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The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space  D ′ while their Fourier transforms belong to  Z ′. Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved. [ABSTRACT FROM AUTHOR]

Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of Taylor's series and the composition approach was used. One of the algorithms had an eighth order of convergence and a high-efficiency index of approximately 1.5157, which was higher than that of some existing algorithms, while the other possessed fourth-order convergence. The convergence analysis was carried out in both senses, that is, local and semi-local convergence. Various complex polynomials of different degrees were considered for visual analysis via the basins of attraction. We analyzed and compared the proposed algorithms with other existing algorithms having the same features. The visual results showed that the modified algorithms had a higher convergence rate compared to existing algorithms. Real-life systems related to chemistry, astronomy, and neurology were used in the numerical simulations. The numerical simulations of the test problems revealed that the proposed algorithms surpassed similar existing algorithms established in the literature. [ABSTRACT FROM AUTHOR]

This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j ∈ N) positively continuous and decaying functions in the finite interval a ≤ t ≤ x , the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results. [ABSTRACT FROM AUTHOR]

In the presence of Banach spaces, a novel iterative algorithm is presented in this study using the Chatterjea–Suzuki–C (CSC) condition, and the convergence theorems are established. The efficacy of the proposed algorithm is discussed analytically and numerically. We explain the solution of the Caputo fractional differential problem using our main result and then provide the numerical simulation to validate the results. Moreover, we use MATLAB R (2021a) to compare the obtained numerical results using the new iterative algorithm with some efficient existing algorithms. The work seems to contribute to the current advancement of fixed-point approximation iterative techniques in Banach spaces. [ABSTRACT FROM AUTHOR]

Self-similarity is a common feature among mathematical fractals and various objects of our natural environment. Therefore, escape criteria are used to determine the dynamics of fractal patterns through various iterative techniques. Taking motivation from this fact, we generate and analyze fractals as an application of the proposed Mann iterative technique with h-convexity. By doing so, we develop an escape criterion for it. Using this established criterion, we set the algorithm for fractal generation. We use the complex function f (x) = x n + c t , with n ≥ 2 , c ∈ C and t ∈ R to generate and compare fractals using both the Mann iteration and Mann iteration with h-convexity. We generalize the Mann iterative scheme using the convexity parameter h (α) = α 2 and provide the detailed representations of quadratic and cubic fractals. Our comparative analysis consistently proved that the Mann iteration with h-convexity significantly outperforms the standard Mann iteration scheme regarding speed and efficiency. It is noticeable that the average number of iterations required to perform the task using Mann iteration with h-convexity is significantly less than the classical Mann iteration scheme. Moreover, the relationship between the fractal patterns and the input parameters of the proposed iteration is extremely intricate. [ABSTRACT FROM AUTHOR]

The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them. [ABSTRACT FROM AUTHOR]

Conservation and storage of energy are important issues for human beings in the 21st century. A significant part of the energy is consumed for space cooling in buildings during summer. Passive cooling systems are used due to their low-power consumption; they, however, have some limitations such as lack of sufficient low temperature for the storage of coolth, in various localities. These problems make passive cooling technologies suitable in limited regions and scenarios. In this work, the inclusions of nanomaterials were examined to accelerate the freezing of phase changing material (PCM), with the applications of fins. The base (PCM) is water and CuO nanoparticles were used in this investigation to upgrade the performance. KKL is used to estimate the properties of CuO–water nanofluids. The finite element method (FEM) was employed to simulate this unsteady process. The outcomes indicate that incorporating CuO nanoparticles into pure PCM speeds up solidification process. The total energy and average temperature profiles decrease as the number of undulations upsurges, while the solid fraction profile rises. Furthermore, for d p = 4 0 nm, the uppermost rate of solidification is acquired. Increasing the concentration of CuO can make the freezing time to decrease by about 10.11% and 12.85% when the utilized shape factors are 3 and 5.7, respectively. Using particles with a higher shape factor causes the period to decrease by about 4% and 6.9% when concentration is 0.02 and 0.04, respectively. [ABSTRACT FROM AUTHOR]

Graph theory is a branch of mathematics that is crucial to modelling applicable systems and networks using matrix representations. In this article, a novel graph-theoretic model was used to assess an urban water distribution system (WDS) in Saudi Arabia. This graph model is based on representing its elements through nodes and links using a weighted adjacency matrix. The nodes represent the points where there can be a water input or output (sources, treatment plants, tanks, reservoirs, consumers, connections), and links represent the edges of the graph that carry water from one node to another (pipes, pumps, valves). Four WDS benchmarks, pumps, tanks, reservoirs, and external sources were used to validate the framework at first. This validation showed that the worst-case scenarios for vulnerability were provided by the fault sequence iterating the calculation of the centrality measurements. The vulnerability framework's application to the Saudi Arabian WDS enabled the identification of the system's most vulnerable junctions and zones. As anticipated, the regions with the fewest reservoirs were most at risk from unmet demand, indicating that this system is vulnerable to the removal of junctions and pipes that are intricately associated with their neighbours. Different centrality metrics were computed, from which the betweenness centrality offered the worst vulnerability prediction measures. The aspects and zones of the WDS that can more significantly impact the water supply in the event of a failure were identified by the vulnerability framework utilising attack tactics. [ABSTRACT FROM AUTHOR]