The numerical study of natural convection heat transfer inside a U-shaped enclosure with various thermal profiles by using copper–alumina/water hybrid nanofluid is reported. The study focuses on the effect of constant, linear, quadratic, and sinusoidal with different amplitudes and periods and absolute sinusoidal thermal profiles to the heat transfer efficiency inside the lid. A weighted residual Galerkin triangular finite element method is used alongside the damped Newton–Raphson algorithm to calculate the heat transfer rate of the system in terms of averaged and local Nusselt number. The result from the present model is compared with the previously published experimental and numerical data. The isotherms and streamlines of the fluid flow for each thermal profile are presented with a nanoparticle volume fraction of between 0 and 0.1 for Rayleigh number of 10⁴–10⁶. It is found that the constant heating profile gives the best heat dissipation performance, while the sinusoidal thermal profile performed the worst. The results gained from this paper will be useful in understanding how heat can be transferred effectively as it can be applied to many different industries, including the electronics field.

In this article, we present a new numerical approach for solving a class of systems of fractional initial value problems based on the operational matrix method. We derive the method and provide a convergence analysis. To reduce computational cost, we transform the algebraic problem produced by this approach into a set of $2\times 2$ nonlinear equations, instead of solving a system of 2 m × 2 m equations. We apply our approach to three main applications in science: optimal control problems, Riccati equations, and clock reactions. We compare our results with those of other researchers, considering computational time, cost, and absolute errors. Additionally, we validate our numerical method by comparing our results with the integer model when the fractional order approaches one. We present numerous figures and tables to illustrate our findings. The results demonstrate the effectiveness of the proposed approach.

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and $L_2$ errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of ${10}^{-14}$, while the $L_2$ error is on the order of ${10}^{-13}$. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one.

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.

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 $(j\in \mathbb{N})$ positively continuous and decaying functions in the finite interval $a\le t\le x$, 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.

Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes.