scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

Radial basis functions

Acta Numerica ◽  
2000 ◽  
Vol 9 ◽  
pp. 1-38 ◽  
Author(s):  
M. D. Buhmann
Keyword(s):  
Radial Basis Function ◽  
Radial Basis Functions ◽  
Basis Function ◽  
Convergence Rates ◽  
Basis Functions ◽  
Point Of View ◽  
Theoretical Point ◽  
Grid Data ◽  
Radial Basis ◽  
Recent Developments

Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper.

Power Quality Events Classification using ANN with Hilbert Transform

2018 ◽  
Vol 6 (6) ◽  
pp. 227
Author(s):  
Tarun Kumar Chheepa ◽  
Tanuj Manglani
Keyword(s):  
Neural Network ◽  
Radial Basis Function ◽  
Power Quality ◽  
Hilbert Transform ◽  
Basis Function ◽  
Additive White Gaussian Noise ◽  
Hybrid Approach ◽  
Feed Forward ◽  
Radial Basis ◽  
Distributed Time Delay

With the evolution of Smart Grid, Power Quality issues have become prominent. The urban development involves usage of computers, microprocessor controlled electronic loads and power electronic devices. These devices are the source of power quality disturbances.  PQ problems are characterized by the variations in the magnitude and frequency in the system voltages and currents from their nominal values. To decide a control action, a proper classification mechanism is required to classify different PQ events. In this paper we propose a hybrid approach to perform this task. Different Neural topologies namely Cascade Forward Backprop Neural Network (CFBNN), Elman Backprop Neural Network (EBPNN), Feed Forward Backprop Neural Network (FFBPNN),  Feed Forward Distributed Time Delay Neural Network (FFDTDNN) , Layer Recurrent Neural Network (LRNN), Nonlinear Autoregressive Exogenous Neural Network (NARX),  Radial Basis Function Neural Network (RBFNN)  along with the application of Hilbert Transform are employed to classify the PQ events. A meaningful comparison of these neural topologies is presented and it is found that Radial Basis Function Neural Network (RBFNN) is the most efficient topology to perform the classification task. Different levels of Additive White Gaussian Noise (AWGN) are added in the input features to present the comparison of classifiers.

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