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Continuously differentiable radial basis functions ( C ∞ -RBFs) are the best method to solve numerically higher dimensional partial differential equations (PDEs). Among the reasons are: 1. An n-dimensional problem becomes a one-dimensional radial distance problem, 2. The convergence rate increases with the dimensionality, 3. Such RBFs possess spectral convergence.Finitely supported polynomial methods only converge at polynomial rates. C ∞ -RBFs have global support; the systems of equations may become computationally singular if the condition number exceeds the inverse machine epsilon, ε M . The solution to computational singularity is to decrease the effective ε M by either hardware or software methods. Computer scientists developed rapidly executable multi-precision packages. [ABSTRACT FROM AUTHOR]

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: The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach. [ABSTRACT FROM AUTHOR]

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]

A strictly conservative volume integral formulation of the time dependent conservation equations in terms of meshless radial basis functions (RBFs) is presented. Rotational and translational transformations are considered that simplify the partial differential equations (PDEs) to be solved. As a result, the solutions that are represented at a finite sample of knots, x∈Ω\∂Ω⊂Rd, are permitted to move as the system of equations evolves in time. Knots are inserted, deleted, or rearranged in such a manner to conserve the extensive physical quantities of mass, momentum components, and total energy.Our study consists of the following parts:(A) Local rotational and Galilean translational transformations can be obtained to reduce the conservation equations into steady-state forms for the inviscid Euler equations or Navier–Stokes equations.(B) The entire set of PDEs are transformed into the method of lines approach yielding a set of coupled ordinary differential equations whose homogeneous solution is exact in time.(C) The spatial components are approximated by expansions of meshless RBFs; each individual RBF is volumetrically integrated at one of the sampling knots xi, yielding a collocation formulation of the method of lines structure of the ODEs.(D) Because the volume integrated RBFs increase more rapidly away from the data center than the commonly used RBFs, we use a higher order preconditioner to counter-act the ill-conditioning problem. Domain decomposition is used over each piecewise continuous subdomain. [Copyright &y& Elsevier]

This paper presents a meshless numerical scheme to solve the inverse heat source time dependent problem. Fundamental solutions of heat equations and radial basis functions (RBFs) are used to obtain a numerical solution. Since the coefficient matrix may be ill-conditioned, the Tikhonov regularization (TR) method is employed to solve the resulting system of linear equations. Therefore, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. The accuracy and efficiency of the proposed method is illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the static and free vibration analysis of composite plates are performed, using a layerwise deformation theory and multiquadrics discretization. This meshless discretization method considers radial basis functions as the approximation method for both the equations of motion and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the natural frequencies. [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]