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The purpose of this paper is to introduce an effective strategy for solving boundary layer flow of an Eyring-Powell fluid over a stretching sheet in unbounded domain. This paper introduces a combination of Modified Generalized Laguerre, the quasi-linearization method, and Collocation method for solving boundary layer flow of an Eyring-Powell fluid over a stretching sheet in unbounded domain. Applying this technique leads to obtaining suitable results and less error. Moreover, the effect of different values of the Eyring-Powell fluid material parameters is investigated and the results show that by increasing the parameter (m), velocity increases, and by increasing the parameter (n), velocity decreases.

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Various phenomena in biology, physics, and engineering are modeled by differential equations. These differential equations including partial differential equations and ordinary differential equations can be converted and represented as integral equations. In particular, Volterra-Fredholm-Hammerstein integral equations are the main type of these integral equations and researchers are interested in investigating and solving these equations. In this paper, we propose Legendre Deep Neural Network (LDNN) for solving nonlinear Volterra-Fredholm-Hammerstein integral equations (V-F-H-IEs). LDNN utilizes Legendre orthogonal polynomials as activation functions of the Deep structure. We present how LDNN can be used to solve nonlinear V-F-H-IEs. We show using the Gaussian quadrature collocation method in combination with LDNN results in a novel numerical solution for nonlinear V-F-H-IEs. Several examples are given to verify the performance and accuracy of LDNN.