In the present study, some perturbation methods are applied to Duffing equations having a displacementtime-delayed variable to study the stability of such systems. Two approaches are considered to analyze Duffingoscillator having a strong delayed variable. The homotopy perturbation method is applied through the frequencyanalysis and nonlinear frequency is formulated as a function of all the problem’ s parameters. Based on the Multiplescales homotopy perturbation method, a uniform second-order periodic solution having a damping part isformulated. Comparing these two approaches reveals the accuracy of using the second approach and further allowsstudying the stability behavior. Numerical simulations are carried out to validate the analytical finding.

Fluid contact structures as a branch of the Fluid–Structure Interaction (FSI) are among the physical models that have been able to present themselves as a new example of dynamic systems due to their very rich dynamics and Pay close attention to scientists. The dynamic behavior of two rigid straight articulated pipes conveying fluid is studied. The flow rate in the pipe is harmonic. The numerical results are compared with the present method and Runge–kutta 4th order for validation and an acceptable match between them is obtained. The method of Multiple time scales is used to drive the time response and phase plane curves. The influence of the initial velocity , ratio fluid mass per total fluid mass and mass of pipes and flow frequency on the time response and phase plane curves are examined. The results show, by increasing "u" _"0" and and decreasing the system is closer to loss of stability and increasing dynamic chaos.

In this paper, we introduce  a single integral transform that  defines all known time scales generalized integral transforms in the family of Laplace transform as the new general integral transform on time scales. As a result, a unified approach is developed for the use of integral transforms representing the family of Laplace transform for solving problems on continuous and discrete cases dynamics. The convergence  conditions and some principal properties accompanying the convolution theorem are given. It is shown that all generalized integral transforms on time scales included in the family of the Laplace transform are special cases of a new general integral transform. The applicability of this transform is demonstrated by solving certain ordinary dynamic equations and integral equations.

In this paper some new Hardy–type inequalities on time scales are derived and proved using the concept of superquadratic functions. Also, we extend Hardy–type inequalities involving superquadratic functions with general kernels to the case with arbitrary time scales. Several consequences of our results are given and their connection with recent results in the literature are pointed out and discussed.

In this paper, we present a generalization of the Montgomery Identity to various time scale versions, including the discrete case, continuous case, and the case of quantum calculus. By obtaining this generalization of Montgomery Identity  we establish results about the generalization of Ostrowski-Gr\"{u}ss like inequality to the several time scales, namely discrete case, continuous case and the case of quantum calculus. Additionally, we recapture several published results from different authors in various papers, thus unifying the corresponding discrete and continuous versions. Furthermore, we demonstrate the applicability of our derived consequence to the case of quantum calculus.

In this paper, a new technique for multi-scale smoothing of a free-form3-D surface is presented. This technique is a non-trivial generalization of the Curvature Scale Space (CSS) representation for 2-D contours. The (SS shape descriptor has been selected to be a part of the MPEG-7 package of standards. Complete triangulated models of 3-D objects are constructed (through fusion of range images) and. then, described at Multiple scales. This is achieved by convolving local parameterizations of the surface with 2-D Gaussian filters iteratively. The method, presented here for local parameterization makes use of semigeodesic or geodesic polar coordinates as a natural and efficient way of sampling the local surface shape. It is demonstrated that smoothing techniques using semi geodesic coordinates and geodesic polar coordinates produce similar results. The smoothing eliminates surface noise and details gradually. During the smoothing process, some surfaces can become very thin locally. Application of decimation followed by refinement removes very small or thin triangles and segments the modified surfaces into parts which are then smoothed separately. The technique presented here for 3-D multi-scale surface smoothing is independent of the underlying triangulation. It is also argued that the proposed technique is preferable to volumetric smoothing or level set methods. Since it is applicable to incomplete surface data which occurs during occlusion. Also surfaces that are not simply connected or have holes do not pose any problem. Furthermore due to employing 2-D convolutions rather than 3-D. this method is more efficient than other techniques.

In this research, the behavior of nonlinear vibrations of the viscoelastic Euler-Bernoulli beam under the influence of external fluid flow has been studied. The governing equations of motion are obtained by assuming Von-Karman nonlinear strain-displacement relations and considering the interaction between structure and fluid. To consider more realistic hypotheses, contrary to previous researches, the effect of viscoelastic behavior has been evaluated using a more complete and practical model called the Standard linear solid model. After non-dimensionalizing the motion equations, the governing nonlinear differential equations are discretized using the Galerkin method. Then, the system's analytical response is acquired through the method of Multiple time scales. After verifying the results and confirming the semi-analytical method's accuracy with the numerical solution results, different parameters' effect on the system's dynamic behavior has been analyzed. The results indicate that the viscoelastic behavior and the nonlinear model significantly affect the lock-in area and the maximum amplitude of the viscoelastic beam vibrations. In most studies on viscoelastic beams' vibrations, the damping effect in nonlinear terms has been neglected. However, this study demonstrates that the effect of damping on terms related to the nonlinearity of strain fields is substantial.

In this paper, we study the concept of generalized differentiability for fuzzy-valued functions on time scales. Using the derivative of the product of two functions, we provide solutions to first order linear fuzzy dynamic equations. We present some examples to illustrate our results.