To date, researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partial\ndifferential equations and also based on polynomial basis. But the principal importance of this paper is to develop the expansion\napproach based on general basis functions (in particular case polynomial basis) for solving general operator equations, wherein the\nparticular cases of our development are integral equations, ordinary differential equations, difference equations, partial differential\nequations, and fractional differential equations. In other words, this paper presents the expansion approach for solving general\noperator equations in the form Lu + Nu = g(x), x ? ?, with respect to boundary condition Bu = ?, where L, N and B are\nlinear, nonlinear, and boundary operators, respectively, related to a suitable Hilbert space, ? is the domain of approximation, ???? is\nan arbitrary constant, and g(x) ? L2(?) is an arbitrary function. Also the other importance of this paper is to introduce the general\nversion of pseudospectral method based on general interpolation problem. Finally some experiments show the accuracy of our\ndevelopment and the error analysis is presented in L2(?) norm.
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