Appl. Math. J. Chinese Univ. 2021, 36(3): 354-369 Laguerre reproducing kernel method in Hilbert spaces for unsteady stagnation point flow over a stretching/shrinking sheet M. R. Foroutan A. S. Gholizadeh Sh. Najafzadeh R. H. Haghi Abstract. This paper investigates the nonlinear boundary value problem resulting from the exact reduction of the Navier-Stokes equations for unsteady magnetohydrodynamic boundary layer flow over the stretching/shrinking permeable sheet submerged in a moving fluid. To solve this equation, a numerical method is proposed based on a Laguerre functions with reproducing kernel Hilbert space method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. §1 Introduction The magnetohydrodynamic flow and incompressible fluid over a stretching/shrinking sheet has attracted the attention of many researchers recently in view of its applications in multiple engineering problems such as magnetohydrodynamic generators, manufacturing processes of polymer, nuclear reactors, glass fiber production and geothermal energy extractions. Crane [10] was the first who studied the two-dimensional steady flow of an incompressible viscous fluid caused by a linearly stretching plate and obtained an exact solution in closed analytical form. Numerous studies have been conducted afterward to explore various aspects of the flow over a steady stretching sheet [1, 13, 30]. Recently, Mohamed et al. [24] studied the stagnation point flow over a stretching sheet and Hayat et al. [25] investigated the flow of a second grade fluid over a stretching surface with Newtonian heating. Most scientific problems like two-dimensional viscous flow between some expanding or contracting walls with various permeability rates and other fluids in mechanic are essentially nonlinear. There are many theoretical and experimental methods used to solve these Received: 2019-02-19. Revised: 2019-09-11. MR Subject Classification: 34B15, 76D03, 65M70. Keywords: nonlinear boundary value problem, Laguerre reproducing kernel method, operational matrix of derivative, existence and nonexistence of solutions, approximate solution. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-021-3761-2.