The primary purpose of this research is to demonstrate an efficient replacement double transform named the Laplace–Sumudu transform (DLST) to unravel integral differential equations. The theorems handling fashionable properties of the Laplace–Sumudu transform are proved; the convolution theorem with an evidence is mentioned; then, via the usage of these outcomes, the solution of integral differential equations is built.

In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted Rössler system, particularly for the detection of chaotic behavior. The enhanced NCFD method is used for reliable and accurate numerical simulations by capturing the intricate dynamics of chaotic systems. Further, analytical solutions are obtained using the H-LM for the fractional-order inverted Rössler system. This method is popular due to its simplicity, numerical stability, and ability to handle most initial values, yielding very accurate results. Combining analytical insights from the H-LM with the robust numerical accuracy of the NCFD approach yields a comprehensive understanding of this system’s dynamics. The advantages of the NCFD method include its high numerical accuracy and ability to capture complex chaotic dynamics. The H-LM offers simplicity and stability. The proposed methods prove to be capable of detecting chaotic attractors, estimating their behavior correctly, and finding accurate solutions. These findings confirm that NCFD- and H-LM-based approaches are promising methods for the modeling and solution of complex systems. Since these results provide improved numerical simulations and solutions for a broad class of fractional-order models, they will thus be of greatest use in forthcoming applications in engineering and science.

This paper investigates the dynamical properties of the fractional-order Arneodo system using a Grünwald–Letnikov-based numerical discretization. Fractional-order operators introduce memory and hereditary effects, enabling a more realistic description than classical integer-order models. The local stability of equilibrium points is examined through eigenvalue analysis of the Jacobian matrix, along with dissipativity conditions and the emergence of complex attractors. A comprehensive dynamical investigation is presented through phase portraits, time series, Lyapunov exponents, and bifurcation diagrams for varying fractional orders. Numerical findings demonstrate the emergence of new chaotic and hyperchaotic attractors. The results confirm that the fractional order strongly influences the system’s stability, sensitivity, and complexity. Our results confirm the relevance of fractional-order modeling in applications, such as secure communication, random number generation, and complex system analysis.

This paper presents an analytical solution framework for the coupled fractional Whitham–Broer–Kaup (FWBK) equations using the Sumudu transform

This study introduces a unified analytical framework for investigating the transient thermoelastic vibration of a micro-scale beam resting on dual-parameter foundations. We apply the fractional three-phase-lag (FTPL) generalized thermoelastic model to accurately characterize scale-dependent coupled thermal and elastic responses by incorporating complex thermal relaxation effects through the fractional derivative order. By employing the Laplace transform technique and its numerical inversion, we derive the coupled distributions of temperature, displacement, bending moment, and deflection within the beams. A comprehensive parametric analysis is conducted to quantify the distinct influence of the fractional factor and the foundation’s shear and stiffness parameters on the beam’s dynamic stability and propagation characteristics. The calculated results are systematically compared with established classical theories to validate the model’s robustness while simultaneously demonstrating the enhanced predictive capacity of the (FTPL) approach, particularly for characterizing thermal wave dispersion at the micro-scale. This research provides critical design criteria for advanced micro-electromechanical systems (MEMSs) where foundation stiffness and thermal inertial effects are intrinsically linked, offering novel insights into the tailored design of microstructural components.

In this paper we introduce a new class of semi generalized open sets, generalized semi closed sets in topological spaces, and studied some of its basic properties. Moreover we define approximately semi generalized open sets and approximately generalized semi closed sets in topological spaces. Further we obtained some properties of closure, semi generalized open sets and generalized semi closed sets in topological spaces.