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Abstract: Many important problems in physics, quantum chemistry, biology, economics, etc., are expressed as multi-dimensional (MD) partial differential equations (PDEs) that are difficult to solve with the dominant numerical techniques such as finite elements, difference, and volume methods. The main problem with multi-dimensional problems is the curse of dimensionality requiring increasingly more computer memory and speed. A radial basis function (RBF) method was used that possess the exponential convergence and is combined with overlapping domain decomposition to solve the inviscid time-dependent Burgers'' equations. Using a power law distribution of shape parameters, it was observed that for increasingly flat shape parameters, the maximum eigenvalues of the time advance matrix tend toward unity from above. Thus two goals were accomplished: (1) minimization of the number of discretization points and (2) stability of the time marching scheme. Domain decomposition methods simplifies the complexity of large domains into simpler structures. A literature search did not yield any previous solutions of the inviscid 4D Burgers''s equations. The exact solution of the inviscid Burgers'' equations have either a cosine or an exponential dependency. The max norm error average for each case was 0.002 and 0.0028 with a crude discretization pushing the MQ shape parameters to very flat limits. [Copyright &y& Elsevier]

Abstract: This study is a progress report that examines the numerical solution of inviscid hyperbolic partial differential equations (PDEs) without the need for upwind differencing and other numerical artifacts. The fixed frame PDEs are locally transformed by rotating and translating the coordinate system at each local discretization point. These transformations yield a simpler PDE system that is effectively linearized. It is assumed that in this transformed local frame within a time interval, , the dependent variables are products of the spatial dependent radial basis functions (RBFs), and the time dependent expansion coefficients, . This linearization is exploited by transforming the PDEs into systems of linear ordinary differential equations (ODEs) in terms of the expansion coefficients. The affine space decomposition is used to obtain an ODE system of ODEs in unknowns that can be integrated exactly in time. Then the entire set of N expansion coefficients is found. Numerical results show that hyperbolic PDEs can be integrated in time without upwinding and the root mean square errors between the exact and numerical solutions are indeed very small. [Copyright &y& Elsevier]

Abstract: A subdomain variational inequality and its meshless linear complementary formulation are developed in the present paper for solving two-dimensional contact problems. The subdomain variational inequality will be defined in detail. The meshless method is based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain and radial basis functions as trial functions for interpolation. Three different radial basis functions (RBFs), i.e. Multiquadrics (MQ), Gaussian (EXP) and Thin Plate Splines (TPS) are examined and the selection of their shape parameters is studied based on 2D solid stress problems with closed-form solutions. The developed meshless/linear complementary method is applied to solve two frictionless contact problems. For the RBFs, it has been found that the TPS shape parameter is not sensitive to nodal distance and a value of 4 is found as a good choice for TPS from this research. [Copyright &y& Elsevier]