This article investigates the problem of simultaneous attitude and vibration control of a flexible spacecraft to perform high precision attitude maneuvers and reduce vibrations caused by the flexible panel excitations in the presence of external disturbances, system uncertainties, and actuator faults. Adaptive integral sliding mode control is used in conjunction with an attitude actuator fault iterative learning observer (based on sliding mode) to develop an active fault tolerant algorithm considering rigid-flexible body dynamic interactions. The discontinuous structure of fault-tolerant control led to discontinuous commands in the control signal, resulting in chattering. This issue was resolved by introducing an adaptive rule for the sliding surface. Furthermore, the utilization of the sign function in the iterative learning observer for estimating actuator faults has not only enhanced its robustness to external disturbances through a straightforward design, but has also led to a decrease in computing workload. The strain rate feedback control algorithm has been employed with the use of piezoelectric sensor/actuator patches to minimize residual vibrations caused by rigid-flexible body dynamic interactions and the effect of attitude actuator faults. Lyapunov's law ensures finite-time overall system stability even with fully coupled rigid-flexible nonlinear dynamics. Numerical simulations demonstrate the performance and advantages of the proposed system compared to other conventional approaches.

In this research paper, we introduce a new transform belonging to the class of Laplace transforms, called the Yasser-Jassim Transform (YJ Transform). We explore its properties and compare it to the classical Laplace transform. Furthermore, we provide proofs for the key properties associated with this transform and demonstrate its application in solving differential and integral equations. By employing this new transform, we can reduce the original problem to an algebraic equation that can be solved directly, followed by applying the inverse transform to obtain the solution to the original problem.

Weighted integral inequalities for general integral operators on monotone positive functions with parameters $p$ and $q$ are established in [4]. The aim of this work is to extend the results to different cases of these parameters, in particular for negative $p$ and $q$. We give some new lemmas which will be frequently used in the proofs of the main theorems.

These days, integral equations govern a wide range of physical processes, including those related to electricity, mechanics, fluid dynamics, ecology, soil moisture dynamics, shallow water wave propagation, and chemical science. As a result, creating new techniques and putting them into practice for solving integral equations becomes more crucial. In this paper, the Upadhyaya integral transform is employed for determining the analytical solutions of the non-linear Volterra integral equations (NVIEs) of the first kind. Four examples suggest the effectiveness of the Upadhyaya integral transform, particularly for NVIEs of the first kind. The calculation results suggest that the proposed method provides accurate solutions to the original problem and this method is a valuable tool for researchers and scientists working on the broader range of problems involving NVIEs of the first kind.

In this paper we specify the class of integral circulant graphs ICG(n; D), which can be characterized by their order n and the set D of positive divisors of n in such a way that they have the vertex set Zn and the edge set (Formola). This group of graphs is called BMM graphs because of the form of its set of edges. A bipartite G graph is a graph whose vertex set can be divided into two subsets X, and Y such that no two vertices in X and no two vertices in Y are adjacent. The duplicate graph is called complete if each vertex in X is connected to all vertices in Y. This graph is represented by K_ (m, n), if | X | = m and | Y | = n. Multipartite graphs are also defined as bipartite graphs.

The matching polynomial of a graph has coefficients that give the number of matchings in the graph. In this paper, we determine all connected graphs on eight vertices whose matching polynomials have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We show that there are exactly two matching integral graphs on eight vertices.

This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Block-pulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results.