on functors and graded modules have also been reviewed. The Mayer-Vietoris-Vietorisfor persistent homology has been derived as a property of persistent homology. Subsequently, a long, exact sequence for persistent homology has been constructed. Furthermore, the stability of persistent homology has been examined carefully. Finally, the Diamond principle of persistent homology has been explained briefly. This study can be used to investigate new properties of persistent homology, among other benefits

A Two-Step Lagrange–Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region—Part I: Flat Bottom Topography

A two-step Lagrange–Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC) is presented. First, we show the experimental order of convergence to see the second-order accuracy in time realized by the two-step methods for conservative and non-conservative material derivatives along the trajectory of fluid particles. Second, we observe the effect of the TBC in a simple domain, and the artificial reflection is removed significantly when the wave touches the TBC. Third, we apply the scheme to a practical domain with islands, namely, the Bay of Bengal region, and observe the effect of the TBC again for the practical domain; the artificial reflections are removed significantly from the transmission boundaries on open sea boundaries. We also study the effect of a position of an open sea boundary with the TBC and reveal that it is sufficiently small to neglect. The numerical results in this study show that the scheme has the following properties: (i) the same advantages of Lagrange–Galerkin methods (the CFL-free robustness for convection-dominated problems and the symmetry of the matrices for the system of linear equations); (ii) second-order accuracy in time by the two-step methods; (iii) mass preservation of the function for the water level from the reference height (until the contact with the transmission boundaries of the wave); and (iv) no significant artificial reflection from the transmission boundaries. The numerical results by the scheme presented in this paper are for the flat bottom topography of the domain. In the next part of this work, Part II, the scheme will be applied to rapidly varying bottom surfaces and a real bottom topography of the Bay of Bengal region