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For the problem of a nonlinear beam equation under nonlinear boundary conditions of moments, a fast iterative method is developed by transforming the ordinary differential equation into an integral one. The sinusoidal functions are used subtly as test functions as well as the bases of numerical solution in the calculation. Due to the orthogonality of the sinusoidal functions, the expansion coefficients of numerical solution in closed form can be found easily. Hence, the iterative scheme converges very fast to find numerical solutions with high accuracy.

A Fast New Algorithm for Solving a Nonlinear Beam Equation under Nonlinear Boundary Conditions