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This paper proves convergence of variations of the unsymmetric kernel-based collocation method introduced by E. Kansa in 1986. Since then, this method has been very successfully used in many applications, though it may theoretically fail in special situations, and though it had no error bound or convergence proof up to now. Thus it is necessary to add assumptions or to make modifications. Our modifications will prevent numerical failure and allow a rigorous mathematical analysis proving errordoi:10.1137/050633366 fatcat:3zy6molmyrhabjkkgffleu3ogi