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This article reports a new numerical procedure, which is based on integrated radial basis functions (IRBFs) and Cartesian grids, for solving time-dependent differential problems that can be defined on non-rectangular domains. For space discretizations, compact five-point IRBF stencils are utilized. For time discretizations, a two-point IRBF scheme is proposed, where the time derivative is approximated in terms of not only nodal function values at the current and previous time levels but also nodal derivative values at the previous time level. This allows functions other than a linear one to also be captured well on a time step. The use of the RBF width as an additional parameter to enhance the approximation quality with respect to time is also explored. Various kinds of test problems of heat transfer and fluid flows are conducted to demonstrate the the attractiveness of the present compact approximations. [ABSTRACT FROM AUTHOR]