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Abstract The main intention of the present work is to develop a numerical scheme based on radial basis functions (RBFs) to solve fractional stochastic integro-differential equations. In this paper, the solution of fractional stochastic integro-differential equation is approximated by using strictly positive definite RBFs such as Gaussian and strictly conditionally positive definite RBFs such as thin plate spline. Then, the quadrature methods are used to approximate the integrals which are appeared in this scheme. When we use thin plate spline to approximate the solution of mentioned equation, we encounter logarithm-like singular integrals which cannot be computed by common quadrature formula. To overcome this difficulty, we introduce the non-uniform composite Gauss–Legendre integration rule and employ it to estimate the singular logarithm integral appeared in this case. This method transforms the solution of linear fractional stochastic integro-differential equations to the solution of linear system of algebraic equations which can be easily solved. We also discuss the error analysis of the proposed method and demonstrate that the rate of convergence of this approach is arbitrary high for infinitely smooth RBFs. Finally, the efficiency and accuracy of the proposed method are checked by some numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents a new meshless numerical method for solving partial differential equations of the second and fourth orders with variable coefficients. The key idea of the method is the use of modified particular solutions which satisfy the homogeneous boundary conditions of the problem. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem. As a result we separate the approximation of the boundary conditions and the PDE inside the solution domain. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions or other numerical methods indicates that the proposed method is accurate in dealing with PDEs with variable coefficients. [Copyright &y& Elsevier]

Abstract: In this paper, the local radial point interpolation method (LRPIM) is developed for the investigation of time dependent problems in solid mechanics. By a new integration scheme considered for the obtained meshless weak form, integrands are approximated up to the second order of the Taylor series and the integrals are evaluated on some points, which are located inside the local quadrature domains, called integration points. In order to show the efficiency of the suggested method, some time dependent mechanical problems are considered for the engineering structures such as beams and plates, which are subjected to dynamical loads, the deflections and stresses are evaluated. Finally, it has been shown that using the proposed method greatly reduces the number of integration points without affecting the accuracy of the results. [Copyright &y& Elsevier]

Abstract: The radial basis functions (RBFs) is an efficient tool in multivariate approximation, but it usually suffers from an ill-conditioned interpolation matrix when interpolation points are very dense or irregularly spaced. The RBFs with variable shape parameters can usually improve the interpolation matrix condition number. In this paper a new shape parameter variation scheme is implemented. Comparison studies with the constant shaped RBF on convergence and stability are made. Results show that under the same accuracy level, the interpolation matrix condition number by our scheme grows much slower than that of the constant shaped RBF interpolation matrix with increase in the number of interpolation points. As an application example, the dual reciprocity method equipped with the new RBF is combined with the boundary face method to solve boundary value problems governed by Poisson equations. Numerical results further demonstrate the robustness and better stability of the new RBF. [ABSTRACT FROM AUTHOR]

Abstract: The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF–PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators. In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson''s, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance. [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]

The boundary node method (BNM) takes the advantages of both the boundary integral equation in dimension reduction and the moving least-square (MLS) approximation in elements elimination. However, the BNM inherits the deficiency of the MLS approximation, in which the shape functions lack the delta function property. As a result boundary conditions could not be exactly implemented for the BNM. In this paper, a radial boundary node method (RBNM) is proposed. The RBNM uses radial basis functions (RBFs) instead of the MLS to construct its interpolation. Consequently, the interpolation function could pass through nodes exactly, and the shape functions are of the delta function property. The exponential (EXP) and the multiquadric (MQ) RBFs are used in the current RBNM, and their shape parameters are studied in detail through some analyses of two-dimensional elastic problems. The suitable ranges of the shape parameters are proposed for both the EXP and the MQ basis functions. It is found that the RBNM is as accurate as or even more precise than the BEM. This suggests that the current RBNM could be robust and applicable. [Copyright &y& Elsevier]