In this paper, we investigate the existence, uniqueness, and Ulam stability of solutions for impulsive type integro-differential equations with generalized fractional derivative. The arguments are based upon the Banach contraction principle and Schaefer's fixed point theorem. \begin{keyword}Integro-differential equations \sep impulsive differential equations \sep generalized fractional derivative \sep existence \sep Ulam-Hyers stablity. \MSC{26A33\sep 34D10\sep 45N05.}

This manuscript presents Hyers-Ulam stability and Hyers-Ulam-Rassias stability results of non-linear Volterra integro-delay dynamic system on time scales with non-instantaneous impulses. Picard fixed point theorem is used for obtaining existence and uniqueness of solutions. By means of abstract Gronwall lemma, Gronwall's inequality on time scales and applications of Gronwall's inequality on time scales, we establish Hyers-Ulam stability and Hyers-Ulam-Rassias stability results. There are some primary lemmas, inequalities and relevant assumptions that helps in our stability results.

In this present study, the response characteristics of a flexible member carrying harmonic moving load are investigated. The beam is assumed to be of uniform cross section and has simple support at both ends. The moving concentrated force is assumed to move with constant velocity type of motion. A versatile mathematical approximation technique often used in structural mechanics called assumed mode method is in first instance used to treat the fourth order partial differential equation governing the motion of the slender member to obtain a sequence of second order ordinary differential equations. Integral transform method is further used to treat this sequence of differential equations describing the motion of the beam-load system. Various results in plotted curves show that, the presence of the vital structural parameters such as the axial force \(N\), rotatory inertia correction factor \(r^0\), the foundation modulus \(F_0\), and the shear modulus \(G_0\) significantly enhances the stability of the beam when under the action of moving load. Dynamic effects of these parameters on the critical speed of the dynamical system are carefully studied. It is found that as the values of these parameters increase, the critical speed also increase. Thereby reducing the risk of resonance and thus the safety of the occupant of this structural member is guaranteed.

In this paper, we study the self-similar surfaces in 4-dimensional Euclidean space \(\mathbb{E}^{4}\). We give an if and only if condition for a generalized rotational surfaces in \( \mathbb{E}^4 \) to be self-similar. In addition we examine self-similarity of some special surfaces in \( \mathbb{E}^4 \). Furthermore we investigate the self-similar condition of Tensor Product surfaces and Meridian surfaces in \(\mathbb{E}^{4}\).

In this work we present a fast and accurate numerical approach for the higher-order boundary value problems via Bernoulli collocation method. Properties of Bernoulli polynomial along with their operational matrices are presented which is used to reduce the problems to systems of either linear or nonlinear algebraic equations. Error analysis is included. Numerical examples illustrate the pertinent characteristic of the method and its applications to a wide variety of model problems. The results are compared to other methods.