Cosine Radial Basis Function Neural Networks for Solving Fractional Differential Equations

2017 ◽  
Vol 9 (3) ◽  
pp. 667-679 ◽  
Author(s):  
Haidong Qu
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Radial Basis Function ◽  
Fractional Differential Equations ◽  
Basis Function ◽  
Numerical Solutions ◽  
Langevin Equations ◽  
Initial Value ◽  
Radial Basis ◽  
Fractional Differential

AbstractIn this paper, we first apply cosine radial basis function neural networks to solve the fractional differential equations with initial value problems or boundary value problems. In the examples, we successfully obtained the numerical solutions for the fractional Riccati equations and fractional Langevin equations. The computer graphics and numerical solutions show that this method is very effective.

scholarly journals A Numerical Method for Solving Fractional Differential Equations by Using Neural Network

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Haidong Qu ◽  
Xuan Liu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Initial Value ◽  
The Neural Networks ◽  
Fractional Differential

We present a new method for solving the fractional differential equations of initial value problems by using neural networks which are constructed from cosine basis functions with adjustable parameters. By training the neural networks repeatedly the numerical solutions for the fractional differential equations were obtained. Moreover, the technique is still applicable for the coupled differential equations of fractional order. The computer graphics and numerical solutions show that the proposed method is very effective.

scholarly journals A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 270
Author(s):  
Cheng-Yu Ku ◽  
Jing-En Xiao ◽  
Chih-Yu Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Numerical Solutions ◽  
Polynomial Basis ◽  
Partial Differential ◽  
Minimum Order ◽  
Radial Basis

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

scholarly journals A Spectral Deferred Correction Method for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jia Xin ◽  
Jianfei Huang ◽  
Weijia Zhao ◽  
Jiang Zhu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Correction Method ◽  
Initial Value ◽  
Deferred Correction ◽  
Fractional Differential ◽  
Spectral Deferred Correction

A spectral deferred correction method is presented for the initial value problems of fractional differential equations (FDEs) with Caputo derivative. This method is constructed based on the residual function and the error equation deduced from Volterra integral equations equivalent to the FDEs. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDEs without the penalty of a huge computational cost due to the nonlocality of Caputo derivative. Finally, preliminary numerical experiments are given to verify the efficiency and accuracy of this method.

scholarly journals Numerical Simulations for Solving Fuzzy Fractional Differential Equations by Max-Min Improved Euler Methods

2015 ◽  
Vol 7 (1) ◽  
pp. 53-83 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Oyoon Abdul Razzaq ◽  
Fatima Riaz
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Value ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Euler Methods ◽  
Fuzzy Fractional Differential Equations

Abstract In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Two modified fractional Euler type methods have been proposed and investigated to obtain numerical solutions of linear and nonlinear FFDEs. The proposed algorithms are tested on various illustrative examples. Exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained. Comparatively, tables and graphs results reveal the complete reliability, efficiency and accuracy of the proposed methods.

Comparison of back propagation network and radial basis function network in Departure from Nucleate Boiling Ratio (DNBR) calculation

Kerntechnik ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. 15-25
Author(s):  
A. Safavi ◽  
M. H. Esteki ◽  
S. M. Mirvakili ◽  
M. Khaki
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Radial Basis Function ◽  
Basis Function ◽  
Numerical Solutions ◽  
Back Propagation ◽  
Nucleate Boiling ◽  
Inlet Temperature ◽  
Radial Basis ◽  
Propagation Network

Abstract Since estimating the minimum departure from nucleate boiling ratio (MDNBR) requires complex calculations, an alternative method has always been considered. One of these methods is neural network. In this study, the Back Propagation Neural network (BPN) and Radial Basis Function Neural network (RBFN) are introduced and compared in order to estimate MDNBR of the VVER-1000 light water reactor. In these networks, the MDNBR were predicted with the inputs including core mass flux, core inlet temperature, pressure, reactor power level and position of the control rods. To obtain the data required to design these neural networks, an externally coupledcode was developed and its ability to estimate the thermo-hydraulic parameters of the VVER-1000 reactor was compared with other numerical solutions of this benchmark and the Final Safety Analysis Report (FSAR). After ensuring the accuracy of this coupled-code, MDNBR was calculated for 272 different conditions of reactor operating, and it was used to design BPN and RBFN. Comparison of these two neural networks revealed that when the output SMEs of the two systems were approximately the same, the training process in RBFN was much faster than in BPN and the maximum network error in RBFN was less than in BPN.

Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Neda Khodabakhshi ◽  
S. Mansour Vaezpour ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Fractional Differential

AbstractIn this paper, we extend a reliable modification of the Adomian decomposition method presented in [34] for solving initial value problem for fractional differential equations.In order to confirm the applicability and the advantages of our approach, we consider some illustrative examples.

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