Abstract. This paper presents mesh-free procedures for solving linear di erential equations (ODEs and elliptic PDEs) based on Multiquadric (MQ) Radial Basis Function Networks (RBFNs). Based on our study of approximation of function and its derivatives using RBFNs that was reported in an earlier paper (Mai-Duy and Tran-Cong, 1999), new RBFN approximation procedures are developed in this paper for solving DEs, which can also be classi ed into two t ypes: a direct (DRBFN) and an indirect (IRBFN) RBFN procedure. In the present procedures the width of the RBFs is the only adjustable parameter according to a (i) = d (i) , where d (i) is the distance from the ith centre to the nearest centre. The IRBFN method is more accuarte than the DRBFN one and experience so far shows that can be chosen in the range 7 10 for the former. Di erent c o m binations of RBF centres and collocation points (uniformly and randomly distributed) are tested on both regularly and irregularly shaped domains. The results for a 1D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1:0e ; 4) and O(1:0e ; 8), respectively, with a centre density of 50. Similarly, the results for a 2D Poisson's equation show t h a t t h e DRBFN and the IRBFN procedures achieve a norm of error of at least O(1:0e;3) and O(1:0e ; 6), respectively, with a centre density o f 1 2 12.

This paper presents a numerical approach, based on Radial Basis Function Networks (RBFNs), for the approximation of a function and its derivatives (scattered data interpolation). The approach proposed here is called the indirect radial basis function network (IRBFN) approximation which is compared with the usual direct approach. In the direct method (DRBFN) the closed form RBFN approximating function is rst obtained from a set of training points and the derivative functions are then calculated directly by di erentiating such closed form RBFN. In the indirect method (IRBFN) the formulation of the problem starts with the decomposition of the derivative of the function into RBFs.

SUMMARYA numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a 'universal approximator' based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of 'finite element-type' discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re= 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of 11×11, Re=400 for the driven cavity flow with a density of 33×33 and Rayleigh number Ra= 1 000 000 for the natural convection flow with a density of 27 ×27.

SUMMARYThis paper is concerned with the application of radial basis function networks (RBFNs) for numerical solution of high order ordinary differential equations (ODEs).Two unsymmetric RBF collocation schemes, namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process, are developed to solve high order ODEs directly and the latter is found to be considerably superior to the former. Good accuracy and high rate of convergence are obtained with the proposed indirect method.

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Purpose -To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.Design/methodology/approach -The proposed method is based on a Cartesian grid and a one-dimensional integrated-radial-basis-function (1D-IRBF) scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme.Findings -The proposed method is verified by considering several test problems governed by second-and fourth-order PDEs; very accurate solutions are achieved using relatively coarse grids.Research limitations/implications -Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further development is needed for the case of non-rectangular domains.Originality/value -The contribution of this paper is a new effective collocation formulation based on RBFs for solving PDEs.

This paper presents a new effective radial basis function (RBF) collocation technique for the free vibration analysis of laminated composite plates using the first order shear deformation theory (FSDT). The plates, which can be rectangular or non-rectangular, are simply discretised by means of Cartesian grids. Instead of using conventional differentiated RBF networks, onedimensional integrated RBF networks (1D-IRBFN) are employed on grid lines to approximate the field variables. A number of examples concerning various thickness-to-span ratios, material properties and boundary conditions are considered. Results obtained are compared with the exact solutions and numerical results by other techniques in the literature to investigate the performance of the proposed method.

This paper is concerned with the use of oscillating particles instead of the usual frozen particles to model a suspended particle in the Dissipative Particle Dynamics (DPD) method. A suspended particle is represented by a set of basic DPD particles connected to reference sites by linear springs of very large stiffness. The reference sites, collectively modelling a rigid body, move as a rigid body motion calculated through their Newton-Euler equations, using data from the previous time step, while the velocities of their associated DPD particles are found by solving the DPD equations at the current time step. In this way, a specified Boltzmann temperature (specific kinetic energy of the particles) can be maintained throughout the computational domain, including the region occupied by the suspended particles. This parameter can also be used to adjust the size of the suspended and solvent particles, which in turn affect the strength of the shear-thinning behaviour and the effective maximal packing fraction. Furthermore, the suspension, comprised of suspended particles in a set of solvent particles all interacting under a quadratic soft repulsive potential, can be simulated using a relatively large time step. Several numerical examples are presented to demonstrate attractiveness of the proposed model.