Identifying common errors in polynomials of eighth grade students

This research aims to study and classify errors in polynomials made by secondary school students. The data for error identification was collected from exercise books of 72 eighth grade students. Three types of errors were examined: careless, computational, and conceptual errors. The errors were considered according to four topics in polynomials: similar terms of monomials; addition of polynomials; subtraction of polynomials; and multiplication of polynomials. It is found that students made the highest computational errors in identifying monomials’ similarity, which accounts for 17.86%. They have the highest percentage of making computational errors in the addition and subtraction of polynomials, which account for 10.88% and 12.04%, respectively. Lastly, they have the highest percentage of making careless errors in the multiplication of polynomials, which accounts for 14.44%. Furthermore, it can be seen that the source of errors is learners’ carelessness when writing the question and its answer. In addition, the basic knowledge of computing addition, subtraction, and multiplication of integers is the most crucial factor that leads to incorrect answers. Nevertheless, most students understand the principle of polynomials, but frequently make errors on other issues

Time-varying sliding mode controller for heat exchanger with dragonfly algorithm

This article proposes the design of a sliding mode controller with a time-varying sliding surface for the plate heat exchanger. A time-varying sliding mode controller (TVSMC) combines the benefit of the control system’s robustness and convergence rate. Using Lyapunov stability theory, the stability of the designed controller is proved. In addition, the controller parameters of the designed controller are specified optimally via the dragonfly algorithm (DA). The input constraint’s effect is considered in the controller design process by applying the concept of the auxiliary system. The bounded disturbances are applied to investigate the robustness of the proposed techniques. Moreover, the quasi-sliding mode controller (QSMC) is developed as a benchmark to evaluate the convergence behavior of the proposed TVSMC technique. The simulation results demonstrate the proposed TVSMC with the optimal parameters provided by the DA algorithm (TVSMC+DA) can regulate the temperature to the desired level under bounded disturbances. When compared to the QSMC method, the TVSMC+DA performs significantly faster convergence speed and greater reduction in chattering occurrence. The results clearly indicate that the proposed controller can enhance convergence properties while being robust to disturbances

Domain perturbation for parabolic equations

DOI: 10.1017/S000497271100293

Optimal control for obstacle problems involving time-dependent variational inequalities with Liouville–Caputo fractional derivative

Abstract We consider an optimal control problem for a time-dependent obstacle variational inequality involving fractional Liouville–Caputo derivative. The obstacle is considered as the control, and the corresponding solution to the obstacle problem is regarded as the state. Our aim is to find the optimal control with the properties that the state is closed to a given target profile and the obstacle is not excessively large in terms of its norm. We prove existence results and establish necessary conditions of obstacle problems via the approximated time fractional-order partial differential equations and their adjoint problems. The result in this paper is a generalization of the obstacle problem for a parabolic variational inequalities as the Liouville–Caputo fractional derivatives were used instead of the classical derivatives

An accelerated variant of the projection based parallel hybrid algorithm for split null point problems

In this paper, we consider an accelerated shrinking projection based parallel hybrid algorithm to study the split null point problem (SNPP) associated with the maximal monotone operators in Hilbert spaces. The analysis of the proposed algorithm provides strong convergence results under suitable set of control conditions as well as viability with the help of a numerical experiment. The results presented in this paper improve various existing results in the current literature

A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings

In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field

Analysis of a Fractional Variational Problem Associated with Cantilever Beams Subjected to a Uniformly Distributed Load

In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as a boundary value problem and obtain existence and uniqueness results in both L2 and L∞ settings. Additionally, we characterize the continuous dependence of the minimizers on varying loads in the energy functional. Moreover, an approximate solution is derived via the homotopy perturbation method, which is numerically demonstrated in various examples. The results show that the deformations are larger for smaller orders of the fractional derivative