Due to the high approximation power and simplicity of computation of smooth radial basis functions (RBFs), in recent decades they have received much attention for function approximation. These RBFs contain a shape parameter that regulates their approximation power and stability but its optimal selection is challenging. To avoid this difficulty, this paper follows a novel and computationally efficient strategy to propose a space of radial polynomials with even degree that well approximates flat RBFs. The proposed space, $\mathcal{H}_n$, is the shifted radial polynomials of degree $2n$. By obtaining the dimension of $\mathcal{H}_n$ and introducing a basis set, it is shown that $\mathcal{H}_n$ is considerably smaller than $\mathcal{P}_{2n}$ while the distances from RBFs to both $\mathcal{H}_n$ and $\mathcal{P}_{2n}$ are nearly equal. For computation, by introducing new basis functions, two computationally efficient approaches are proposed. Finally, the presented theoretical studies are verified by the numerical results.

In this paper, an adaptive meshless method of line is applied to distributethe nodes in the spatial domain. In many cases in meshless methods, it isalso necessary for the chosen nodes to have certain smoothness properties. The set of nodes is also required to satisfy certain constraints. In this paper, one of these constraints is investigated. The aim of this manuscript is theimplementation of an algorithm for selection of the nodes satisfying a givenconstraint, in the meshless method of line. This algorithm is applied to someillustrative examples to show the e ciency of the algorithm and its ability toincrease the accuracy.

In this paper, we decide to select the best center nodes of radial basis functions by applying the Multiple Criteria Decision Making (MCDM) techniques. Two methods based on radial basis functions to approximate the solution of partial differential equation by using collocation method are applied. The first is based on the Kansa’ s approach, and the second is based on the Hermite interpolation. In addition, by choosing five sets of center nodes: Uniform grid, Cartesian, Chebyshev, Legendre and Legendre-Gauss-Lobato (LGL) as alternatives and achieving the error, condition number of interpolation matrix and memory time as criteria, rating of cases with the help of PROMETHEE technique is obtained. In the end, the best center nodes and method is selected according to the rankings. This ranking shows that Hermite interpolation by using non-uniform nodes as center nodes is more suitable than Kansa’ s approach with each center nodes.

A numerical technique based on hybrid of radial basis functions including Guassians (GAs) and Multiquadrics (MQs) is proposed to obtain the solution of nonlinear Fredholm integral equations. Zeros of the shifted Legendre polynomials are used as the collocation points. The integral involved in the formulation of the problems are approximated based on Legendre-Gauss-Lobatto integration rule. Some numerical examples illustrate the accuracy and validity of the proposed method.

In this paper, we use the rational radial basis functions ( RRBFs) method to solve the Korteweg-de Vries (KdV) equation, particularly when the equation has a solution with steep front or sharp gradients. We approximate the spatial derivatives by RRBFs method then we apply an explicit fourth-order Runge-Kutta method to advance the resulting semi-discrete system in time. Numerical examples show that the presented scheme preserves the conservation laws and the results obtained from this method are in good agreement with analytical solutions.

In this work, we apply the radial basis functions for solving the time fractional diffusion-wave equation defined by Caputo sense for (1<a£2). The problem is discretized in the time direction based on finite difference scheme and is continuously approximated by using the radial basis functions in the space direction which achieves the semi-discrete solution. Numerical results show the accuracy and efficiency of the presented method.

In this paper, we propose radial basis functions (RBF) to solve the two dimensional flow of fluid near a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow can be reduced to an ordinary differential equation of third order using similarity transformation. Because of its wide applications the flow near a stagnation point has attracted many investigations during the past several decades. We satisfy boundary conditions such as infinity condition, by using Gaussian radial basis function through the both differential and integral operations. By choosing center points of RBF with shift on one point in uniform grid, we increase the convergence rate and decrease the collocation points.