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In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of [Formula: see text] equations, we have simplified the problem to finding the solution for a set of [Formula: see text] nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam–Hyers stability and generalized Ulam–Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method.

A curved enclosure in attendance of MHD was evaluated in this article. Both outer cold and inner hot surfaces have curved shapes which boosts the convective rate. The impact of MHD in equations has been applied and there is one negative term is x direction if horizontal magnetic fields have been applied. Presence of such a field leads to generation of Lorentz force which prevents the nanofluid from moving faster. Such reduction in velocity with growth of Ha makes Nu decrease. Adding nanoparticles with greater “m” and φ can increase the convective flow. When Ra = 700, dispersing nano-powders can enhance the Nu in existence of MHD about 41.9%. When Ra = 150, Nu can improve about 12.01% if “m” is enhanced. Nu decreases around 40.1% with growth of Ha while it increases about 43.36% in existence of greater Ra.

The current article aims to simulate the promising influence of porous zones on treatment of working fluid within curved shaped containers under the impact of MHD. The MHD was applied in x direction which produces one negative source term in the y velocity equation. The features of base fluid have been improved with dispersing Al2O3 nanoparticles. The formulation based on Ψ was used to simplify the equations and finally the equations were solved via CVFEM. Results showed that dispersing nano-powders boost the convective rate about 39.34% in the presence of MHD. Increase in shape factor can make Nu increase about 11.46%. The applying MHD causes Nu to reduce about 54.93%. Augment of Ra makes Nu augments around 68.91%.

To evaluate the efficiency of a system involving wavy shapes of wall, numerical technique has been utilized in this article. The region between two wavy complex surfaces was fabricated from porous material and the working fluid is nanofluid (mixture of water with various shapes of alumina nanoparticles). The new terms related to porous media have been added according to Darcy law. The pressure terms were removed with defining stream function. The last form of equations contains two variables (θ and Ψ) which were solved via CVFEM. The results from the present code have been compared with previous article and good agreement was reported. With augmenting Ha, the speed of nanofluid decreases and Nu drops about 50.48% when Ra = 700. Considering greater buoyancy force makes Nu to increase about 61.36% when Ha = 0. The shape factor has a direct relation with the amount of conductivity, thus, Nu enhances about 11.97% with an increase of this factor when Ha = 15, Ra = 150.

The new computational approach has been incorporated for modeling the nanofluid flow through a cavity involving a wavy inner wall which experiences uniform flux. The fluid was created by loading alumina powders with various shapes within the water. The stream function formulation was applied after employing Darcy law and final equations have been generated. The container is under the impact of Lorentz force in addition to buoyancy forces. With loading additives, Nu enhances about 17.07% and 41.57% depending on the amount of Ha. The greater value of Ha makes the effect of adding nanoparticles become more sensible. With increasing the value of shape factor, Nu increases about 11.76%. As Ha grows, the rate of cooling decreases about 41.31%. With augment of Ra, Nu enhances about 50.41% in absence of Ha.

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 2×2 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 L2 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 L2 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.

For q∈0,1, we introduce the q-time deformed wave equation; using the q-derivative, the solution of the q-time deformed wave equation is given. Also, we introduce the free-time wave equation which is a limit case q⟶0 of the q-time deformed wave equation.

Let q∈0,1. We know that, when q tends to 1, we recover “classical” quantum mechanics. However, when q is not equal to one, we have a theory of quantum mechanics in a spacetime, i.e., a theory where the vacuum has a nonzero energy density. As a q-analogue on the space variable of the wave equation, we introduce the q-space wave equation. Solutions of these equations are given for q∈0,1 and for q⟶0. Since some states have more spin degrees of freedom than the physical particle has, the q-space wave equations remove the additional degrees of freedom.

In this paper, we introduce different equivalent formulations of variational principle. The language of differential forms and manifold has been utilized to deduce Euler–Lagrange equations in free coordinates. Thus, the expression is simple and global.