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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 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 present a numerical solution of a family of generalized fifth‐order Korteweg‐de Vries equations using a meshless method of lines. This method uses radial basis functions for spatial derivatives and Runge‐Kutta method as a time integrator and exhibits high accuracy as seen from the comparison with the exact solutions.

The study uses a meshless method of lines. This method uses radial basis functions for spatial derivatives and Runge‐Kutta method as a time integrator.

The paper reveals that this method exhibits high accuracy as seen from the comparison with the exact solutions.

This method is efficient method as it is easy to implement for the numerical solutions of PDEs.

This study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains.

The main aim of the current paper is to propose a local meshless collocation method to solve the two-dimensional Klein-Kramers equation with a fractional derivative in the Riemann-Liouville sense, in the time term. This equation describes the sub-diffusion in the presence of an external force field in phase space.

First, the authors use two finite difference schemes to discrete temporal variables and then the radial basis function-differential quadrature method has been used to estimate the spatial direction. To discrete the time-variable, the authors use two different strategies with convergence orders O(τ1+γ) and O(τ2−γ) for 0 < γ < 1. Finally, some numerical examples have been presented to show the high accuracy and acceptable results of the proposed technique.

The proposed numerical technique is flexible for different computational domains.

This paper compares three different radial basis function neural networks, as well as the diffuse element method, according to their ability of approximation. This is very useful for the optimization of electromagnetic devices. Tests are done on several analytical functions and on the TEAM workshop problem 25.

Radial basis function (RBF) neural networks form an essential category of architectures of neurocomputing. They exhibit interesting and useful properties of stable and fast learning associated with significant generalization capabilities. This successful performance of RBF neural networks can be attributed to the use of a collection of properly selected RBFs. In this way this category of the networks strongly relies on some domain knowledge about a classification problem at hand. Following this vein, this study introduces fuzzy clustering, and fussy isodata, in particular, as an efficient tool aimed at constructing receptive fields of RBF neural networks. It is shown that the functions describing these fields are completely derived as a by‐product of fuzzy clustering and do not require any further tedious refinements. The efficiency of the design is illustrated with the use of synthetic two‐dimensional data as well as real‐world highly dimensional ECG patterns. The classification of the latter data set clearly points out advantages of RBF neural networks in pattern recognition problems.

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 paper is to assess the feasibility of analytical models, specifically the radial basis function method, Akbari–Ganji method and Gaussian method, in conjunction with the finite element method. The aim is to examine the impact of processing parameters on temperature history.

Through analytical investigation and finite element simulation, this research examines the influence of processing parameters on temperature history. Simufact software with a thermomechanical approach was used for finite element simulation, while radial basis function, Akbari–Ganji and Gaussian methods were used for analytical modeling to solve the heat transfer differential equation.

The accuracy of both finite element and analytical methods was validated with about 90%. The findings revealed direct relationships between thermal conductivity (from 100 to 200), laser power (from 400 to 800 W), heat source depth (from 0.35 to 0.75) and power absorption coefficient (from 0.4 to 0.8). Increasing the values of these parameters led to higher temperature history. On the other hand, density (from 7,600 to 8,200), emission coefficient (from 0.5 to 0.7) and convective heat transfer (from 35 to 90) exhibited an inverse relationship with temperature history.

The application of analytical modeling, particularly the utilization of the Akbari–Ganji, radial basis functions and Gaussian methods, showcases an innovative approach to studying directed energy deposition. This analytical investigation offers an alternative to relying solely on experimental procedures, potentially saving time and resources in the optimization of DED processes.

The purpose of this paper is to obtain accurate numerical solutions of two-dimensional (2-D) and 3-dimensional (3-D) Klein–Gordon–Schrödinger (KGS) equations.

The use of linear barycentric interpolation differentiation matrices facilitates the computation of numerical solutions both in 2-D and 3-D space within reasonable central processing unit times.

Numerical simulations corroborate the efficiency and accuracy of the proposed method.

Linear barycentric interpolation method is applied to 2-D and 3-D KGS equations for the first time, and good results are obtained.