Due to the nonlinear diffusion term, it is hard to use the collocation method to solve the unsaturated soil water movement equation directly. In this paper, a nonmesh Hermite collocation method with radial basis functions was proposed to solve the nonlinear unsaturated soil water movement equation with the Neumann boundary condition. By preprocessing the nonlinear diffusion term and using the Hermite radial basis function to deal with the Neumann boundary, the phenomenon that the collocation method cannot be used directly is avoided. The numerical results of unsaturated soil moisture movement with Neumann boundary conditions on the regular and nonregular regions show that the new method improved the accuracy significantly, which can be used to solve the low precision problem for the traditional collocation method when simulating the Neumann boundary condition problem. Moreover, the effectiveness and reliability of the algorithm are proved by the one-dimensional and two-dimensional engineering problem of soil water infiltration in arid area. It can be applied to engineering problems.

/pdf/hermite-radial-basis-collocation-method-for-unsaturated-soil-2f3nez9ih1.pdf

Hermite Radial Basis Collocation Method for Unsaturated Soil Water Movement Equation

The study of differential games has yielded a number of substantial results. However, numerical method for the saddle point strategy is still open question. In this paper, a new method with radial basis functions for the two-point boundary value problems of differential games is developed. Moreover, the method is applied to numerical examples to examine its appropriateness. Therefore, it is a useful way to solving the numerical solutions of saddle point strategy.

Approximation method with radial basis functions for TPBVP of differential games

In this paper, a meshless method with ridge basis functions for solving the time fractional two-flow domain model problem is proposed The method uses the L1 approximation formula based on piecewise linear interpolation to discretize the Caputo time fractional derivative 
$$ (0 < \alpha < 1) $$

, and by means of the ridge basis function to construct the approximation function, and uses the collocation method to discretize the governing equation The existence and uniqueness of the numerical solution are analyzed The error between the proposed method and the finite difference method is compared by numerical examples; then the affecting factors of the calculation accuracy are discussed The results show that the proposed method is feasible and simple

Meshless method with ridge basis functions for time fractional two-flow domain model

The moving least squares (MLS) and moving total least squares (MTLS) are two of the most popular methods used for reconstructing measurement data, on account of their good local approximation accuracy. However, their reconstruction accuracy and robustness will be greatly reduced when there are outliers in measurement data. This article proposes an improved MTLS method (IMTLS), which introduces an improved random sample consensus (RANSAC) algorithm and a correction parameter in the support domain, to deal with the outliers and random errors. Based on the nodes within the support domain, firstly the improved RANSAC is used to generate a model for establishing the group of pre-interpolation and calculating the residual of each node. Subsequently, the abnormal degree of the node with the largest residual is evaluated by the correction parameter associated with the node residual and random errors. The node with certain abnormal degree will be eliminated and the remaining nodes are used to obtain the approximation coefficients. The correction parameter can be used for data reconstruction without insufficient or excessive elimination. The results of numerical simulation and measurement experiment show that the reconstruction accuracy and robustness of the IMTLS method is superior to the MLS and MTLS method.

/pdf/surface-reconstruction-method-for-measurement-data-with-1c7vxw2z.pdf

Surface reconstruction method for measurement data with outlier detection by using improved RANSAC and correction parameter

A meshless method for modified Helmholtz equations has been developed by utilizing the collocation method and the ridge basis function interpolation. This method is a truly meshless technique without mesh discretization: it neither needs the computation of integrals, nor requires a partition of the region and its boundary. Moreover, the method is applied to 2-dimensional equations to examine its appropriateness, numerical results are compared to that obtained from other (meshless or mesh-dependent) methods, and influence factors of accuracy for numerical solutions are analyzed.

Meshless method with ridge basis functions for modified Helmholtz equations

Meshless method with ridge basis functions

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Meshless method with ridge basis functions