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.
Recently, fractional differential equations have gained considerable importance due to their frequent appearance applications in fluid flow, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, electrical networks, probability and statistics, control theory of dynamical systems, viscoelasticity, electrochemistry of corrosion, chemical physics, optics and signal processing [
Now, many effective methods for solving fractional differential equations have been presented, such as nonlinear functional analysis method including monotone iterative technique [
The first neural network (NU) is applied to linear and nonlinear fractional differential equations of the form
The second neural network (NU) is applied to the fractional coupled differential equations of the form
The Riemann-Liouville fractional integral of order
The Riemann-Liouville and Caputo fractional derivatives of order
The classical Mittag-Leffler function is defined by
The functions
If
The beta function was defined by
To describe the method, we consider (
Let
Let
We first consider the following linear fractional differential equation:
Weights (
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4857 | 6310 | 6665 | 8226 | 9941 | 8880 | 5415 | 4070 | 5101 |
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4080 | 0219 | 2914 |
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0680 |
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1680 | 1372 | 3715 |
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1896 | 4204 | 5290 |
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3295 | 3947 | 6096 |
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5534 |
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0222 | 1895 | 0482 | 1372 | 2815 | 2901 |
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0182 |
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0424 | 0273 |
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(a) Comparison of results for the solution of Example
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Numerical solution | Accuracy | ||||||
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GL | PSO | GA | NU | GL | PSO | GA | NU | ||
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0.01 | 0.0101 |
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0.04 | 0.0401 | 0.0404 | 0.0396 | 0.0407 |
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0.09 | 0.0901 | 0.0907 | 0.0917 |
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0.16 | 0.1601 | 0.1604 | 0.1596 | 0.1621 |
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0.25 | 0.2501 | 0.2496 | 0.2505 |
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0.36 | 0.3602 | 0.3583 | 0.3573 | 0.3571 |
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0.49 | 0.4902 | 0.4869 | 0.4853 |
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0.64 | 0.6402 | 0.6362 | 0.6352 | 0.6397 |
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0.81 | 0.8102 | 0.8069 | 0.8186 |
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1 | 0.1001 | 0.1000 | 0.1004 | 0.1003 |
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Numerical solution | Accuracy | ||||
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GL | PSO | NU | GL | PSO | NU | ||
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0.01 | 0.0107 | 0.0103 | 0.0092 |
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0.04 | 0.0413 | 0.0414 | 0.0377 |
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0.09 | 0.0918 | 0.0928 | 0.0875 |
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0.16 | 0.1622 | 0.1636 | 0.1592 |
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0.25 | 0.2527 | 0.2538 | 0.2511 |
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0.36 | 0.3631 | 0.3631 | 0.3609 |
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0.49 | 0.4934 | 0.4918 | 0.4884 |
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0.64 | 0.6438 | 0.6402 | 0.6373 |
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0.81 | 0.8141 | 0.8091 | 0.8106 |
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1 | 1.0044 | 0.9991 | 1.0020 |
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The learning curve for Example 1.
The inspection curve for Example 1.
The error curve for Example 1.
We secondly consider the following linear fractional differential equation:
Exact solution, approximate solution, and accuracy for Example
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Numerical solution | Accuracy | |||||
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The learning curve for Example 2.
The inspection curve for Example 2.
The error curve for Example 2.
We thirdly consider the following nonlinear fractional differential equation:
Exact solution, approximate solution, and accuracy for Example
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Numerical solution | Accuracy | |||||
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0.1 | 0.0031 | 0.0022 | 0.0055 | 0.0066 |
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0.2 | 0.0178 | 0.0133 | 0.0234 | 0.0266 |
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0.3 | 0.0492 | 0.0426 | 0.0566 | 0.0603 |
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0.4 | 0.1011 | 0.0972 | 0.1075 | 0.1093 |
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0.5 | 0.1767 | 0.1773 | 0.1783 | 0.1772 |
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0.6 | 0.2788 | 0.2797 | 0.2733 | 0.2711 |
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0.7 | 0.4099 | 0.4055 | 0.4010 | 0.4009 |
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0.8 | 0.5724 | 0.5643 | 0.5712 | 0.5738 |
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0.9 | 0.7684 | 0.7670 | 0.7832 | 0.7847 |
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1 | 1 | 1.0064 | 1.0105 | 1.0056 |
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To describe the method, we consider (
Let
Let
We first consider the following linear coupled fractional differential equations:
Weights obtained along with the solution of Examples
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Example |
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Exact solution, approximate solution, and accuracy for Example
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Numerical solution | Accuracy | ||||
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0.1 | 0.01 | 0.001 | 0.0101 | 0.0020 |
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0.2 | 0.04 | 0.008 | 0.0408 | 0.0104 |
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0.3 | 0.09 | 0.027 | 0.0926 | 0.0301 |
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0.4 | 0.16 | 0.064 | 0.1641 | 0.0664 |
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0.5 | 0.25 | 0.125 | 0.2529 | 0.1233 |
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0.6 | 0.36 | 0.216 | 0.3584 | 0.2071 |
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0.7 | 0.49 | 0.343 | 0.4847 | 0.3290 |
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0.8 | 0.64 | 0.512 | 0.6389 | 0.5033 |
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0.9 | 0.81 | 0.729 | 0.8215 | 0.7361 |
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1 | 1 | 1 | 1.0126 | 1.0076 |
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The learning curve for Example 4.
The inspection curve for Example 4.
The error curve for Example 4.
The graphics of the function
We second consider the following nonlinear fractional coupled differential equations:
Exact solution, approximate solution, and accuracy for Example
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Numerical solution | Accuracy | ||||
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0.1 | 0.9950 | 0.001 | 0.9926 | 0.0042 |
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0.2 | 0.9800 | 0.008 | 0.9834 | 0.0148 |
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0.3 | 0.9553 | 0.027 | 0.9625 | 0.0351 |
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0.4 | 0.9210 | 0.064 | 0.9267 | 0.0689 |
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0.5 | 0.8775 | 0.125 | 0.8773 | 0.1220 |
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0.6 | 0.8253 | 0.216 | 0.8198 | 0.2040 |
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0.7 | 0.7648 | 0.343 | 0.7604 | 0.3276 |
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0.8 | 0.6967 | 0.512 | 0.7004 | 0.5049 |
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0.9 | 0.6216 | 0.729 | 0.6336 | 0.7375 |
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1 | 0.5403 | 1 | 0.5475 | 1.0056 |
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In this paper, by using the neural network, we obtained the numerical solutions for single fractional differential equations and the systems of coupled differential equations of fractional order. The computer graphics demonstrates that numerical results are in well agreement with the exact solutions. In (
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for their many constructive comments and suggestions to improve the paper. This work was partly supported by the Special Funds of the National Natural Science Foundations of China under Grant no. 11247310, the Foundations for Distinguished Young Talents in Higher Education of Guangdong under Grant no. 2012LYM0096, and the Fund of Hanshan Normal University under Grant nos. LY201302 and LF201403.