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Abstract
In this paper I introduce a new method for solving partial and ordinary differential equations with large first, second, and third derivatives of the solution in some part of the domain using the finite-element technique (here called the Galerkin-Gokhman method) The method is based on the application of the Galerkin method to a modified differential equation. The exact solution of the modified equation is the Galerkin approximation for the unknown function with exact values of the unknown at the nodal points.
An application of the Galerkin-Gokhman method to a general second-order nonlinear ordinary differential equation and to Navier-Stokes equations in the case of developing flow in a pipe is formulated. I also include the results of an application of the Galerkin-Gokhman method to two specific ordinary differential equations. One is y — dy/dx = 0, the other one is a second-order nonlinear equation describing fully developed turbulent flow in a pipe.