| Citation: |
Neha Yadav, Thi Thuy Ngo, Joong Hoon Kim. AN ALGORITHM FOR NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS USING HARMONY SEARCH AND NEURAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1277-1293. doi: 10.11948/20200377
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AN ALGORITHM FOR NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS USING HARMONY SEARCH AND NEURAL NETWORKS
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Abstract
In this article, an algorithm based on artificial neural networks (ANN) and harmony search algorithm (HSA) is presented for the numerical solution of ordinary and partial differential equations. The power of ANN is used to construct an approximate solution of differential equations (DEs) such that it satisfies the DEs initial conditions (ICs) or boundary conditions (BCs) automatically. An automated design parameter selection approach is utilised to pick the optimum ANN ensemble from various combinations of ANN design parameters, random beginning weights, and biases. An unsupervised error is constructed in order to approximate the DE solution and HSA is used to minimize this error by training the neural network design parameters. A few test problems of various types are considered for verifying the algorithm's accuracy, convergence, and efficacy. The proposed algorithm is assessed using the results of statistical analysis obtained from a large number of independent runs for each model equation. The correctness and validity of the algorithm is also verified by comparing the obtained numerical results to the exact solution.
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References
[1] W. E. Boyce, R. C. DiPrima and D. B. Meade, Elementary differential equations and boundary value problems, John Wiley & Sons, 2021. [2] J. P. Boyd, One-point pseudospectral collocation for the one-dimensional bratu equation, Applied Mathematics and Computation, 2011, 217(12), 5553-5565. doi: 10.1016/j.amc.2010.12.029 [3] M. A. Darwish and B. S. Kashkari, Numerical solutions of second order initial value problems of bratu-type via optimal homotopy asymptotic method, American Journal of Computational Mathematics, 2014, 4(2), 47. doi: 10.4236/ajcm.2014.42005 [4] E. Deeba, S. Khuri and S. Xie, An algorithm for solving boundary value problems, Journal of Computational Physics, 2000, 159(2), 125-138. doi: 10.1006/jcph.2000.6452 [5] M. A. El-Tawil, A. A. Bahnasawi and A. Abdel-Naby, Solving riccati differential equation using adomian's decomposition method, Applied Mathematics and Computation, 2004, 157(2), 503-514. doi: 10.1016/j.amc.2003.08.049 [6] X. Feng, L. Mei and G. He, An efficient algorithm for solving troesch's problem, Applied Mathematics and Computation, 2007, 189(1), 500-507. doi: 10.1016/j.amc.2006.11.161 [7] Z. W. Geem, Music-inspired harmony search algorithm: theory and applications, 191, Springer, 2009. [8] F. Ghomanjani and S. Shateyi, Numerical solution for fractional bratu's initial value problem, Open Physics, 2017, 15(1), 1045-1048. doi: 10.1515/phys-2017-0131 [9] H. N. Hassan and M. A. El-Tawil, An efficient analytic approach for solving two-point nonlinear boundary value problems by homotopy analysis method, Mathematical methods in the applied sciences, 2011, 34(8), 977-989. doi: 10.1002/mma.1416 [10] A. Kattan, R. Abdullah and R. A. Salam, Harmony search based supervised training of artificial neural networks, in 2010 International conference on intelligent systems, modelling and simulation, IEEE, 2010, 105-110. [11] W. Khan and I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, International journal of heat and mass transfer, 2010, 53(11-12), 2477-2483. doi: 10.1016/j.ijheatmasstransfer.2010.01.032 [12] S. Kulluk, L. Ozbakir and A. Baykasoglu, Training neural networks with harmony search algorithms for classification problems, Engineering Applications of Artificial Intelligence, 2012, 25(1), 11-19. doi: 10.1016/j.engappai.2011.07.006 [13] M. Kumar and N. Yadav, Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: a survey, Computers & Mathematics with Applications, 2011, 62(10), 3796-3811. [14] I. E. Lagaris, A. C. Likas and D. G. Papageorgiou, Neural-network methods for boundary value problems with irregular boundaries, IEEE Transactions on Neural Networks, 2000, 11(5), 1041-1049. doi: 10.1109/72.870037 [15] K. S. McFall, Automated design parameter selection for neural networks solving coupled partial differential equations with discontinuities, Journal of the Franklin Institute, 2013, 350(2), 300-317. doi: 10.1016/j.jfranklin.2012.11.003 [16] K. S. McFall and J. R. Mahan, Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions, IEEE Transactions on Neural Networks, 2009, 20(8), 1221-1233. doi: 10.1109/TNN.2009.2020735 [17] S. Momani and N. Shawagfeh, Decomposition method for solving fractional riccati differential equations, Applied Mathematics and Computation, 2006, 182(2), 1083-1092. doi: 10.1016/j.amc.2006.05.008 [18] L. Özbakır, A. Baykasoğlu and S. Kulluk, A soft computing-based approach for integrated training and rule extraction from artificial neural networks: Difaconn-miner, Applied Soft Computing, 2010, 10(1), 304-317. doi: 10.1016/j.asoc.2009.08.008 [19] H. Qu, X. Liu and Z. She, Neural network method for fractional-order partial differential equations, Neurocomputing, 2020, 414, 225-237. doi: 10.1016/j.neucom.2020.07.063 [20] M. A. Z. Raja, Stochastic numerical treatment for solving troesch's problem, Information Sciences, 2014, 279, 860-873. doi: 10.1016/j.ins.2014.04.036 [21] M. A. Z. Raja, J. Khan and I. Qureshi, Swarm intelligent optimized neural networks for solving fractional differential equations, International Journal of Innovative Computing, Information and Control, 2011, 7(11), 6301-6318. [22] M. A. Z. Raja, J. A. Khan and I. M. Qureshi, A new stochastic approach for solution of riccati differential equation of fractional order, Annals of Mathematics and Artificial Intelligence, 2010, 60(3), 229-250. [23] S. Roberts and J. Shipman, On the closed form solution of troesch's problem, Journal of Computational Physics, 1976, 21(3), 291-304. doi: 10.1016/0021-9991(76)90026-7 [24] B. Troesch, A simple approach to a sensitive two-point boundary value problem, Journal of Computational Physics, 1976, 21(3), 279-290. doi: 10.1016/0021-9991(76)90025-5 [25] A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the bratu-type equations, Applied Mathematics and Computation, 2005, 166(3), 652-663. doi: 10.1016/j.amc.2004.06.059 [26] N. Yadav, T. T. Ngo, A. Yadav and J. H. Kim, Numerical solution of boundary value problems using artificial neural networks and harmony search, in International Conference on Harmony Search Algorithm, Springer, 2017, 112-118. [27] N. Yadav, A. Yadav and J. H. Kim, Numerical solution of unsteady advection dispersion equation arising in contaminant transport through porous media using neural networks, Computers & Mathematics with Applications, 2016, 72(4), 1021-1030. [28] N. Yadav, A. Yadav, M. Kumar and J. H. Kim, An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear troesch's problem, Neural Computing and Applications, 2017, 28(1), 171-178. doi: 10.1007/s00521-015-2046-1 [29] Y. Yang, M. Hou and J. Luo, A novel improved extreme learning machine algorithm in solving ordinary differential equations by legendre neural network methods, Advances in Difference Equations, 2018, 2018(1), 1-24. doi: 10.1186/s13662-017-1452-3 [30] Y. Yang, M. Hou, H. Sun et al., Neural network algorithm based on legendre improved extreme learning machine for solving elliptic partial differential equations, Soft Computing, 2020, 24(2), 1083-1096. doi: 10.1007/s00500-019-03944-1 -
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Neha Yadav, Thi Thuy Ngo, Joong Hoon Kim. AN ALGORITHM FOR NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS USING HARMONY SEARCH AND NEURAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1277-1293. doi: 10.11948/20200377
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- Figure 1. Procedure to compute approximate solution of DEs using the ANN and HSA
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Figure 2. Boxplots for MAE in the solution with 100 runs for solving test problem 1 with (a)
$ \mu=0.5 $ and (b)$ \mu=1.0 $ -
Figure 3. Exact and ANN solutions for test problem 1 for (a)
$ \mu=0.5 $ and (b)$ \mu=1.0 $ - Figure 4. Boxplots for MAE in the solution for 100 runs for solving test problem 2
- Figure 5. Exact and ANN solutions for test problem 2
- Figure 6. Boxplots for MAE in the solution for 100 runs for solving test problem 3
- Figure 7. Exact and ANN solutions for test problem 3
- Figure 8. Boxplots for MAE in the solution for 100 runs for solving test problem 4
- Figure 9. Exact and ANN solutions for test problem 4
- Figure 10. Boxplots for MAE in the solution for 100 runs for solving test problem 5
- Figure 11. Exact and ANN solutions for test problem 5
- Figure 12. Summary of minimization of EFs over the number of iterations (a) for test problems 1–3 and (b) for test problems 4, 5