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Solving Partial Differential Equation Based on Bernstein Neural Network and Extreme Learning Machine Algorithm

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Abstract

In this paper, we introduce a new method based on Bernstein Neural Network model (BeNN) and extreme learning machine algorithm to solve the differential equation. In the proposed method, we develop a single-layer functional link BeNN, the hidden layer is eliminated by expanding the input pattern by Bernstein polynomials. The network parameters are obtained by solving a system of linear equations using the extreme learning machine algorithm. Finally, the numerical experiment is carried out by MATLAB, results obtained are compared with the existing method, which proves the feasibility and superiority of the proposed method.

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References

  1. Butcher JC (1987) The numerical analysis of ordinary differential equations: Runge Kutta and general linear methods. Math Comput 51(183):693

    Google Scholar 

  2. Verwer JG (1996) Explicit Runge–Kutta methods for parabolic partial differential equations. Appl Numer Math 22(1–3):359–379

    Article  MathSciNet  MATH  Google Scholar 

  3. Hamming RW (1959) Stable predictor corrector methods for ordinary differential equations. J ACM 6(1):37–47

    Article  MathSciNet  MATH  Google Scholar 

  4. Tseng AA, Gu SX (1989) A finite difference scheme with arbitrary mesh system for solving high order partial differential equations. Comput Struct 31(3):319–328

    Article  MathSciNet  MATH  Google Scholar 

  5. Wu B, White RE (2004) One implementation variant of finite difference method for solving ODEs/DAEs. Comput Chem Eng 28(3):303–309

    Article  Google Scholar 

  6. Kumar M, Kumar P (2009) Computational method for finding various solutions for a quasilinear elliptic equations with periodic solutions. Adv Eng Softw 40(11):1104–1111

    Article  MATH  Google Scholar 

  7. Thomee V (2001) From finite difference to finite elements:a short history of numerical analysis of partial differential equations. J Comput Appl Math 128(1–2):1–54

    Article  MathSciNet  MATH  Google Scholar 

  8. Sallam S, Ameen W (1990) Numerical solution of general nth order differential equations via splines. Appl Numer Math 6(3):225–238

    Article  MathSciNet  MATH  Google Scholar 

  9. Ei-Hawary HM, Mahmoud SM (2003) Spline collocation methods for solving delay differential equations. Appl Math Comput 146(2–3):359–372

    MathSciNet  MATH  Google Scholar 

  10. Tsoulos IG, Gavrilis D, Glavas E (2009) Solving differential equations with constructed neural networks. Neurocomputing 72(10–12):2385–2391

    Article  Google Scholar 

  11. Liu J, Hou G (2011) Numerical solutions of the space-and time-fractional coupled Burgers equations by generalized differential transform method. Appl Math Comput 217(16):7001–7008

    MathSciNet  MATH  Google Scholar 

  12. Darnia P, Ebadian A (2007) A method for the numerical solution of the integro-differential equations. Appl Math Comput 188(1):657–668

    MathSciNet  Google Scholar 

  13. Kumar M, Singh N (2009) A collection of computational techniques for solving singular boundary value problems. Adv Eng Softw 40(4):288–297

    Article  MATH  Google Scholar 

  14. Wang Q, Cheng D (2005) Numerical solution of damped nonlinear Klein–Gordon equations using variational method and finite element approach. Appl Math Comput 162(1):381–401

    MathSciNet  MATH  Google Scholar 

  15. Kumar M, Singh N (2010) Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems. Comput Chem Eng 34(11):1750–1760

    Article  Google Scholar 

  16. Yang X, Liu Y, Bai S (2006) A numerical solution of second order linear partial differential equation by differential transform. Appl Math Comput 173(2):792–802

    MathSciNet  MATH  Google Scholar 

  17. Erturk VS, Momani S (2008) Solving system of fractional differential equations using differential transform method. J Comput Appl Math 215(1):142–151

    Article  MathSciNet  MATH  Google Scholar 

  18. Coronel-Escamilla A, Gomez-Aguilar JF, Torres L, Escobar-Jimenez RF, Valtierra-Rodriguez M (2017) Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order. Phys A 487:1–21

    Article  MathSciNet  Google Scholar 

  19. Coronel-Escamilla A, Gomez-Aguilar JF, Torres L, Escobar-Jimenez RF (2018) A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A 491:406–424

    Article  MathSciNet  Google Scholar 

  20. Zuniga-Aguilar CJ, Gomez-Aguilar JF, Escobar-Jimenez RF, Romero-Ugalde HM (2018) Robust control for fractional variable-order chaotic systems with non-singular kernel. Eur Phys J Plus 133(1):1–13

    Article  MATH  Google Scholar 

  21. Yazdi HS, Pakdaman M, Modaghegh H (2011) Unsupervised kernel least mean square algorithm for solving ordinary differential equations. Neurocomputing 74(12–13):2062–2071

    Article  Google Scholar 

  22. Lagaris IE, Likas A (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5):987–1000

    Article  Google Scholar 

  23. Lagaris IE, Likas AC (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11(5):1041

    Article  Google Scholar 

  24. Mai-Duy N, Tran-Cong T (2001) Numerical solution of differential equations using multiquadric radial basis function networks. Neural Netw 14(2):185–199

    Article  MATH  Google Scholar 

  25. Aarts LP, Veer PVD (2001) Neural network method for solving partial differential equations. Neural Process Lett 14(3):261–271

    Article  MATH  Google Scholar 

  26. Mai-Duy N, Tran-Cong T (2003) Approximation of function and its derivatives using radial basis function networks. Appl Math Model 27(3):197–220

    Article  MATH  Google Scholar 

  27. Li J, Luo S, Qi Y, Huang Y (2003) Numerical solution of elliptic partial differential equation using radial basis function neural networks. IJCNN 16(5–6):85–90

    Google Scholar 

  28. Malek A, Beidokhti RS (2006) Numerical solution for high order differential equations using a hybrid neural network—optimization method. Appl Math Comput 183(1):260–271. https://doi.org/10.1016/j.amc.2006.05.068

    Article  MathSciNet  MATH  Google Scholar 

  29. Xu LY, Wen H, Zeng ZZ (2007) The algorithm of neural networks on the initial value problems in ordinary differential equations. In: IEEE conference on industrial electronics & applications, pp 813–816

  30. Aein MJ, Talebi HA (2009) Introducing a training methodology for cellular neural networks solving partial differential equations. IJCNN, Atlanta, Georgia, USA, June 14–19, pp 71–75

  31. Beidokhti RS, Malek A (2009) Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques. J Franklin Inst 346(9):898–913

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu C, Wang C, Ji F, Yuan X (2012) Finite-element neural network-based solving 3-D differential equations in MFL. IEEE Trans Magn 48(12):4747–4756

    Article  Google Scholar 

  33. Mcfall KS (2013) Automated design parameter selection for neural networks solving coupled partial differential equations with discontinuities. J Franklin Inst 350(2):300–317

    Article  MathSciNet  MATH  Google Scholar 

  34. Zuniga-Aguilar CJ, Romero-Ugalde HM, Gomez-Aguilar JF, Escobar-Jimenez RF, Valtierra-Rodriguez M (2017) Solving fractional differential equations of variable-order involving operators with Mittag–Leffler kernel using artificial neural networks. Chaos Soliton Fract 103:382–403

    Article  MathSciNet  MATH  Google Scholar 

  35. Pao YH, Phillips SM (1995) The functional link net and learning optimal control. Neurocomputing 9(2):149–164

    Article  MATH  Google Scholar 

  36. Lee TT, Jeng JT (1998) The Chebyshev-polynomials based unified model neural networks for function approximation. IEEE Trans Syst Man Cybern 28(6):925–935

    Article  Google Scholar 

  37. Yang SS, Tseng CS (1996) An orthogonal neural network for function approximation. IEEE Trans Syst Man Cybern 26(5):779

    Article  Google Scholar 

  38. Patra JC, Juhola M, Meher PK (2008) Intelligent sensors using computationally efficient Chebyshev neural networks. IET Sci Meas Technol 2(2):68–75

    Article  Google Scholar 

  39. Weng WD, Yang CS, Lin RC (2007) A channel equalizer using reduced decision feedback Chebyshev functional link artificial neural networks. Inf Sci 177(13):2642–2654

    Article  MATH  Google Scholar 

  40. Patra JC, Kot AC (2002) Nonlinear dynamic system identification using Chebyshev functional link artificial neural network. IEEE Trans Syst Man Cybern 32(4):505–511

    Article  Google Scholar 

  41. Purwar S, Kar IN, Jha AN (2007) Online system identification of complex systems using Chebyshev neural network. Appl Soft Comput 7(1):364–372

    Article  Google Scholar 

  42. Mall S, Chakraverty S (2016) Application of Legendre neural network for solving ordinary differential equations. Appl Soft Comput 43(C):347–356. https://doi.org/10.1016/j.asoc.2015.10.069

    Article  Google Scholar 

  43. Mall S, Chakraverty S (2017) Single layer Chebyshev neural network model for solving elliptic partial differential equations. Neural Process Lett 45(3):825–840

    Article  Google Scholar 

  44. Huang GB, Zhu QY, Siew CK (2006) Extreme learning machine: theory and applications. Neurocomputing 70(1–3):489–501

    Article  Google Scholar 

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Acknowledgements

This study was funded by the National Natural Science Foundation of China under Grants 61375063, 61271355, 11301549 and 11271378. This study was also funded by the Central South University Caitian Xuanzhu student innovation and business Projects 201710533542.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Hongli Sun, Muzhou Hou, Yunlei Yang, Tianle Zhang & Futian Weng

  2. College of Science, Hunan University of Technology, Hunan, China

    Feng Han

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  1. Hongli Sun
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  2. Muzhou Hou
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  3. Yunlei Yang
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  4. Tianle Zhang
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  5. Futian Weng
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  6. Feng Han
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Correspondence to Muzhou Hou.

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Sun, H., Hou, M., Yang, Y. et al. Solving Partial Differential Equation Based on Bernstein Neural Network and Extreme Learning Machine Algorithm. Neural Process Lett 50, 1153–1172 (2019). https://doi.org/10.1007/s11063-018-9911-8

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