scholarly journals Radial Basis Function Pseudospectral Method for Solving Standard Fitzhugh-Nagumo Equation

2020 ◽  
Vol 5 (6) ◽  
pp. 1488-1497
Author(s):  
Geeta Arora ◽  
Gurpreet Singh Bhatia
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Shape Parameter ◽  
Cross Validation ◽  
Pseudospectral Method ◽  
Mesh Free ◽  
Radial Basis ◽  
Nagumo Equation ◽  
The Stability ◽  
Leave One Out

In this article, a pseudospectral approach based on radial basis functions is considered for the solution of the standard Fitzhugh-Nagumo equation. The proposed radial basis function pseudospectral approach is truly mesh free. The standard Fitzhugh-Nagumo equation is approximated into ordinary differential equations with the help of radial kernels. An ODE solver is applied to solve the resultant ODEs. Shape parameter which decides the shape of the radial basis function plays a significant role in the solution. A cross-validation technique which is the extension of the statistical approach leave-one-out-cross-validation is used to find the shape parameter value. The presented method is demonstrated with the help of numerical results which shows a good understanding with the exact solution. The stability of the proposed method is demonstrated with the help of the eigenvalues method numerically.

scholarly journals A New Radial Basis Function Approach Based on Hermite Expansion with Respect to the Shape Parameter

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 979
Author(s):  
Saleh A. Bawazeer ◽  
Saleh S. Baakeem ◽  
Abdulmajeed A. Mohamad
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Hermite Polynomial ◽  
Shape Parameter ◽  
Hermite Expansion ◽  
Node Distribution ◽  
Radial Basis ◽  
The Stability ◽  
Stability And Accuracy ◽  
Truncation Order

Owing to its high accuracy, the radial basis function (RBF) is gaining popularity in function interpolation and for solving partial differential equations (PDEs). The implementation of RBF methods is independent of the locations of the points and the dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is mainly affected by the basis function and the node distribution. If the shape parameter has a small value, then the RBF becomes accurate but unstable. Several approaches have been proposed in the literature to overcome the instability issue. Changing or expanding the radial basis function is one of the most commonly used approaches because it addresses the stability problem directly. However, the main issue with most of those approaches is that they require the optimization of additional parameters, such as the truncation order of the expansion, to obtain the desired accuracy. In this work, the Hermite polynomial is used to expand the RBF with respect to the shape parameter to determine a stable basis, even when the shape parameter approaches zero, and the approach does not require the optimization of any parameters. Furthermore, the Hermite polynomial properties enable the RBF to be evaluated stably even when the shape parameter equals zero. The proposed approach was benchmarked to test its reliability, and the obtained results indicate that the accuracy is independent of or weakly dependent on the shape parameter. However, the convergence depends on the order of the truncation of the expansion. Additionally, it is observed that the new approach improves accuracy and yields the accurate interpolation, derivative approximation, and PDE solution.

A Meshfree Numerical Technique Based on Radial Basis Function Pseudospectral Method for Fisher’s Equation

2020 ◽  
Vol 21 (1) ◽  
pp. 37-49
Author(s):  
Geeta Arora ◽  
Gurpreet Singh Bhatia
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Numerical Technique ◽  
Pseudospectral Method ◽  
Numerical Results ◽  
Fisher’S Equation ◽  
Optimal Value ◽  
Radial Basis ◽  
Fisher's Equation ◽  
Leave One Out

AbstractThis paper concerns with the implementation of radial basis function pseudospectral (RBF-PS) method for solving Fisher’s equation. Pseudospectral methods are well known for being highly accurate but are limited in terms of geometric flexibility. Radial basis function (RBF) in combination with the pseudospectral method is capable to overcome this limitation. Using RBF, Fisher’s equation is approximated by transforming it into a system of ordinary differential equations (ODEs). An ODE solver is used to solve the resultant ODEs. In this approach, the optimal value of the shape parameter is discussed with the help of leave-one out cross validation strategy which plays an important role in the accuracy of the result. Several examples are given to demonstrate the accuracy and efficiency of the method. RBF-PS method is applied using different types of basis functions and a comparison is done based upon the numerical results. A two-dimensional problem that generalizes the Fisher’s equation is also solved numerically. The obtained numerical results and comparisons confirm that the use of RBF in pseudospectral mode is in good agreement with already known results in the literature.

scholarly journals New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter

Algorithms ◽  
2020 ◽  
Vol 14 (1) ◽  
pp. 1
Author(s):  
Saleh A. Bawazeer ◽  
Saleh S. Baakeem ◽  
Abdulmajeed A. Mohamad
Keyword(s):  
Radial Basis Function ◽  
Taylor Series ◽  
Basis Function ◽  
Shape Parameter ◽  
Partition Of Unity ◽  
Taylor Series Expansion ◽  
Qr Factorization ◽  
New Approach ◽  
Radial Basis ◽  
The Stability

Radial basis function (RBF) is gaining popularity in function interpolation as well as in solving partial differential equations thanks to its accuracy and simplicity. Besides, RBF methods have almost a spectral accuracy. Furthermore, the implementation of RBF-based methods is easy and does not depend on the location of the points and dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is primarily impacted by the basis function and the node distribution. At a small value of shape parameter, the RBF becomes more accurate, but unstable. Several approaches were followed in the open literature to overcome the instability issue. One of the approaches is optimizing the solver in order to improve the stability of ill-conditioned matrices. Another approach is based on searching for the optimal value of the shape parameter. Alternatively, modified bases are used to overcome instability. In the open literature, radial basis function using QR factorization (RBF-QR), stabilized expansion of Gaussian radial basis function (RBF-GA), rational radial basis function (RBF-RA), and Hermite-based RBFs are among the approaches used to change the basis. In this paper, the Taylor series is used to expand the RBF with respect to the shape parameter. Our analyses showed that the Taylor series alone is not sufficient to resolve the stability issue, especially away from the reference point of the expansion. Consequently, a new approach is proposed based on the partition of unity (PU) of RBF with respect to the shape parameter. The proposed approach is benchmarked. The method ensures that RBF has a weak dependency on the shape parameter, thereby providing a consistent accuracy for interpolation and derivative approximation. Several benchmarks are performed to assess the accuracy of the proposed approach. The novelty of the present approach is in providing a means to achieve a reasonable accuracy for RBF interpolation without the need to pinpoint a specific value for the shape parameter, which is the case for the original RBF interpolation.

A novel local radial basis function collocation method for multi-dimensional piezoelectric problems

2021 ◽  
pp. 1045389X2110572
Author(s):  
Amir Noorizadegan ◽  
Der Liang Young ◽  
Chuin-Shan Chen
Keyword(s):  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Shape Parameter ◽  
Piezoelectric Medium ◽  
The Finite Element Method ◽  
Mechanical Field ◽  
Radial Basis ◽  
2D And 3D

The local radial basis function collocation method (LRBFCM), a strong-form formulation of the meshless numerical method, is proposed for solving piezoelectric medium problems. The proposed numerical algorithm is based on the local Kansa method using variable shape parameter. We introduce a novel technique for the determination of shape parameter in the LRBFCM, which leads to greater accuracy, and simplicity. The implemented algorithm is first verified with a 2D Poisson equation. Then, we employed LRBFCM in a numerical simulation for 2D and 3D piezoelectric problems involving mutual coupling of the electric field and elastodynamic equations for mechanical field. The presented meshless method is verified using corresponding results obtained from the finite element method and moving least squares meshless local Petrov–Galerkin method. In particular, the 2D piezoelectric problem is verified with an exact solution.

Shape Parameter Optimization of Gaussian Radial Basis Function Using Genetic Algorithm for Natural Frequencies of Laminated Composite Plates

2014 ◽  
Vol 709 ◽  
pp. 153-156
Author(s):  
Guo Qing Zhou ◽  
Wei Ping Zhao ◽  
Song Xiang
Keyword(s):  
Genetic Algorithm ◽  
Radial Basis Function ◽  
Basis Function ◽  
Shape Parameter ◽  
Natural Frequencies ◽  
Laminated Composite ◽  
Composite Plates ◽  
Laminated Composite Plates ◽  
Simply Supported ◽  
Radial Basis

Natural frequencies of simply supported laminated composite plates are calculated by the meshless global collocation method based on Gaussian radial basis function. The accuracy of meshless global radial basis function collocation method depends on the choice of shape parameter of radial basis function. In present paper, the shape parameter of Gaussian radial basis function is optimized using the genetic algorithm. Gaussian radial basis function with optimal shape parameter is utilized to analyze the natural frequencies of simply supported laminated composite plates. The present results are compared with the results of available literatures which verify the accuracy of present method.

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