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Year 2015, Volume: 36 Issue: 3, 2584 - 2589, 13.05.2015

Abstract

References

  • Table The optimal values of weights and biases. i 1 2 3 4 5 vi -0.7624 0038 -8458 5550 -0.6750 wi b 9275 2401 -5926 i -0138 0.0401 7595 EP(min) = 6.8078e-09
  • Table Comparison of the exact yand approximated y solutions. aand approximated t 6 5 4 3 2 1 i x 0.5 0.4 0.3 0.2 0.1 i 4256 2141 0151 0.8293 0.6574 0.5000 yt 4256 2141 0151 0.8293 0.6574 0.5000 ya 11 1 0.9 xt 0.8 0.7 0.6 6409 3802 1272 8831 6489 yt ya 6409 3802 1272 8831 6489
  • Meade Jr, A.J. and Fernandez, A.A., The numerical solution of linear ordinary differential equations by feed forward neural networks, Mathematical and Computer Modelling 19 (12) (1994), 1-25.
  • Meade Jr, A.J. and Fernandez, A.A., Solution of nonlinear ordinary differential equations by feedforward neural networks, Mathematical and Computer Modelling 20 (9) (1994) 19
  • Dissanayake, M.W.M.G. and Phan-Thien, N., Neuralnetwork based approximations for solving partial differential equations, Communications in Numerical Methods in Engineering 10 (1994) 195-201.
  • Lagaris, I.E. and Likas, Fotiadis, A, D.I., Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks 9 (5) (1998) 987-1000.
  • Liu, Bo-An. and Jammes, B., Solving ordinary differential equations by neural networks, in: Proceeding of 13th European Simulation Multi-Conference Modelling and Simulation: A Tool for the Next Millennium, Warsaw, Poland, June, (1999) 14.
  • Malek A., Beidokhti R. Shekar.i, Numerical solution for high order differential equations, using a hybrid neural networkOptimization method, Applied Mathematics and Computation 183, (2006) 260-271.
  • Effati, S. and Pakdaman, M., Artificial neural network approach for solving fuzzy differential equations Information Sciences 180 (2010) 1434-1457.
  • Ezadi, S. and Parandin, N., An application of neural networks to solve ordinary differential equations, International Journal of Mathematical Modelling & Computations 3 pp. (2013) 245- 252.
  • Ezadi, S. and Parandin, N. and GHomashi, A., Numerical solution of fuzzy differential equations based on Semi- Taylor by using neural network, Journal of Basic and Applied Scientific Research 3(1s) pp. (2013) 477-482.
  • Hornick K., Stinchcombe M., White H. Multilayer feedforward networks are universal approximators, Neural Networks, 2 pp. (1989) 359-366.
  • Liu, C. and Nocedal, J., On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45(3) pp. (1989) 503-528.

Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement

Year 2015, Volume: 36 Issue: 3, 2584 - 2589, 13.05.2015

Abstract

Abstract. In this paper, a new approach is proposed in order to solve the differential equations of ordinary initial value based on the feed-forward neural network and Semi-Taylor series ordinary differential equation is first replaced by a system of ordinary differential equations. A trial Solution of this System consists of two parts. The first part, that is, Semi-Taylor series contains no adjustable parameters. And the second part includes the neural network and adjustable parameters (the weights). Using modified neural network requires that training points be selected over the open interval (a, b) without training the network in the range of the first and end points. Therefore, the calculating volume containing computational error is reduced. In fact, the training points depending on the distance [a, b] selected for training neural networks are converted into similar points in the open interval (a, b) using the new approach, then the network is trained in these similar areas.In comparison with existing similar neural networks, the proposed model provides solutions with high accuracy. Numerical examples with simulation results illustrate the effectiveness of the proposed model.

References

  • Table The optimal values of weights and biases. i 1 2 3 4 5 vi -0.7624 0038 -8458 5550 -0.6750 wi b 9275 2401 -5926 i -0138 0.0401 7595 EP(min) = 6.8078e-09
  • Table Comparison of the exact yand approximated y solutions. aand approximated t 6 5 4 3 2 1 i x 0.5 0.4 0.3 0.2 0.1 i 4256 2141 0151 0.8293 0.6574 0.5000 yt 4256 2141 0151 0.8293 0.6574 0.5000 ya 11 1 0.9 xt 0.8 0.7 0.6 6409 3802 1272 8831 6489 yt ya 6409 3802 1272 8831 6489
  • Meade Jr, A.J. and Fernandez, A.A., The numerical solution of linear ordinary differential equations by feed forward neural networks, Mathematical and Computer Modelling 19 (12) (1994), 1-25.
  • Meade Jr, A.J. and Fernandez, A.A., Solution of nonlinear ordinary differential equations by feedforward neural networks, Mathematical and Computer Modelling 20 (9) (1994) 19
  • Dissanayake, M.W.M.G. and Phan-Thien, N., Neuralnetwork based approximations for solving partial differential equations, Communications in Numerical Methods in Engineering 10 (1994) 195-201.
  • Lagaris, I.E. and Likas, Fotiadis, A, D.I., Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks 9 (5) (1998) 987-1000.
  • Liu, Bo-An. and Jammes, B., Solving ordinary differential equations by neural networks, in: Proceeding of 13th European Simulation Multi-Conference Modelling and Simulation: A Tool for the Next Millennium, Warsaw, Poland, June, (1999) 14.
  • Malek A., Beidokhti R. Shekar.i, Numerical solution for high order differential equations, using a hybrid neural networkOptimization method, Applied Mathematics and Computation 183, (2006) 260-271.
  • Effati, S. and Pakdaman, M., Artificial neural network approach for solving fuzzy differential equations Information Sciences 180 (2010) 1434-1457.
  • Ezadi, S. and Parandin, N., An application of neural networks to solve ordinary differential equations, International Journal of Mathematical Modelling & Computations 3 pp. (2013) 245- 252.
  • Ezadi, S. and Parandin, N. and GHomashi, A., Numerical solution of fuzzy differential equations based on Semi- Taylor by using neural network, Journal of Basic and Applied Scientific Research 3(1s) pp. (2013) 477-482.
  • Hornick K., Stinchcombe M., White H. Multilayer feedforward networks are universal approximators, Neural Networks, 2 pp. (1989) 359-366.
  • Liu, C. and Nocedal, J., On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45(3) pp. (1989) 503-528.
There are 13 citations in total.

Details

Journal Section Special
Authors

Somayeh Ezadi

Sahar Aksarı This is me

Mitra Jasemı This is me

Publication Date May 13, 2015
Published in Issue Year 2015 Volume: 36 Issue: 3

Cite

APA Ezadi, S., Aksarı, S., & Jasemı, M. (2015). Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi, 36(3), 2584-2589.
AMA Ezadi S, Aksarı S, Jasemı M. Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi. May 2015;36(3):2584-2589.
Chicago Ezadi, Somayeh, Sahar Aksarı, and Mitra Jasemı. “Numerical Solution of Ordinary Differential Equations Based on Semi-Taylor by Neural Network Improvement”. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi 36, no. 3 (May 2015): 2584-89.
EndNote Ezadi S, Aksarı S, Jasemı M (May 1, 2015) Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi 36 3 2584–2589.
IEEE S. Ezadi, S. Aksarı, and M. Jasemı, “Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement”, Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi, vol. 36, no. 3, pp. 2584–2589, 2015.
ISNAD Ezadi, Somayeh et al. “Numerical Solution of Ordinary Differential Equations Based on Semi-Taylor by Neural Network Improvement”. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi 36/3 (May 2015), 2584-2589.
JAMA Ezadi S, Aksarı S, Jasemı M. Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi. 2015;36:2584–2589.
MLA Ezadi, Somayeh et al. “Numerical Solution of Ordinary Differential Equations Based on Semi-Taylor by Neural Network Improvement”. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi, vol. 36, no. 3, 2015, pp. 2584-9.
Vancouver Ezadi S, Aksarı S, Jasemı M. Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement. Cumhuriyet Üniversitesi Fen Edebiyat Fakültesi Fen Bilimleri Dergisi. 2015;36(3):2584-9.