scholarly journals A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 113 ◽  
Author(s):  
Muaz Seydaoğlu
Keyword(s):  
Lie Group ◽  
Kinematic Viscosity ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Time Integration ◽  
Group Scheme ◽  
Basis Functions ◽  
One Dimensional ◽  
Radial Basis ◽  
The One

An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made with the exact solutions and the reported results to demonstrate the efficiency and accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and the proposed algorithm is efficient and can be easily implemented.

scholarly journals Computer modeling system for the numerical solution of the one-dimensional non-stationary Burgers’ equation

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solution ◽  
Computer Modeling ◽  
Burgers Equation ◽  
Boundary Value ◽  
Three Dimensional ◽  
Benchmark Problems ◽  
One Dimensional ◽  
Radial Basis ◽  
The One

The computer modeling system for numerical solution of the nonlinear one-dimensional non-stationary Burgers’ equation is described. The numerical solution of the Burgers’ equation is obtained by a meshless scheme using the method of partial solutions and radial basis functions. Time discretization of the one-dimensional Burgers’ equation is obtained by the generalized trapezoidal method (θ-scheme). The inverse multiquadric function is used as radial basis functions in the computer modeling system. The computer modeling system allows setting the initial conditions and boundary conditions as well as setting the source function as a coordinate- and time-dependent function for solving partial differential equation. A computer modeling system allows setting such parameters as the domain of the boundary-value problem, number of interpolation nodes, the time interval of non-stationary boundary-value problem, the time step size, the shape parameter of the radial basis function, and coefficients in the Burgers’ equation. The solution of the nonlinear one-dimensional non-stationary Burgers’ equation is visualized as a three-dimensional surface plot in the computer modeling system. The computer modeling system allows visualizing the solution of the boundary-value problem at chosen time steps as three-dimensional plots. The computational effectiveness of the computer modeling system is demonstrated by solving two benchmark problems. For solved benchmark problems, the average relative error, the average absolute error, and the maximum error have been calculated.

scholarly journals On the Numerical Solution of One-Dimensional Nonlinear Nonhomogeneous Burgers’ Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Maryam Sarboland ◽  
Azim Aminataei
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
Meshfree Methods ◽  
Radial Basis Function Networks ◽  
One Dimensional ◽  
Temporal Derivative ◽  
Spatial Derivatives ◽  
The Stability ◽  
The One ◽  
Nonlinear Nature

The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based on the multiquadric (MQ) quasi-interpolation operatorℒ𝒲2and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency of the present methods, several experiments are considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are efficient, robust, and reliable for solving Burgers’ equation.

The United Adaptive Learning Algorithm for the Link Weights and Shape Parameter in RBFN for Pattern Recognition

1997 ◽  
Vol 11 (06) ◽  
pp. 873-888 ◽  
Author(s):  
De-Shuang Huang
Keyword(s):  
Adaptive Learning ◽  
Shape Parameter ◽  
Learning Algorithm ◽  
Basis Functions ◽  
Recursive Least Squares ◽  
Training Method ◽  
One Dimensional ◽  
Radial Basis ◽  
Radar Targets ◽  
The One

This paper proposes a united training method of the link weights of the Gaussian radial basis function networks (GRBFN) and the shape parameter α of the RBF. The training method corresponding to the former is a kind of recursive least squares backpropagation (RLS-BP) learning algorithm which is an accurately recursive method, the training method corresponding to the latter is an adaptive gradient descending (AGD) searching algorithm which is an approximately approaching method. We use the one-dimensional images of radar targets to study the effect of the shape parameter α on the rate of recognition, and survey the changes of the shape parameter αs of radial basis functions corresponding to different hidden nodes, and present the judgement confidence curves of different radar targets. In addition, the forgotten factor λ which makes the effects on the speed of convergence is also discussed. The experimental results are presented.

scholarly journals Taylor's Meshless Petrov-Galerkin Method for the Numerical Solution of Burger's Equation by Radial Basis Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Maryam Sarboland ◽  
Azim Aminataei
Keyword(s):  
Galerkin Method ◽  
Numerical Solution ◽  
Radial Basis Functions ◽  
Taylor Series Expansion ◽  
Time Discretization ◽  
Basis Functions ◽  
One Dimensional ◽  
Radial Basis ◽  
The One ◽  
Rms Errors

During the last two decades, there has been a considerable interest in developing efficient radial basis functions (RBFs) algorithms for solving partial differential equations (PDEs). In this paper, we introduce the Petrov-Galerkin method for the numerical solution of the one-dimensional nonlinear Burger equation. In this method, the trial space is generated by the multiquadric (MQ) RBF and the test space is generated by the compactly supported RBF. In the time discretization of the equation, the Taylor series expansion is used. This method is applied on some test experiments, and the numerical results have been compared with the exact solutions. The , , and root-mean-square (RMS) errors in the solutions show the efficiency and the accuracy of the method.

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.

Wavelets in the Solution of Nongray Radiative Heat Transfer Equation

1998 ◽  
Vol 120 (1) ◽  
pp. 133-139 ◽  
Author(s):  
Y. Bayazitoglu ◽  
B. Y. Wang
Keyword(s):  
Transfer Equation ◽  
Solution Method ◽  
Radiative Heat ◽  
Radiative Transfer Equation ◽  
Basis Functions ◽  
Wavelet Basis ◽  
System Of Equations ◽  
Heat Transfer Equation ◽  
One Dimensional ◽  
The One

The wavelet basis functions are introduced into the radiative transfer equation in the frequency domain. The intensity of radiation is expanded in terms of Daubechies’ wrapped-around wavelet functions. It is shown that the wavelet basis approach to modeling nongrayness can be incorporated into any solution method for the equation of transfer. In this paper the resulting system of equations is solved for the one-dimensional radiative equilibrium problem using the P-N approximation.

Sign in / Sign up

Export Citation Format

Share Document