In the current manuscript, the flow of a thin film of a power-law fluid over a slippery topographical substrate is discussed.The von Karman–Pohlhausen method is used to develop the integral boundary layer (IBL)—model for the film thickness using an analytical approach.The findings are supported by earlier research on the resonance phenomena in gravity-driven Newtonian films with and without slippery properties. The novelty of this study lies in the influence of the slip condition at various power-law index n values.The study of the linear and non-linear resonance interactions between the free surface and the slippery undulating substrate yields expressions for the resonant Reynolds number for the largest free-surface amplitudes. The effects of shear-thinning and shear-thickening on the model system’s stability are taken into account when treating the slippery offer. The free surface amplitude is significantly affected by the slip length parameter’s rise, which has a major passive effect on the model system’s instability.

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

Model of the extended Dunkl oscillator based on Milovanović generalized Hermite polynomials on radial rays is discussed. Simple explicit realization of creation and annihilation operators in terms of difference-differential operators, coherent states are investigated.

Water pollution poses a significant threat to public health, particularly in developing nations. This study aims to enhance our understanding of water pollution transmission dynamics by introducing a novel mathematical model within a general fractional framework. The model offers a comprehensive analysis, including assessing solution feasibility and the stability of equilibrium points. To effectively implement this model, we develop an efficient numerical scheme based on the trapezoidal method. Furthermore, we conduct a thorough error discussion and convergence analysis. Notably, we devise and evaluate a new strategy to effectively tackle the associated fractional optimal control problem concerning water pollutant transmission. Applying this approach to a real case demonstrates its substantial impact, notably in reducing non-soluble contaminants within the water transmission system.

We consider a system of PDE’s describing the steady flow of an electrically conducting fluid in the presence of a magnetic field. The system of governing equations composes of the stationary non-Newtonian incompressible MHD equations coupled to the heat equation wherein the influence of buoyancy is taken into account in the momentum equation and the Joule heating and viscous heating terms are included. We proved the existence of C1,γ(Ω¯)×W2,r(Ω)×W2,2(Ω) solutions of the systems for 1<p<2 corresponding to a small data and we show that this solution is unique in case 6/5<p<2. Moreover, we also proved the higher regularity properties of this solution.

After the discovery of fractal structures of financial markets, fractional partial differential equations (fPDEs) became very popular in studying dynamics of financial markets. Available research results involves two key modelling aspects; firstly, derivation of tractable asset pricing models, those that closely reflects the actual dynamics of financial markets. Secondly, the development of robust numerical solution methods. Often times, most the effective models are of a nonlinear nature, and as such, reliable analytical solution methods are seldomly available. On the other hand, the accurate value of American options strongly lies on the unknown free boundaries associated with these types of derivative contracts. The free boundaries emanates from the flexibility of the early exercise features with American options. To the best of our knowledge, the approach of pricing American options under the fractional calculus framework has not been extensively explored in literature, and an obvious wider research gap still exist on the design of robust solution methods for pricing American option problems formulated under the fractional calculus framework. Therefore, this paper serve to propose a robust numerical scheme for solving time-fractional Black–Scholes PDEs for pricing American put option problems. The proposed scheme is based on the front-fixing algorithm, under which the early exercise boundaries are transformed into fixed boundaries, allowing for a simultaneous computation of optimal exercise boundaries and their corresponding fair premiums. Results herein indicate that, the proposed numerical scheme is consistent, stable, convergent with order O(h2,k), and also does guarantee positivity of solutions under all possible market conditions.

The terrible operating constraints of many real-world events cause systems to malfunction regularly. The failure of systems to perform their intended duties when they reach their lowest, highest, or both extreme operating conditions is a phenomenon that researchers rarely focus on. The multi-stress strength reliability R=P(W<X<Z) is deemed in this study for a component whose strength X falls between two stresses, W, and Z, where X, W, and Z are independently inverted Kumaraswamy distributed. Both maximum likelihood and maximum product spacing procedures are employed to obtain the reliability estimator under simple random sampling (SRS) and ranked set sampling (RSS) methodologies. Four scenarios for reliability estimators are considered. The reliability estimator in the first and second cases can be determined by applying the same sample design (RSS/SRS) to the strength and stress distributions. When the sample data for W and Z originate from RSS while those for X are acquired from SRS, the third reliability estimator is calculated. The drawn data of the strength and stress random variables, which are obtained from SRS and RSS, respectively, are taken into consideration in the final scenario. The effectiveness of the suggested estimators is compared using a comprehensive computer simulation. Lastly, three real data sets have been used to determine reliability estimators.

This research explores the complex interaction between piezoelectric waves and heat-moisture diffusion within a semi-infinite piezoelectric material under hygro-thermal conditions. By employing a two-dimensional Cartesian framework, novel governing equations for a thermo-piezoelectrically orthotropic medium influenced by moisture effects are developed. Accurate representations for key parameters are obtained by utilizing normal mode analysis. The investigation examines the influence of critical factors like moisture content, diffusivity, and temperature diffusivity on the spatial distribution of various physical fields. Additionally, a particular scenario of significance is highlighted. These results have the potential to improve sensor, actuator, and energy-harvesting device performance and dependability.

In this paper, we consider the nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including peakon, periodic peakon, solitary wave solutions and compactons) under different parameter conditions.

The inverse problem concerns how to reconstruct the operator from given spectral data. The main goal of this paper is to address nonlinear inverse Sturm–Liouville problem with multiple delays. By using a new technique and method: zero function extension, we establish the uniqueness result and practical method for recovering the nonlinear inverse problem from two spectra.