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In this article, two radial basis functions based collocation schemes, differentiated and integrated methods (DRBF and IRBF), are extended to solve a class of nonlinear fractional initial and boundary value problems. Before discretization, the nonlinear problem is linearized using generalized quasilinearization. An interesting proof via generalized monotone quasilinearization for the existence and uniqueness for fractional order initial value problem is given. This convergence analysis also proves quadratic convergence of the generalized quasilinearization method. Both the schemes are compared in terms of accuracy and convergence and it is found that IRBF scheme handles inherent RBF ill-condition better than corresponding DRBF method. Variety of numerical examples are provided and compared with other available results to confirm the efficiency of the schemes. [ABSTRACT FROM AUTHOR]

Abstract: A ‘local’ radial basic function (RBF) based gridfree scheme has been developed to solve unsteady, incompressible Navier–Stokes equations in primitive variables. The velocity–pressure decoupling is obtained by making use of a fractional step algorithm. The scheme is validated over a variety of benchmark problems and found a very good agreement with the existing results. Comparisons with the benchmark solutions show that the developed local RBF gridfree scheme is stable and produces accurate results on domains discretized even with non-uniform distribution of nodal points. [Copyright &y& Elsevier]

In this paper, we prove an existence and uniqueness theorem for solving the nonlinear fractional differential equation of Caputo’s type of order q ∈ (0, 1] using the method of modified quasilinearization. The main theorem has been illustrated numerically using appropriate examples which shows that the proposed quasilinearization method is robust and easy to apply. [ABSTRACT FROM AUTHOR]