This article focuses on the study of fractional integral operators involving the ‐function. Two main theorems are established that present new fractional integral formulas associated with the ‐function. Moreover, several well‐known results related to various special functions can be derived as particular cases by assigning suitable parameter values to the general formulas. The proposed results encompass and extend the fractional integral operators previously investigated by Saxena and Kumbhat, Saigo, Erdélyi‐Kober, and Riemann‐Liouville, thereby providing a broader framework within the theory of fractional calculus and special functions. MSC2020 Classification: 26A33, 33B15, 33C05, 33C20, 44A10, 44A20.

To satisfy the power needs of human beings without affecting the environment, scientists have been working on the efficient practices of renewable energy sources such as solar, water, and wind. As a source of energy resources, solar radiation is preferred because of its unlimited availability and low ecological impact. Thus, this article examines the heat transfer of unsteady electrically conducting viscous nanofluid flowing over a stretchable cylindrical surface in the presence of solar radiation. The entropy generation is also analyzed with the velocity slip and convective heat transfer boundary conditions. Considering Beer’s law for representing solar radiation, the horizontal cylindrical surface for nanofluid flow, and solving the model by the homotopy analysis method (HAM) can be the novelty of this study. The governing nonlinear partial differential equations (PDEs) are transformed into systems of higher‐order nonlinear ordinary differential equations (ODEs) using appropriate similarity transformations. These ODEs are then solved via the HAM, applying the BVPh2.0 package on Mathematica 12.1. Comparisons with previously published studies confirm the validity of the method and highlight its consistency. The results reveal that the presence of a magnetic field interaction slows down the flow while increasing both local skin friction and the temperature of the nanofluid. Solar radiation and the Eckert and Biot numbers enhance the nanofluid’s temperature, whereas the Prandtl number and the unsteady parameter do not. Variations in temperature and velocity slip are found to reduce entropy generation. When the magnetic field interaction increased by 0.2, the local Nusselt number decreased by 0.1%, whereas the local skin friction rose by 6%. However, both the local Nusselt number and skin friction rose by 0.3% when the curvature parameter increased by 0.2. The findings demonstrate that the flow of nanofluid can be used to transfer heat from solar, which has practical applications in cooking, water heating, and electricity generation. Therefore, the global demand for energy can be partly met by harnessing solar energy effectively.

This article aims to present a new fixed point (FP) result for interpolative Kannan type and Ćirić‐Reich‐Rus type cyclic contractions (CRRTCCs) in dislocated quasi‐rectangular b‐metric spaces. We go through rigorous steps to prove the existence of a unique FP for the stated mapping in the setting of dislocated quasi‐rectangular b‐metric spaces. Our results generalize recent and related findings in literature. We provided a nontrivial example that justifies our findings. We also demonstrate the application of our result Theorem 3 on the existence of a unique solution for a nonlinear Fredholm integral equation. MSC2020 Classification 47H10, 54H25.

This article discusses a few different types of singular integral equations of the convolution type with Carleman shift in class {0}. By using the theory of Fourier analysis, these equations under consideration are transformed into Riemann–Hilbert boundary value problems for analytic functions with shift and discontinuous coefficients. For such problems, we propose a method different from the classical ones, and we obtain the analytic solutions and the conditions of Noether solvability.MSC2010 Classification: 45E10, 45E05, 30E25

Brucellosis remains a significant public health and economic burden, especially in developing nations where livestock–human interactions are frequent. This paper presents a partial differential equation (PDE) model to capture the transmission dynamics of brucellosis, with a focus on age‐specific impacts in humans and domestic animals. Using the integrated semigroup approach, the positivity and boundedness of the model were established ensuring biologically meaningful solutions. The basic reproduction number (R0) was derived through the Lotka–Sharpe–McKendrick integral equation, and the disease‐free equilibrium (DFE) was proven to be locally asymptotically stable when R0 < 1. Global sensitivity analysis was performed using the PAWN method to identify key parameters influencing disease dynamics. Results revealed that humans aged 0–72 years are carriers of the disease, while domestic animals, particularly those aged 0–5 and 11–15 years, act as significant reservoirs. Furthermore, environmental contamination was found to be a critical driver of transmission, emphasizing the urgent need for improved sanitation and hygiene measures. This study underscores the importance of targeting specific age groups and addressing environmental factors to enhance brucellosis control strategies, providing valuable insights for policymakers and public health practitioners.

This study introduces a novel extension of the Wright function using the Macdonald function as an extension of the Pochhammer symbol. We establish integral, differential, and generating function formulas for this new function. Furthermore, we apply it to fractional differential equations, providing integral transformations of Cauchy‐type problems with graphical representation. The Mellin and Rishi transformations are also derived for the extended Wright function. These results offer new insights and potential applications in fractional calculus (FC), mathematical physics, and engineering.MSC2020 Classification: 26A33, 33B15, 33C05, 65D20, 33E20, 91B25

We consider an advection‐diffusion equation involving a fractional Laplace operator of order s ∈]0; 1]∖{1/2}. Using a combination of fractional left and right Riemann–Liouville derivatives of order 2 s to approximate the fractional Laplace operator, we construct a numerical scheme using the Crank–Nicolson method. Using the Crank–Nicolson scheme, we succeeded in putting the numerical scheme of the problem under consideration in the form of a strictly and diagonally dominant positive definite matrix. This has allowed us to prove that the numerical scheme is stable and converges to first order in time and space for s ∈]0; 1]∖{1/2}. Numerical tests are performed to illustrate the results. MSC2020 Classification : 35R11; 35S15; 65M12.