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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]

Radial basis functions play an increasingly prominent role in modern approximation. They are widely used in scattered data fitting, numerical solution of partial differential equations, machine learning and others. Although radial basis functions have excellent approximation properties, they often produce highly ill-conditioned discrete algebraic system and lead to a high computational cost. The paper introduces local multilevel scattered data interpolation method, which employ nested scattered data sets and scaled compactly supported radial basis functions with varying support radii. We will provide convergence theory for Sobolev target functions. And several numerical experiments will be provided to conform the efficiency of new method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an adaptive fast multipole boundary face method is introduced to implement acoustic problems analysis of 3D solids with open-end small tubular shaped holes. The fast multipole boundary face method is referred as FMBFM. These holes are modeled by proposed tube elements. The hole is open-end and intersected with the outer surface of the body. The discretization of the surface with circular inclusions is achieved by applying several special triangular elements or quadrilateral elements. In the FMBFM, the boundary integration and field variables approximation are both performed in the parametric space of each boundary face exactly the same as the B-rep data structure in standard solid modeling packages. Numerical examples for acoustic radiation in this paper demonstrated the accuracy, efficiency and validity of this method. [Copyright &y& Elsevier]

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: The element-free Galerkin (EFG) method is a promising method for solving partial differential equations in which trial and test functions employed in the discretization process result from moving least-squares (MLS) approximation. In this paper, by employing the improved moving least-squares (IMLS) approximation, we derive formulae for an improved element-free Galerkin (IEFG) method for the modified equal width (MEW) wave equation. A variation of the method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method. Therefore, the IEFG method may result a better computing speed. In this paper, the effectiveness of the IEFG method for modified equal width (MEW) wave equation is investigated by numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method. [Copyright &y& Elsevier]

Summary: The accurate numerical solution of hypersingular boundary integral equations necessitates the precise evaluation of free terms, which are required to counter discontinuous and often unbounded behaviour of hypersingular integrals at a boundary. The common approach for the evaluation of free terms involves integration over a portion of a spherical shaped surface centred at a singularity and allowing the radius of the sphere to tend to zero. In this paper two alternative methods, which are shape invariant, are proposed and investigated for the determination of free terms. One approach, the point-limiting method, involves moving a singularity towards a shrinking integration domain at a faster rate than the domain shrinks. Issues surrounding the choice of approach and shrinkage rates, and path dependency are examined. A related approach, the boundary-limiting method, involves moving an invariant but shrinking boundary toward the singularity again at a faster rate than the shrinkage of the domain. The latter method can be viewed as a vanishing exclusion zone approach but the actual boundary shape is used for the boundary of the exclusion zone. Both these methods are shown to provide consistent answers and can be shown to be directly related to the result obtained by moving a singularity towards a boundary, i.e. by comparison with the direct method. Unlike the spherical approach the two methods involve integration over the actual boundary shape and consequently shape dependency is not a concern. A particular highlight of the point limiting approach, as a result of field approximations being restricted to the boundary, is the ability to obtain free terms in a mixed formulation without reference to the underpinning constitutive equations, which is not available to the spherical method. Focus in the paper is on the 2-D potential equation as this is shown to be sufficient to demonstrate the concepts involved. [Copyright &y& Elsevier]

Abstract: In the present paper, a simple mushy zone model is used to track the moving boundaries in an evaporation problem in which the vapor is removed upon formation. Two main parameters for the mushy zone model are analyzed as well as their effect on the movement of the moving boundaries and the thickness of the mushy zone. A new approximate method is developed for analysis and tracking the moving boundaries appears throughout the process. The proposed method mainly based on applying the boundary integral equation corresponding to each phase in such a way that the associated boundary and initial conditions as well as energy equations at the moving boundaries achieved with minimum error and low number of iterations. The results of the present paper seem to be good because there are neither analytical or numerical solutions available. [Copyright &y& Elsevier]

The purpose of this paper is to extend the boundary approximation method proposed by Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], i.e. the collocation Trefftz method called in this paper, for biharmonic equations with singularities. First, this paper derives the Green formulas for biharmonic equations on bounded domains with a non-smooth boundary, and corner terms are developed. The Green formulas are important to provide all the exterior and interior boundary conditions which will be used in the collocation Trefftz method. Second, this paper proposes three crack models (called Models I, II and III), and the collocation Trefftz method provides their most accurate solutions. In fact, Models I and II resemble Motz''s problem in Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], and Model III with all the clamped boundary conditions originated from Schiff et al. [The mathematics of finite elements and applications III, 1979]. Moreover, effects on d1 of different boundary conditions are investigated, and a brief analysis of error bounds for the collocation Trefftz method is made. Since accuracy of the solutions obtained in this paper is very high, they can be used as the typical models in testing numerical methods. The computed results show that as the singularity models, Models I and II are superior to Model III, because more accurate solutions can be obtained by the collocation Trefftz method. [Copyright &y& Elsevier]

In this paper two numerical procedures are presented for solving a class of Turing system. Firstly, we obtain a time discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and radial point interpolation and also obtain their shape functions. We use the element free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element free Galerkin method test and trial functions are moving least squares (MLS) approximation shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. The element free Galerkin method is a global method while finite elements method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. The coefficient matrix of the element free Galerkin is symmetric. Also, using numerical algorithms, we can conclude that the eigenvalues of the coefficient matrix are positive. Thus, for solving the obtained linear system of equations from the discretization, we use the conjugant gradient method. To keep away from solving a nonlinear algebraic system of equations and obtaining the acceptable numerical results, we use a predictor–corrector algorithm. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

This paper formulates the flat-top partition of unity (PU)-based numerical manifold method (NMM). Through modifying the finite element PU into the flat-top PU, the linear dependence problem involved in finite element PU-based high-order polynomial approximations has been successfully alleviated. Moreover, the procedure to construct the flat-top PU is substantially simplified by taking full advantage of the unique features of the NMM, compared to that in other flat-top PU-based methods. The proposed method could also be treated as an improved version of the discontinuous deformation analysis (DDA) by completely avoiding the additional strategies previously used to enforce the displacement compatibility between adjacent elements. The formulations are presented in details for 1-D, 2-D and 3-D cases. The discrete equations of the flat-top PU-based NMM are derived based on the minimum potential energy principle. The simplex integration technique and its variation are employed to evaluate the integration of the matrices of the discrete equations over arbitrarily shaped manifold elements. Seven representative examples have validated the proposed method in the aspects of approximation accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Recently, real options have gained more importance in computational finance studies. It has already been shown that the compound option pricing can be formulated as a two-pass boundary PDE arising from Black–Scholes model. Radial basis function (RBF) as a meshfree approximation method is widely used for numerical study of the time dependent PDEs. In this paper, the aim is to introduce the robust numerical approach based on RBF-QR to compute the price of European compound options such as the popular put on put options. We also extend the proposed approach to American compound option pricing. The numerical experiments will show the efficiency of the performance for European and American compound option with single asset and multi-asset cases. [ABSTRACT FROM AUTHOR]

Abstract: This paper discusses the efficient extraction of acoustic resonances in 2D open cavities using a meshless method, the method of particular solutions. A first order, local non-reflecting boundary condition is chosen to account for the opening and Fourier-Bessel functions are employed to approximate the pressure at the borders of the cavity and inside. A minimization problem is then solved for the complex frequency range of interest. For the investigated cavity, the minimum values obtained match those found in previously published studies or by other numerical methods. But, unlike the perfectly matched layer absorbing boundary conditions now usually employed, this approach is free from spurious eigenfrequencies. Moreover, the specific treatments of the geometric singularities allow this method to be particularly efficient in the presence of corner singularities. [Copyright &y& Elsevier]

Abstract: The MFS is a popular meshless boundary collocation method but encounters troublesome fully-populated interpolation matrix, whose operation count and memory requirement for interpolation matrices buildup are of order O(N 2). This is computationally prohibitively expensive. The adaptive cross approximation (ACA), a kernel-independent fast algorithm, was devised to reduce both CPU time and memory requirement in solving dense matrix systems from O(N 2) to O(N log β N) by a pseudo-skeletal entries approximation. This paper makes a first attempt to apply the ACA to remedy the expensive matrix computation and accelerate the solution of the MFS in potential problems, and further improves the ACA in generating the admissible cluster pairs. The MFS accelerated by the present improved ACA are tested to three benchmark cases. Numerical results show that the proposed methodology is computationally more efficient than the conventional scheme. [Copyright &y& Elsevier]

Abstract: Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward problem of time-fractional diffusion equation (BTFDE). The kernels used in the approximation are the fundamental solutions of the time-fractional diffusion equation which can be expressed in terms of the M-Wright functions. To stably and accurately solve the resultant highly ill-conditioned system of equations, we successfully combine the standard Tikhonov regularization technique and the L-curve method to obtain an optimal choice of the regularization parameter and the location of source points. Several 1D and 2D numerical examples are constructed to demonstrate the superior accuracy and efficiency of the proposed method for solving both the classical backward heat conduction problem (BHCP) and the BTFDE. [Copyright &y& Elsevier]

Abstract: This paper presents a meshless implementation of dual analysis for 2D linear elasticity problems. The derivation of the governing systems of equations for the discretized compatible and equilibrated models is detailed and crucial implementation issues of the proposed algorithm are discussed: (i) arising of deficiencies associated with the independent approximation field used for the imposition of the essential boundary conditions (EBC) for the two parts of the boundary sharing a corner and (ii) determination of the Lagrange multipliers functional space used to impose EBC. An attempt to implement the latter resulted in an approximation which is nothing more than the trace on the essential boundary of the domain nodal functions. The difficulties posed by such approximation are explained using the inf–sup condition. Several examples of global (energy) and local (displacements) quantities of interest and their bounds determination are used to demonstrate the validity of the presented meshless approach to dual analysis. Numerical assessment of the convergence rates obtained for both models is made, for different polynomial basis degrees. [Copyright &y& Elsevier]

Abstract: In this paper, an efficient Kansa-type method of fundamental solutions (MFS-K) is extended to the solution of two-dimensional time fractional sub-diffusion equations. To solve initial boundary value problems for these equations, the time dependence is removed by time differencing, which converts the original problems into a sequence of boundary value problems for inhomogeneous Helmholtz-type equations. The solution of this type of elliptic boundary value problems can be approximated by fundamental solutions of the Helmholtz operator with different test frequencies. Numerical results are presented for several examples with regular and irregular geometries. The numerical verification shows that the proposed numerical scheme is accurate and computationally efficient for solving two-dimensional fractional sub-diffusion equations. [Copyright &y& Elsevier]

Abstract: In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach. [Copyright &y& Elsevier]

Abstract: Quasi-interpolation is a very powerful tool in the field of approximation theory and its applications, which can avoid solving large scale ill-conditioned linear system arising in approximating an unknown function by means of radial basis functions. In this paper, we use an univariate multi-quadrics(MQ) quasi-interpolation scheme to solve one-dimensional nonlinear Schrödinger equation. In this novel numerical scheme, the spatial derivatives are approximated by using the derivative of the quasi-interpolation and the temporal derivative is approximated by finite difference method. The main advantage of this proposed scheme is its simplicity. Two numerical examples are given and compared with the finite difference method (FDM) to verify the good accuracy and easy implementation of this method. [Copyright &y& Elsevier]

Abstract: The element free Galerkin (EFG) method is a promising method for solving flow problems, but it meets the difficulty of volumetric locking for solving the incompressible flow problems. In this paper, a mixed EFG method is proposed for solving the steady incompressible flow problems, which avoids the volumetric locking and inherits the meshfree properties. The method employs two sets of nodes, one for the velocity approximation and the other for the pressure approximation. Specially, the ratio between the velocity node number and the pressure node number is taken as the only indicator for the locking behavior of the mixed EFG method. And inf–sup tests are carried out to investigate the relationship between the ratio and locking behavior. By two numerical examples, the accuracy, rate of convergence and efficiency of the mixed EFG method are also carefully studied. The results show that the accuracy, convergence and efficiency of the mixed EFG method are superior to that of the time-related fractional step methods. [Copyright &y& Elsevier]

Abstract: We investigate in this paper a Cauchy problem for the time-fractional diffusion equation (TFDE). Based on the idea of kernel-based approximation, we construct an efficient numerical scheme for obtaining the solution of a Cauchy problem of TFDE. The use of M-Wright functions as the kernel functions for the approximation space allows us to express the solution in terms of M-Wright functions, whose numerical evaluation can be accurately achieved by applying the inverse Laplace transform technique. To handle the ill-posedness of the resultant coefficient matrix due to the noisy Cauchy data, we adapt the standard Tikhonov regularization technique with the L-curve method for obtaining the optimal regularization parameter to give a stable numerical reconstruction of the solution. Numerical results indicate the efficiency and effectiveness of the proposed scheme. [Copyright &y& Elsevier]

Abstract: This paper presents a new meshless method developed by combining the quasi-linear method of fundamental solution (QMFS) and the finite difference method to analyze wave equations. The method of fundamental solution (MFS) is an efficient numerical method for solution Laplace equation for both two- and three-dimensional problems. The method has also been applied for the solution of Poisson equations and transient Poisson-type equations by finding the particular solution to the non-homogeneous terms. In general, approximate particular solutions are constructed using the interpolation of the non-homogeneous terms by the radial basis functions (RBFs). The interpolation in terms of RBFs often leads to a badly conditioned problem which demands special cares. The current work suggests a linearization scheme for the non-homogeneous term in terms of the dependent variable and finite differencing in time resulting in Helmholtz-type equations whose fundamental solutions are available. Consequently, the particular solution is no longer needed and the MFS can be directly applied to the new linearized equation. The numerical examples illustrate the effectiveness of the presented method. [Copyright &y& Elsevier]

Abstract: The Complex Variable Boundary Element Method or CVBEM is a numerical technique for approximating particular partial differential equations such as the Laplace or Poisson equations (which frequently occur in physics and engineering problems, among many other fields of study). The advantage in using the CVBEM over traditional domain methods such as finite difference or finite element based methods includes the properties that the resulting CVBEM approximation is a function: (i) defined throughout the entire plane, (ii) that is analytic throughout the problem domain and almost everywhere on the problem boundary and exterior of the problem domain union boundary; (iii) is composed of conjugate two-dimensional real variable functions that are both solutions to the Laplace equation and are orthogonal such as to provide the “flow net” of potential and stream functions, among many other features. In this paper, a procedure is advanced that locates CVBEM nodal point locations on and exterior of the problem boundary such that error in matching problem boundary conditions is reduced. That is, locating the nodal points is part of modeling optimization process, where nodes are not restricted to be located on the problem boundary (as is the typical case) but instead locations are optimized throughout the exterior of the problem domain as part of the modeling procedure. The presented procedure results in nodal locations that achieve considerable error reduction over the usual methods of placing nodes on the problem boundary such as at equally spaced locations or other such procedures. Because of the significant error reduction observed, the number of nodes needed in the model is significantly reduced. It is noted that similar results occur with the real variable boundary element method (or BEM). The CVBEM and relevant nodal location optimization algorithm is programmed to run on program Mathematica, which provides extensive internal modeling and output graphing capabilities, and considerable levels of computational accuracy. The Mathematica source code is provided. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical scheme to approximate water wave diffraction, refraction and frictional dissipation over an axi-symmetric pit. Based on an improved extended mild-slope equation (EMSE) including bottom friction effect, as the elliptic governing differential equation, dual reciprocity boundary element method (DRBEM) is employed to model water wave propagation over an axi-symmetric pit. To the authors'' knowledge, this is the first application of DRBEM for water wave scattering over a pit. In order to promote accuracy of the model, not only effects of the bottom curvature and the slope-squared terms which are neglected in the mild-slope equation (MSE), are considered, but also effect of the bottom friction is measured by the improved EMSE. Numerical results are compared with existing analytical or numerical solutions or with experimental data by several examples. Through these numerical experiments reliability and efficiency of present DRBEM model for determining the total wave field over an uneven bottom is approved. [Copyright &y& Elsevier]

Abstract: The wave based method (WBM), which is based on an indirect Trefftz approach, is a deterministic prediction method posed as an alternative to the element-based methods. It uses wave functions, which are exact solutions of the underlying differential equation, to describe the dynamic field variables. In this way, it can avoid the pollution errors associated with the polynomial element-based approximations. As a consequence, a dense element discretization is no longer required, yielding a smaller numerical system. The resulting enhanced computational efficiency of the WBM as compared to the element-based methods has been proven for the analysis of both bounded and unbounded acoustic problems. This paper extends the applicability of the WBM to semi-infinite domains. An appropriate function set is proposed, together with a calculation procedure for both semi-infinite radiation and scattering problems, and transmission or diffraction problems containing a rigid baffle. The resulting technique is validated on two numerical examples. [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: In the present paper problem of natural convection in a cubic porous cavity is studied numerically, using an algorithm based on a combination of single domain and subdomain boundary element method (BEM). The modified Navier–Stokes equations (Brinkman-extended Darcy formulation with inertial term included) were adopted to model fluid flow in porous media, coupled with the energy equation using the Boussinesq approximation. The governing equations are transformed by the velocity–vorticity variables formulation which separates the computation scheme into kinematic and kinetic parts. The kinematics equation, vorticity transport equation and energy equation are solved by the subdomain BEM, while the boundary vorticity values, needed as a boundary conditions for the vorticity transport equation, are calculated by single domain BEM solution of the kinematics equation. Computations are performed for steady state cases, for a range of Darcy numbers from 10−6 to 10−1, and porous Rayleigh numbers ranging from 50 to 1000. The heat flux through the cavity and the flow fields are analyzed for different cases of governing parameters and compared to the results in some published studies. [Copyright &y& Elsevier]

Abstract: This paper presents a formulation of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of two-dimensional mixed boundary-value problems (BVP) for a second-order linear elliptic partial differential equation (PDE) with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the BVP to a BDIE or BDIDE. The numerical formulation of the BDIDE employs an approximation for the boundary fluxes in terms of the potential function within the domain cells; therefore, the solution is fully described in terms of the variation of the potential function along the boundary and domain. Linear basis functions localised on triangular elements and standard quadrature rules are used for the calculation of boundary integrals. For the domain integrals, we have implemented Gaussian quadrature rules for two dimensions with Duffy transformation, by mapping the triangles into squares and eliminating the weak singularity in the process. Numerical examples are presented for several simple problems with square and circular domains, for which exact solutions are available. It is shown that the present method produces accurate results even with coarse meshes. The numerical results also show that high rates of convergence are obtained with mesh refinement. [Copyright &y& Elsevier]

Abstract: In this paper, three dimensional (3D) static and dynamic analysis of thick functionally graded plates based on the Meshless Local Petrov–Galerkin (MLPG) is presented. Using the kinematics of a three-dimensional continuum, the local weak form of the equilibrium equations is derived. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains using a Heaviside step function as test function. In this case, governing equations corresponding to the stiffness matrix do not involve any domain integration or singular integrals. Nodal points are distributed in the 3D analyzed domain and each node is surrounded by a cubic sub-domain to which a local integral equation is applied. The meshless approximation based on the three-dimensional Moving Least-Square (MLS) is employed as shape function to approximate the field variable of scattered nodes in the problem domain. The Newmark time integration method is used to solve the system of coupled second-order ODEs. Effective material properties of the plate, made of two isotropic constituents with volume fractions varying only in the thickness direction, are computed using the Mori–Tanaka homogenization technique. Numerical examples for solving the static and dynamic response of elastic thick functionally graded plates are demonstrated. As a result, the distributions of the deflection and stresses through the plate thickness are presented for different material gradients and boundary conditions. The effects of the volume fractions of the constituents on the centroidal deflection are also investigated. The numerical efficiency of the proposed meshless method is illustrated by the comparison of results obtained from previous literatures. [Copyright &y& Elsevier]

Abstract: In this paper, basis functions generated from B-spline or Non-Uniform Rational B-spline (NURBS), are used for approximating the boundary variables to solve the 3D linear elasticity Boundary Integral Equations (BIEs). The implementation is based on the BFM framework in which both boundary integration and variable approximation are performed in the parametric spaces of the boundary surfaces to keep the exact geometric information in the BIEs. In order to reduce the influence of tensor product of B-spline and make the discretization of a body surface easier, the basis functions defined in global intervals are translated into local form. B-spline fitting function built with the local basis functions is converted into an interpolation type of function in which the nodal values of the boundary variables are used for control points. Numerical tests for 3D linear elasticity problems show that the BFM with B-spline basis functions outperforms that with the well-known Moving Least Square (MLS) approximation. [Copyright &y& Elsevier]

Abstract: In this paper, a new approach using analytical expressions in the radial integration boundary element method (RIBEM) is presented for solving variable coefficient heat conduction problems. This approach can improve the computational efficiency considerably and can overcome the time-consuming deficiency of RIBEM in computing involved radial integrals. The fourth-order spline RBF is employed to approximate unknowns appearing in domain integrals arising from the varying heat conductivity. The radial integration method is utilized to convert domain integrals to the boundary resulting in a pure boundary discretization algorithm. Numerical examples are given to demonstrate the efficiency of the presented approach. [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: Based on the idea of radial basis functions approximation and the method of particular solutions, we develop in this paper a new meshless computational method to solve nonhomogeneous backward heat conduction problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several benchmark problems in both two- and three-dimensions. Numerical results indicate that this novel approach can achieve an efficient and accurate solution even when the final temperature data is almost undetectable or disturbed with large noises. It has also been shown that the proposed method is stable to recover the unknown initial temperature from scattered final temperature data. [Copyright &y& Elsevier]

Abstract: In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction in layered materials, where the thermal diffusivity is piecewise constant. Recently, in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction. Eng Anal Boundary Elem 2008;32:697–703], a MFS was proposed with the sources placed outside the space domain of interest, and we extend that technique to numerically approximate the heat flow in layered materials. Theoretical properties of the method, as well as numerical investigations are included. [Copyright &y& Elsevier]

Abstract: In this paper, a meshless method of lines (MOL) is presented for the numerical solution of the Korteweg–de Vries (KdV) equation. This novel method has an advantage over the traditional method of lines which approximates the spatial derivatives using finite difference method (FDM) or finite element method (FEM), because it does not need the mesh in the domain, and it approximates the solution using the radial basis functions (RBFs) on a set of node scattered in problem domain. A comparison among some RBFs is made in numerical examples. Numerical examples demonstrate the accuracy and easy implementation of this novel method and it is an efficient method for the nonlinear time-dependent partial differential equations (PDEs). [Copyright &y& Elsevier]

Abstract: In this paper, the method of time-dependent fundamental solutions is extended to solving transient heat conduction problems in inhomogeneous media. The solution of the problem under investigation is split into two parts, namely the particular and homogenous solutions. The novelty of the proposed approach lies in that an approximation of the particular solution is derived by using the fundamental solutions of the associated eigenvalue equations. Numerical results for one- and two-dimensional geometries are presented to verify the efficacy of the proposed method. The effects of the numbers of source and collocation points, the eigenvalues and the parameter T on the accuracy of the numerical solution are also investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a regularization method for determining a moving boundary from Cauchy data in one-dimensional heat equation with a multilayer domain. The numerical scheme is based on the use of the method of fundamental solutions and a discrete Tikhonov regularization technique. The generalized cross validation rule for the choice of a regularization parameter is applied to obtain a stable numerical approximation to the moving boundary. Numerical experiments for five examples show that our proposed method is effective and stable. [Copyright &y& Elsevier]

Abstract: A meshless local natural neighbor interpolation method (MLNNIM) is utilized to build shape functions for solving elastoplastic problems in this paper. This method searches natural neighbor nodes of the given point and defines local Delaunay triangles. Total Lagrangian (TL) and Updated Lagrangian (UL) formulae have been derived and analyzed. Consideration of the constitutive law and numerical efficiency indicates that the UL formulation is more expedient and attractive in practical problems of solid mechanics. Combined with the linearized UL formulation, the MLNNIM was used to solve problems of two-dimension elastoplastic large deformation. Finally, two representative numerical examples are given to demonstrate the effectiveness and superiority of the MLNNIM combined with the UL formulation for two-dimension incompressible large deformation problems. [Copyright &y& Elsevier]

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: It is shown in this paper that self-consistent boundary conditions for numerical methods based on blurred derivatives can be derived from a suitable change of variables of the fundamental blurred approximation of the differential equation, followed by application of Leibnitz theorem for differentiation of an integral. The simplest scheme obtained in this way resembles the weak Local Petrov-Galerkin approximation, although interpretation of the operators appearing in the final equations is quite different—as is the derivation itself. Subsequent transformation leads to integral equations similar to the starting point for boundary integral methods of solution. In this way, a number of well-known computational methods are shown to be derivable from adequate manipulation of the blurred derivative technique. However, other approximations, which are not derivable with standard methods can also be obtained, hinting at a greater generality of blurred derivatives. [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]

This paper presents the application of a Trefftz type method with approximate trial functions for time-dependent problems of the parabolic type. At first it was suggested for elliptic equations only. The method was developed for PDEs with variable coefficients when a total differential operator of a problem can be represented as a sum of the one-dimensional ones. A comparison of the numerical solutions with analytic solutions is performed. [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]

A local numerical approach to cope with the singular and hypersingular boundary integral equations (BIEs) in non-regularized forms is presented in the paper for 2D elastostatics. The approach is based on the fact that the singular boundary integrals can be represented approximately by the mean values of two nearly singular boundary integrals and on the techniques of distance transformations developed primarily in previous work of the authors. The nearly singular approximations in the present work, including the normal and the tangential distance transformations, are designed for the numerical evaluation of boundary integrals with end-singularities at junctures between two elements, especially at corner points where sufficient continuity requirements are met. The approach is very general, which makes it possible to solve the hypersingular BIE numerically in a non-regularized form by using conforming C0 quadratic boundary elements and standard Gaussian quadratures without paying special attention to the corner treatment.With the proposed approach, an infinite tension plate with an elliptical hole and a pressurized thick cylinder were analyzed by using both the formulations of conventional displacement and traction boundary element methods, showing encouragingly the efficiency and the reliability of the proposed approach. The behaviors of boundary integrals with end- and corner-singular kernels were observed and compared by the additional numerical tests. It is considered that the weakly singularities remain but should have been cancelled with each other if used in pairs by the corresponding terms in the neighboring elements, where the corner information is included naturally in the approximations. [Copyright &y& Elsevier]