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The novel learning solutions to nonlinear differential models on a semi-infinite domain

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Abstract

The aim of this paper is to introduce a new numerical approach named least-squares support vector machines based on generalized Laguerre functions collocation quasilinearization method (LS-SVM-GLQ). LS-SVM-GLQ combines collocation Quasilinearization methods based on generalized Laguerre functions and least-squares support vector machines to solve nonlinear differential equations (NDEs) on a semi-infinite domain. Applying LS-SVM-GLQ leads to solve a system of nonlinear/linear equations instead of solving the NDEs. The different types of Lane–Emden, Emden–Fowler and White-dwarf equations are investigated. Comparing numerical results with other numerical techniques declares that the present approach is more accurate and efficient.

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References

  1. Delkhosh M, Parand K, Ganji DD (2019) An efficient numerical method to solve the boundary layer flow of an Eyring-Powell non-Newtonian fluid. J Appl Comput Mech 5(2):454–467

    Google Scholar 

  2. Yonthanthum W, Rattana A, Razzaghi M (2018) An approximate method for solving fractional optimal control problems by the hybrid of block-pulse functions and Taylor polynomials. Optim Control Appl Methods 39(2):873–887

    MathSciNet  MATH  Google Scholar 

  3. Dehestani H, Ordokhani Y, Razzaghi M (2018) Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations. Appl Math Comput 336:433–453

    MathSciNet  MATH  Google Scholar 

  4. Mohammadi V, Dehghan M, Khodadadian A, Wick T (2021) Numerical investigation on the transport equation in spherical coordinates via generalized moving least squares and moving kriging least squares approximations. Eng Comput 37(2):1231–1249

    Google Scholar 

  5. Mashayekhi S, Razzaghi M (2018) An approximate method for solving fractional optimal control problems by hybrid functions. J Vib Control 24(9):1621–1631

    MathSciNet  MATH  Google Scholar 

  6. Mohammadi V, Dehghan M (2019) Simulation of the phase field Cahn-Hilliard and tumor growth models via a numerical scheme: element-free Galerkin method. Comput Methods Appl Mech Eng 345:919–950

    MathSciNet  MATH  Google Scholar 

  7. Keshavarz E, Ordokhani Y, Razzaghi M (2018) The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations. Appl Numer Math 128:205–216

    MathSciNet  MATH  Google Scholar 

  8. Dehghan M, Abbaszadeh M (2019) Error analysis and numerical simulation of magnetohydrodynamics (MHD) equation based on the interpolating element free Galerkin (IEFG) method. Appl Numer Math 137:252–273

    MathSciNet  MATH  Google Scholar 

  9. Abbaszadeh M, Dehghan M (2019) The interpolating element-free Galerkin method for solving Korteweg-de Vries-Rosenau-regularized long-wave equation with error analysis. Nonlinear Dyn 96(2):1345–1365

    MATH  Google Scholar 

  10. Guo BY, Shen J (2000) Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numer Math 86(4):635–654

    MathSciNet  MATH  Google Scholar 

  11. Siyyam HI (2001) Laguerre Tau methods for solving higher-order ordinary differential equations. J Comput Anal Appl 3(2):173–182

    MathSciNet  MATH  Google Scholar 

  12. Wazwaz AM (2001) A new algorithm for solving differential equations of Lane-Emden type. Appl Math Comput 118(2–3):287–310

    MathSciNet  MATH  Google Scholar 

  13. Parand K, Razzaghi M (2004) Rational Legendre approximation for solving some physical problems on semi-infinite intervals. Phys Scr 69(5):353–357

    MATH  Google Scholar 

  14. Guo BY, Shen J, Xu CL (2005) Generalized Laguerre approximation and its applications to exterior problems. J Comput Math 23:113–130

    MathSciNet  MATH  Google Scholar 

  15. Yousefi SA (2006) Legendre wavelets method for solving differential equations of Lane-Emden type. Appl Math Comput 181(2):1417–1422

    MathSciNet  MATH  Google Scholar 

  16. Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71(1):16–30

    MathSciNet  MATH  Google Scholar 

  17. Yıldırım A, Öziş T (2007) Solutions of singular IVPs of Lane-Emden type by Homotopy perturbation method. Phys Lett A 369(1–2):70–76

    MATH  Google Scholar 

  18. Parand K, Dehghan M, Pirkhedri A (2009) Sinc-collocation method for solving the Blasius equation. Phys Lett A 373(44):4060–4065

    MathSciNet  MATH  Google Scholar 

  19. Parand K, Taghavi A (2009) Rational scaled generalized Laguerre function collocation method for solving the Blasius equation. J Comput Appl Math 233(4):980–989

    MathSciNet  MATH  Google Scholar 

  20. Parand K, Shahini M (2009) Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation. Phys Lett A 373(2):210–213

    MathSciNet  MATH  Google Scholar 

  21. Parand K, Taghavi A, Shahini M (2009) Comparison between rational Chebyshev and modified generalized Laguerre functions pseudo spectral methods for solving Lane-Emden and unsteady gas equations. Acta Phys Pol B 40(6):1749–1763

    Google Scholar 

  22. Parand K, Dehghan M, Taghavi A (2010) Modified generalized Laguerre function tau method for solving laminar viscous flow: the Blasius equation. Int J Numer Methods Heat Fluid Flow 20(7):728–743

    MathSciNet  Google Scholar 

  23. Parand K, Nikarya M, Rad JA, Baharifard F (2012) A new reliable numerical algorithm based on the first kind of Bessel functions to solve Prandtl-Blasius laminar viscous flow over a semi-infinite flat plate. Z Naturforschung A 67(12):665–673

    Google Scholar 

  24. Bhrawy AH, Alofi AS (2012) A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations. Commun Nonlinear Sci Numer Simul 17(1):62–70

    MathSciNet  MATH  Google Scholar 

  25. Kazem S, Rad JA, Parand K, Shaban M, Saberi H (2012) The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method. Int J Comput Math 89(16):2240–2258

    MathSciNet  MATH  Google Scholar 

  26. Pandey RK, Kumar N (2012) Solution of Lane-Emden type equations using Bernstein operational matrix of differentiation. New Astron 17(3):303–308

    Google Scholar 

  27. Parand K, Baharifard F, Babolghani FB (2012) Comparison between rational Gegenbauer and modified generalized Laguerre functions collocation methods for solving the case of heat transfer equations arising in porous medium. Int J Ind Math 4(2):107–122

    Google Scholar 

  28. Babolghani FB, Parand K (2013) Comparison between Hermite and Sinc collocation methods for solving steady flow of a third grade fluid in a porous half space. In: Proceedings of the international conference on scientific computing (CSC), pp 124–129

  29. Parand K, Dehghan M, Pirkhedri A (2013) TheSinc-collocation method for solving the Thomas-Fermi equation. J Comput Appl Math 237(1):244–252

    MathSciNet  MATH  Google Scholar 

  30. Yüzbaşı Ş, Sezer M (2013) An improved Bessel collocation method with a residual error function to solve a class of Lane-Emden differential equations. Math Comput Model 57(5–6):1298–1311

    MathSciNet  Google Scholar 

  31. Parand K, Khaleqi S (2016) The rational Chebyshev of second kind collocation method for solving a class of astrophysics problems. Eur Phys J Plus 131(2):24–47

    Google Scholar 

  32. Comte F, Cuenod CA, Pensky M, Rozenholc Y (2017) Laplace deconvolution on the basis of time domain data and its application to dynamic contrast-enhanced imaging. J R Stat Soc Ser B Stat Methodol 79(1):69–94

    MathSciNet  MATH  Google Scholar 

  33. Yüzbaşı Ş (2017) A Laguerre approach for the solutions of singular perturbated differential equations. Int J Comput Methods 14(04):1750034

    MathSciNet  MATH  Google Scholar 

  34. Parand K, Lotfi Y, Rad JA (2017) An accurate numerical analysis of the laminar two-dimensional flow of an incompressible Eyring-Powell fluid over a linear stretching sheet. Eur Phys J Plus 132(9):397–417

    Google Scholar 

  35. Parand K, Moayeri MM, Latifi S, Delkhosh M (2017) A numerical investigation of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet via rational Chebyshev functions. Eur Phys J Plus 132(7):325–335

    Google Scholar 

  36. Parand K, Hemami M (2017) Application of meshfree method based on compactly supported radial basis function for solving unsteady isothermal gas through a micro-nano porous medium. Iran J Sci Technol Trans A Sci 41(3):677–684

    MathSciNet  MATH  Google Scholar 

  37. Parand K, Hajimohammadi Z (2018) Using modified generalized Laguerre functions, QLM and collocation method for solving an Eyring-Powell problem. J Braz Soc Mech Sci Eng 40(4):182–190

    Google Scholar 

  38. Lin CH (2017) Application of a V-belt continuously variable transmission system by using a composite recurrent Laguerre orthogonal polynomial neural network control system and modified particle swarm optimization. J Vib Control 23(9):1437–1462

    MathSciNet  Google Scholar 

  39. Agarwal P, Qi F, Chand M, Jain S (2017) Certain integrals involving the generalized hypergeometric function and the Laguerre polynomials. J Comput Appl Math 313:307–317

    MathSciNet  MATH  Google Scholar 

  40. Xu S, Li Y, Huang T, Chan R (2017) A sparse multiwavelet-based generalized Laguerre-Volterra model for identifying time-varying neural dynamics from spiking activities. Entropy 19(8):425–446

    Google Scholar 

  41. Makarenkov AM, Seregina EV, Stepovich MA (2017) The projection Galerkin method for solving the time-independent differential diffusion equation in a semi-infinite domain. Comput Math Math Phys 57(5):802–814

    MathSciNet  MATH  Google Scholar 

  42. Lin CH (2017) Hybrid recurrent Laguerre-orthogonal-polynomials neural network control with modified particle swarm optimization application for V-belt continuously variable transmission system. Neural Comput Appl 28(2):245–264

    Google Scholar 

  43. Sabouri M, Dehghan M (2018) A hk mortar spectral element method for the p-Laplacian equation. Comput Math Appl 76(7):1803–1826

    MathSciNet  MATH  Google Scholar 

  44. Ilati M, Dehghan M (2019) Dmlpg method for numerical simulation of soliton collisions in multi-dimensional coupled damped nonlinear schrödinger system which arises from bose-einstein condensates. Appl Math Comput 346:244–253

    MathSciNet  MATH  Google Scholar 

  45. Dehghan M, Abbaszadeh M (2019) The solution of nonlinear Green-Naghdi equation arising in water sciences via a meshless method which combines moving kriging interpolation shape functions with the weighted essentially non-oscillatory method. Commun Nonlinear Sci Numer Simul 68:220–239

    MathSciNet  MATH  Google Scholar 

  46. Assari P, Dehghan M (2019) Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace’s equations with nonlinear boundary conditions. Int J Comput Math 96(1):170–198

    MathSciNet  MATH  Google Scholar 

  47. Langsung PTPTS, Majid ZA, Suleiman M (2006) Direct integration implicit variable steps method for solving higher order systems of ordinary differential equations directly. Sains Malays 35(2):63–68

    MATH  Google Scholar 

  48. 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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  50. Mehrkanoon S, Falck T, Suykens JAK (2012) Approximate solutions to ordinary differential equations using least squares support vector machines. IEEE Trans Neural Netw Learn Syst 23(9):1356–1367

    Google Scholar 

  51. Mehrkanoon S, Suykens JAK (2015) Learning solutions to partial differential equations using LS-SVM. Neurocomputing 159:105–116

    Google Scholar 

  52. Mall S, Chakraverty S (2016) Application of Legendre neural network for solving ordinary differential equations. Appl Soft Comput 43:347–356

    Google Scholar 

  53. Ghaderi A, Parand K, Abdar M (2017) Numerical study on Lane-Emden type equation using Bessel artificial neural network approach. In: 2nd National mathematical conference: advanced engineering with math techniques

  54. Jeswal SK, Chakraverty S (2018) Solving transcendental equation using artificial neural network. Appl Soft Comput 73:562–571

    Google Scholar 

  55. Leake C, Johnston H, Smith L, Mortari D (2019) Theory of connections applied to support vector machines to solve differential equations. arXiv preprint arXiv:1812.05571

  56. Lu Y, Yin Q, Li H, Sun H, Yang Y, Hou M (2019) Solving higher order nonlinear ordinary differential equations with least squares support vector machines. J Ind Manag Optim 15:987–993

    Google Scholar 

  57. Raissi M (2018) Deep hidden physics models: Deep learning of nonlinear partial differential equations. J Mach Learn Res 19(1):932–955

    MathSciNet  MATH  Google Scholar 

  58. Hadian-Rasanan AH, Rahmati D, Gorgin S, Parand K (2020) A single layer fractional orthogonal neural network for solving various types of Lane-Emden equation. New Astron 75:101307

    Google Scholar 

  59. Parand K, Shahini M, Dehghan M (2009) Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type. J Comput Phys 228(23):8830–8840

    MathSciNet  MATH  Google Scholar 

  60. Parand K, Dehghan M, Rezaei AR, Ghaderi SM (2010) An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method. Comput Phys Commun 181(6):1096–1108

    MathSciNet  MATH  Google Scholar 

  61. Lakestani M, Dehghan M (2013) Four techniques based on the b-spline expansion and the collocation approach for the numerical solution of the Lane-Emden equation. Math Methods Appl Sci 36(16):2243–2253

    MathSciNet  MATH  Google Scholar 

  62. Wazwaz AM, Rach R, Duan JS (2015) Solving new fourth-order Emden-Fowler-type equations by the Adomian decomposition method. Int J Comput Methods Eng Sci Mech 16(2):121–131

    MathSciNet  Google Scholar 

  63. Wazwaz AM (2015) Solving two Emden-Fowler type equations of third order by the variational iteration method. Appl Math Inf Sci 9(5):2429–2436

    MathSciNet  Google Scholar 

  64. Syam MI (2018) Analytical solution of the fractional initial Emden-Fowler equation using the fractional residual power series method. Int J Appl Comput Math 4(4):106–113

    MathSciNet  MATH  Google Scholar 

  65. Roul P, Madduri H, Agarwal R (2019) A fast-converging recursive approach for Lane-Emden type initial value problems arising in astrophysics. J Comput Appl Math 359:182–195

    MathSciNet  MATH  Google Scholar 

  66. Madduri H, Roul P (2019) A fast-converging iterative scheme for solving a system of Lane-Emden equations arising in catalytic diffusion reactions. J Math Chem 57(2):570–582

    MathSciNet  MATH  Google Scholar 

  67. Ullah A, Shah K (2018) Numerical analysis of lane Emden-Fowler equations. J Taibah Univ Sci 12(2):180–185

    Google Scholar 

  68. Joshi P, Pathak M (2019) A coupled approach for solving a class of singular initial value problems of Lane-Emden type arising in astrophysics. Harmony Search Nat Inspired Optim Algorithms 741:669–678

    Google Scholar 

  69. Suykens JAK, Gestel TV, Brabanter JD, Moor BD, Vandewalle J (2002) Least squares support vector machines. World Scientific Publishing, London

    MATH  Google Scholar 

  70. Vapnik V (2013) The nature of statistical learning theory. Springer, New York

    MATH  Google Scholar 

  71. Parand K, Rezaei AR, Taghavi A (2010) Lagrangian method for solving Lane-Emden type equation arising in astrophysics on semi-infinite domains. Acta Astronaut 67(7–8):673–680

    Google Scholar 

  72. Parand K, Shahini M, Taghavi A (2009) Generalized Laguerre polynomials and rational Chebyshev collocation method for solving unsteady gas equation. Int J Contemp Math Sci 4(21):1005–1011

    MathSciNet  MATH  Google Scholar 

  73. Scholkopf B, Smola AJ (2001) Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT press, Cambridge

    Google Scholar 

  74. Horedt GP (2004) Polytropes: applications in astrophysics and related fields, vol 306. Springer Science & Business Media, Berlin

    Google Scholar 

  75. Yıldırım A, Öziş T (2009) Solutions of singular IVPs of Lane-Emden type by the variational iteration method. Nonlinear Anal Theory Methods Appl 70(6):2480–2484

    MathSciNet  MATH  Google Scholar 

  76. Bataineh AS, Noorani MSM, Hashim I (2009) Homotopy analysis method for singular IVPs of Emden-Fowler type. Commun Nonlinear Sci Numer Simul 14(4):1121–1131

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Data and Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    Z. Hajimohammadi, S. Shekarpaz & K. Parand

  2. Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran

    K. Parand

  3. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada

    K. Parand

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  1. Z. Hajimohammadi
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  3. K. Parand
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Hajimohammadi, Z., Shekarpaz, S. & Parand, K. The novel learning solutions to nonlinear differential models on a semi-infinite domain. Engineering with Computers 39, 2169–2186 (2023). https://doi.org/10.1007/s00366-022-01603-y

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