scholarly journals A meshless method for the numerical solution of the seventh-order Korteweg-de Vries equation

2020 ◽  
Keyword(s):  
Radial Basis Functions ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Basis Functions ◽  
Benchmark Problems ◽  
One Dimensional ◽  
De Vries ◽  
Radial Basis ◽  
Seventh Order

This article describes a meshless method for the numerical solution of the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation. The meshless scheme is based on the use of the collocation method and radial basis functions. In this approach, the solution is approximated by radial basis functions, and the collocation method is used to compute the unknown coefficients. The meshless method uses the following radial basis functions: Gaussian, inverse quadratic, multiquadric, inverse multiquadric and Wu’s compactly supported radial basis function. Time discretization of the nonlinear one-dimensional non-stationary Korteweg-de Vries equation is obtained using the θ-scheme. This meshless method has an advantage over traditional numerical methods, such as the finite difference method and the finite element method, because it doesn’t require constructing an interpolation grid inside the domain of the boundary-value problem. In this meshless scheme the domain of a boundary-value problem is a set of uniformly or arbitrarily distributed nodes to which the basic functions are “tied”. The paper presents the results of the numerical solutions of two benchmark problems which were obtained using this meshless approach. The graphs of the analytical and numerical solutions for benchmark problems were obtained. Accuracy of the method is assessed in terms of the average relative error, the average absolute error, and the maximum error. Numerical experiments demonstrate high accuracy and robustness of the method for solving the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation.

The meshless method of radial basis functions for the numerical solution of time fractional telegraph equation

2014 ◽  
Vol 24 (8) ◽  
pp. 1636-1659 ◽  
Author(s):  
Akbar Mohebbi ◽  
Mostafa Abbaszadeh ◽  
Mehdi Dehghan
Keyword(s):  
Radial Basis Functions ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Telegraph Equation ◽  
Basis Functions ◽  
Content Type ◽  
Fractional Telegraph Equation ◽  
Radial Basis ◽  
Collocation Approach ◽  
Derivatives Of

Purpose – 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. Design/methodology/approach – 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. Findings – The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy. Originality/value – 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.

scholarly journals Entropy stable essentially nonoscillatory methods based on RBF reconstruction

2019 ◽  
Vol 53 (3) ◽  
pp. 925-958 ◽  
Author(s):  
Jan S. Hesthaven ◽  
Fabian Mönkeberg
Keyword(s):  
Radial Basis Functions ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Basis Functions ◽  
One Dimensional ◽  
Hyperbolic Conservation ◽  
Radial Basis ◽  
Sign Property ◽  
Entropy Stable ◽  
Essentially Nonoscillatory

To solve hyperbolic conservation laws we propose to use high-order essentially nonoscillatory methods based on radial basis functions. We introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Thus, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. The main contribution is the construction of an algorithm and a smoothness indicator that ensures an interpolation function which fulfills the sign-property on general one dimensional grids.

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