In this essay, the nonlinear fractional integral equation is studied. Akbari-Ganji’s Method (AGM), Homotopy Perturbation Method (HPM) and Vibrational Iteration Method (VIM) are applied to obtain its solution. We present a new strategy for finding the approximate solutions to Fractional differential equations. We experience Fractional differential equations, which are broadly utilized in fluids. In this article, we have used analytical methods to check the correctness of the answers. Ordinary equations and fractional differential equations are related to entropy and wavelets, and so on. A few examples are employed to appear accurate and simple to implement and demonstrate the method. The solutions are clarified in convergent series. Some well-known models for anticipating the oscillation behavior of the action in a vibrating system are presented, then with the help of fractional calculus which could exceptionally powerful tool in mathematics and modeling of complex systems, a model for the same system is proposed. Compare the models and finally show that the proposed fractional model not only includes non-fractional models but also predicts the behavior of the system more comprehensively.

This study aimed at investigating the variation of heat transfer and velocity changes of the fluid flow along the vertical line on a surface drawn from both sides. In the beginning, the several parameters such as Prandtl number and viscoelastic effect evaluated for heat transfer and fluid velocity by variation Iteration method. The results were compared with the numerical method. The second part of the description relates to the use RSM method in the Design Expert software. In this paper by using the RSM method, optimized the fluid velocity and heat transfer passing from the stretching sheet. By increasing the Prandtl number, the convection heat transfer 43 % increased ratio the minimum Prandtl number. In accordance with balanced modes for Prandtl number and viscoelastic parameter and wall temperature, the best optimization occurred for fluid velocity and fluid temperature with f=0.67 and θ=0.606. The results of variation iteration method are accurate for the nonlinear solution. As the value of k increases, the value of fluid velocity indicates an increase and by increase Prandtl number, the value of Temperature decreases.