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This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time.

In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight.

The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems.

The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two dimensional time fractional telegraph equation.

In this method the authors first approximate the time fractional derivatives of mentioned equation by two schemes of orders O(τ3−α) and O(τ2−α), 1/2<α<1, then the authors will use the Kansa approach to approximate the spatial derivatives.

The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy.

The results show that the meshless method based on the RBFs and collocation approach is also suitable for the treatment of the time fractional telegraph equation.

The purpose of this paper is to develop two meshfree algorithms based on multiquadric radial basis functions (RBFs) and differential quadrature (DQ) technique for numerical simulation and to capture the shocks behavior of Burgers’ type problems.

The algorithms convert the problems into a system of ordinary differential equations which are solved by the Runge–Kutta method.

Two meshfree algorithms are developed and their stability is discussed. Numerical experiment is done to check the efficiency of the algorithms, and some shock behaviors of the problems are presented. The proposed algorithms are found to be accurate, simple and fast.

The present algorithms LRBF-DQM and GRBF-DQM are based on radial basis functions, which are new for Burgers’ type problems. It is concluded from the numerical experiments that LRBF-DQM is better than GRBF-DQM. The algorithms give better results than available literature.

Develop a local radial point interpolation method (LRPIM) to analyze the dissipation process of excess pore water pressure in porous media and verify its numerical capability.

Terzaghi's consolidation theory is used to describe the dissipation process. A local residual form is formulated over only a sub‐domain. This form is spatially discretized by radial point interpolation method (RPIM) with basis of multiquadrics (MQ) and thin‐plate spline (TPS), and temporally discretized by finite difference method. One‐dimensional (1D) and two‐dimensional consolidation problems are numerically analyzed.

The LRPIM is suitable, efficient and accurate to simulate this dissipation process. The shape parameters, q=1.03, R=0.1 for MQ and η=4.001 for TPS, are still valid.

The asymmetric system matrix in LRPIM spends more resources in storage and CPU time.

Local residual form requires no background mesh, thus being a truly meshless method. This provides a fast and practical algorithm for engineering computation.

This paper provides a simple, accurate and fast numerical algorithm for the dissipation process of excess pore water pressure, largely simplifies data preparation, shows that the shape parameters from solid mechanics are also suitable for the dissipation process.

The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.

The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.

Several test problems are provided to confirm the validity and efficiently of the proposed method.

For the first time, some famous examples are solved by using the proposed high-order technique.

The purpose of this paper is to upgrade our previous developments of Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow and electromagnetic problems to thermoelastic problems and to study its numerical performance with the aim to build a multiphysics meshless computing environment based on LRBFCM.

Linear thermoelastic problems for homogenous isotropic body in two dimensions are considered. The stationary stress equilibrium equation is written in terms of deformation field. The domain and boundary can be discretized with arbitrary positioned nodes where the solution is sought. Each of the nodes has its influence domain, encompassing at least six neighboring nodes. The unknown displacement field is collocated on local influence domain nodes with shape functions that consist of a linear combination of multiquadric radial basis functions and monomials. The boundary conditions are analytically satisfied on the influence domains which contain boundary points. The action of the stationary stress equilibrium equation on the constructed interpolation results in a sparse system of linear equations for solution of the displacement field.

The performance of the method is demonstrated on three numerical examples: bending of a square, thermal expansion of a square and thermal expansion of a thick cylinder. Error is observed to be composed of two contributions, one proportional to a power of internodal spacing and the other to a power of the shape parameter. The latter term is the reason for the observed accuracy saturation, while the former term describes the order of convergence. The explanation of the observed error is given for the smallest number of collocation points (six) used in local domain of influence. The observed error behavior is explained by considering the Taylor series expansion of the interpolant. The method can achieve high accuracy and performs well for the examples considered.

The method can at the present cope with linear thermoelasticity. Other, more complicated material behavior (visco-plasticity for example), will be tackled in one of our future publications.

LRBFCM has been developed for thermoelasticity and its error behavior studied. A robust way of controlling the error was devised from consideration of the condition number. The performance of the method has been demonstrated for a large number of the nodes and on uniform and non-uniform node arrangements.

The purpose of this paper is to introduce a local Newton basis functions collocation method for solving the 2D nonlinear coupled Burgers’ equations. It needs less computer storage and flops than the usual global radial basis functions collocation method and also stabilizes the numerical solutions of the convection-dominated equations by using the Newton basis functions.

A meshless method based on spatial trial space spanned by the local Newton basis functions in the “native” Hilbert space of the reproducing kernel is presented. With the selected local sub-clusters of domain nodes, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs).

The method is successfully used for solving the 2D nonlinear coupled Burgers’ equations for reasonably high values of Reynolds number (Re). It is a well-known issue in the analysis of the convection-diffusion problems that the solution becomes oscillatory when the problem becomes convection-dominated if the standard methods are followed without special treatments. In the proposed method, the authors do not detect any instability near the front, hence no technique is needed. The numerical results show that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.

The authors used more stable basis functions than the standard basis of translated kernels for representing of kernel-based approximants for the numerical solution of partial differential equations (PDEs). The local character of the method, having a well-structured implementation including enforcing the Dirichlet and Neuman boundary conditions, and producing accurate and stable results for flow with reasonably high values of Re for the numerical solution of the 2D nonlinear coupled Burgers’ equations without any special technique are the main values of the paper.

The purpose of this research paper is to investigate and model the fused deposition modelling (FDM) process to predict the mechanical attributes of 3D printed specimens.

By exploiting the main effect plots, a Taguchi L18 orthogonal array is used to investigate the effects of such parameters on three mechanical attributes of the 3D printed specimens. A radial-based integrated network is then developed to map the eight FDM parameters to the three mechanical attributes for both PEEK and PEKK. Such an integrated network maps and predicts the mechanical attributes through two consecutive phases that consist of several radial basis functions (RBFs).

Validated on a set of further experiments, the integrated network was successful in predicting the mechanical attributes of the 3D printed specimens. It also outperformed the well-known RBF network with an overall improvement of 24% in the coefficient of determination. The integrated network is also further validated by predicting the mechanical attributes of a medical-surgical implant (i.e. the MidFace Rim) as an application.

The main aim of this paper is to accurately predict the mechanical properties of parts produced using the FDM process. Such an aim requires modelling a highly dimensional space to represent highly nonlinear relationships. Therefore, a radial-based integrated network based on the combination of composition and superposition of radial functions is developed to model FDM using a limited number of data points.

The study aims to present a new meshless method based on the Taylor series expansion. The compact supported radial basis functions (CSRBFs) are very attractive, can be considered as a numerical tool for the engineering problems and used to obtain the trial solution and its derivatives without differentiating the basis functions for a meshless method. A meshless based on the CSRBF and Taylor series method has been developed for the solutions of engineering problems.

This paper is devoted to present a truly meshless method which is called a radial basis Taylor series method (RBTSM) based on the CSRBFs and Taylor series expansion (TSE). The basis function and its derivatives are obtained without differentiating CSRBFs.

The RBTSM does not involve differentiation of the approximated function. This property allows us to use a wide range of CSRBF and weight functions including the constant one. By using a different number of terms in the TSE, the global convergence properties of the RBTSM can be improved. The global convergence properties are satisfied by the RBTSM. The computed results based on the RBTSM shows excellent agreement with results given in the open literature. The RBTSM can provide satisfactory results even with the problem domains which have curved boundaries and irregularly distributed nodes.

The CSRBFs have been widely used for the construction of the basic function in the meshless methods. However, the derivative of the basis function is obtained with the differentiation of the CSRBF. In the RBTSM, the derivatives of the basis function are obtained by using the TSE without differentiating the CSRBF.

The purpose of this paper is to propose an adaptive neural-sliding mode-based observer for the estimation and reconstruction of unknown faults and disturbances for time-varying nonlinear systems such as aircraft, to ensure preciseness in the diagnosis of fault magnitude as well as the shape without enhancement of system complexity and cost. Fault-tolerant control (FTC) strategy based on adaptive neural-sliding mode is also proposed in the existence of faults for ensuring the stability of the faulty system.

In this paper, three strategies are presented: adaptive radial basis functions neural network (ARBFNN), conventional radial basis functions neural network (CRBFNN) and integral-chain differentiator. For the purpose of enhancement of fault diagnosis and isolation, a new sliding mode-based concept is introduced for the weight updating parameters of radial basis functions neural network (RBFNN).The main objective of updating the weight parameters adaptively is to enhance the effectiveness of fault diagnosis and isolation without increasing the computational complexities of the system. Results depict the effectiveness of the proposed ARBFNN approach in fault detection (FD) and approximation compared to CRBFNN, integral-chain differentiator and schemes existing in literature. In the second step, the FTC strategy is presented separately for each observer in the presence of unknown faults and failures for ensuring the stability of the system, which is validated on Boeing 747 100/200 aircraft.

The proposed adaptive neural-sliding mode approach is investigated, which depicts more effectiveness in numerous situations such as faults, disturbances and uncertainties compared to algorithms used in literature. In this paper, both the fault approximation and isolation and the fault tolerance approaches are studied.

For the enhancement of safety level as well as for avoiding any kind of damage, timely FD and fault tolerance have always had a significant role; therefore, the algorithms proposed in this research ensure the tolerance of faults and failures, which plays a vital role in practical life for avoiding any kind of damage.

In this study, a new neural-sliding mode concept is adopted for the adaptive faults approximation and reconstruction, and then the FTC algorithms are studied for each observer separately, whereas in previous studies, only the fault detection and isolation (FDI) or the fault tolerance problems were studied. Results demonstrate the effectiveness of the proposed strategy compared to the approaches given in the literature.