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In this paper, a well-conditioned collocation method is constructed forsolving general $p$-th order linear differential equations with various typesof boundary conditions. Based on a suitable Birkhoff interpolation, we obtain anew set of polynomial basis functions that results in a collocation scheme withtwo important features: the condition number of the linear system isindependent of the number of collocation points; and the underlying boundaryconditions are imposed exactly. Moreover, the new basis leads to exact inverseof the pseudospectral differentiation matrix (PSDM) of the highest derivative(at interior collocation points), which is therefore called the pseudospectralintegration matrix (PSIM). We show that PSIM produces the optimal integrationpreconditioner, and stable collocation solutions with even thousands of points.

A well-conditioned collocation method using pseudospectral integration matrix