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New numerical solutions for solving Kidder equation by using the rational Jacobi functions

SeMA Journal volume 74, pages 569–583 (2017)Cite this article

Abstract

In this paper, a new method based on rational Jacobi functions (RJ) is proposed that utilizes quasilinearization method to solve non-linear singular Kidder equation on unbounded interval. The Kidder equation is a second order non-linear two-point boundary value ordinary differential equation on unbounded interval \([0,\infty )\). The equation is solved without domain truncation and variable changing. First, the quasilinearization method is used to convert the equation to sequence of linear ordinary differential equations. Then, by using RJ collocation method equations are solved. For the evaluation, comparison with some numerical solutions shows that the proposed solution is highly accurate. Using 200 collocation points, the value of initial slope that is important is calculated as \(-1.1917906497194217341228284 \) for \(\kappa =0.5\).

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References

  1. Abbasbandy, S.: Numerical study on gas flow through a micro–nano porous media. Acta Phys. Pol. Ser. A Gen. Phys. 121(3), 581 (2012)

  2. Baharifard, F., Kazem, S., Parand, K.: Rational and exponential legendre tau method on steady flow of a third grade fluid in a porous half space. Int. J. Appl. Comput. Math. 2(4), 678–698 (2016)

  3. Bellman, R.E., Kalaba, R.E.: Quasilinearization and nonlinear boundary-value problems. RAND Corporation (1965)

  4. Bhrawy, A.: An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput. 247, 30–46 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Bhrawy, A., Tharwat, M., Alghamdi, M.: A new operational matrix of fractional integration for shifted Jacobi polynomials. Bull. Malays. Math. Sci. Soc. 37(4), 983–995 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Bhrawy, A.H., Hafez, R.M., Alzaidy, J.F.: A new exponential Jacobi pseudospectral method for solving high-order ordinary differential equations. Adv. Differ. Equ. 2015(1), 1–15 (2015)

    Article  MathSciNet  Google Scholar 

  7. Boyd, J.P.: Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys. 69(1), 112–142 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, New York (2001)

    MATH  Google Scholar 

  9. Bu, W., Tang, Y., Wu, Y., Yang, J.: Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations. J. Comput. Phys. 293, 264–279 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, Y., Li, X., Tang, T.: A note on Jacobi spectral-collocation methods for weakly singular volterra integral equations with smooth solutions. J. Comput. Math. 31, 47–56 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Choi, H.J., Kweon, J.R.: A finite element method for singular solutions of the Navier–Stokes equations on a non-convex polygon. J. Comput. Appl. Math. 292, 342–362 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Christov, C.: A complete orthonormal system of functions in l \(\hat{}\) 2(-8,8) space. SIAM J. Appl. Math. 42(6), 1337–1344 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Costabile, F., Napoli, A.: A new spectral method for a class of linear boundary value problems. J. Comput. Appl. Math. 292, 329–341 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dehghan, M., Fakhar-Izadi, F.: The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves. Math. Comput. Model. 53(9), 1865–1877 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Delkhosh, M., Delkhosh, M., Jamali, M.: Introduction to green’s function and its numerical solution. Middle-East J. Sci. Res. 11(7), 974–981 (2012)

    MATH  Google Scholar 

  16. Doha, E., Baleanu, D., Bhrawy, A., Abdelkawy, M.: A Jacobi collocation method for solving nonlinear burgers-type equations. Abstr Appl Anal 1–12 (2013). doi:10.1155/2013/760542

  17. Doha, E., Bhrawy, A., Abdelkawy, M.: A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions. Open Phys. 12(9), 637–653 (2014)

    Article  Google Scholar 

  18. Doha, E., Bhrawy, A., Baleanu, D., Hafez, R.: A new Jacobi rational-gauss collocation method for numerical solution of generalized pantograph equations. Appl. Numer. Math. 77, 43–54 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Doha, E.H., Bhrawy, A.H.: A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numer. Methods Partial Differ. Equ. 25(3), 712–739 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Doha, E.H., Bhrawy, A.H., Abdelkawy, M.A., Hafez, R.M.: A Jacobi collocation approximation for nonlinear coupled viscous burgers equation. Cent. Eur. J. Phys. 12(2), 111–122 (2014)

    Google Scholar 

  21. Doha, E.H., Bhrawy, A.H., Hafez, R.M., Van Gorder, R.A.: Jacobi rational-gauss collocation method for Lane–Emden equations of astrophysical significance. Nonlinear Anal. Model. Control 19, 1–14 (2014)

    MathSciNet  Google Scholar 

  22. Fakhar-Izadi, F., Dehghan, M.: A spectral element method using the modal basis and its application in solving second-order nonlinear partial differential equations. Math. Methods Appl. Sci. 38(3), 478–504 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Funaro, D.: Computational aspects of pseudospectral Laguerre approximations. Appl. Numer. Math. 6(6), 447–457 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Funaro, D., Kavian, O.: Approximation of some diffusion evolution equations in unbounded domains by hermite functions. Math. Comput. 57(196), 597–619 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gepreel, K.A., Nofal, T.A., Al-Thobaiti, A.A.: The modified rational Jacobi elliptic functions method for nonlinear differential difference equations. J. Appl. Math. 2012, 30 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Guo, B.y.: Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. J. Math. Anal. Appl. 243(2), 373–408 (2000)

  27. Guo, B.Y., Shen, J.: Laguerre–Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numerische Mathematik 86(4), 635–654 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Guo, B.Y., Yi, Y.G.: Generalized Jacobi rational spectral method and its applications. J. Sci. Comput. 43(2), 201–238 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hashim, I., Noorani, M.S.M., Al-Hadidi, M.S.: Solving the generalized Burgers–Huxley equation using the adomian decomposition method. Math. Comput. Model. 43(11), 1404–1411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3), 257–262 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30(3), 700–708 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hojjati, G., Parand, K.: An efficient computational algorithm for solving the nonlinear Lane–Emden type equations. Int. J. Math. Comput. 4(7), 182–187 (2011)

    Google Scholar 

  33. Iacono, R., Boyd, J.P.: The kidder equation: uxx\(+\) 2xux/1\(-\alpha \) u \(=\) 0. Stud. Appl. Math. 135(1), 63–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kalaba, R.: 0n nonlinear differential equations, the maximum operation and monotone convergence. J. Math. Mech. 8, 519–574 (1959)

    MathSciNet  MATH  Google Scholar 

  35. Kazem, S., Rad, J., Parand, K., Shaban, M., Saberi, H.: 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 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Khan, Y., Faraz, N., Yildirim, A.: Series solution for unsteady gas equation via MLDM-Pade technique. World Appl. Sci. J. 10(12), 1452–1456 (2010)

    Google Scholar 

  37. Kidder, R.: Unsteady flow of gas through a semi-infinite porous medium. J. Appl. Mech. 27, 329–332 (1957)

    MathSciNet  MATH  Google Scholar 

  38. Krivec, R., Mandelzweig, V.: Numerical investigation of quasilinearization method in quantum mechanics. Comput. Phys. Commun. 138(1), 69–79 (2001)

    Article  MATH  Google Scholar 

  39. Krivec, R., Mandelzweig, V.: Quasilinearization approach to computations with singular potentials. Comput. Phys. Commun. 179(12), 865–867 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lakshmikantham, V., Vatsala, A.S.: Generalized Quasilinearization for Nonlinear Problems, vol. 440. Springer Science & Business Media, Berlin (2013)

    MATH  Google Scholar 

  41. Liverts, E., Krivec, R., Mandelzweig, V.: Quasilinearization approach to the resonance calculations: the quartic oscillator. Phys. Scr. 77(4), 045,004 (2008)

  42. Mandelzweig, V., Tabakin, F.: Quasilinearization approach to nonlinear problems in physics with application to nonlinear odes. Comput. Phys. Commun. 141(2), 268–281 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mohyud-Din, S.T., Yildirim, A., Hosseini, M.: Variational iteration method for initial and boundary value problems using he’s polynomials. Int. J. Differ. Equ. 2010, 1–28 (2010). doi:10.1155/2010/426213

  44. Na, T.: Computational Methods in Engineering Boundary Value Problems. Academic, New York (1979)

    MATH  Google Scholar 

  45. Noor, M.A., Mohyud-Din, S.T.: Variational iteration method for unsteady flow of gas through a porous medium using Hes polynomials and Pade approximants. Comput. Math. Appl. 58(11), 2182–2189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Noye, B., Dehghan, M.: New explicit finite difference schemes for two-dimensional diffusion subject to specification of mass. Numer. Methods Partial Differ. Equ. 15(4), 521–534 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  47. Parand, K., Abbasbandy, S., Kazem, S., Rezaei, A.: An improved numerical method for a class of astrophysics problems based on radial basis functions. Phys. Scr. 83(1), 015,011 (2011)

  48. Parand, K., Dehghan, M., Taghavi, A.: 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 (2010)

  49. Parand, K., Delafkar, Z., Pakniat, N., Pirkhedri, A., Haji, M.K.: Collocation method using sinc and rational legendre functions for solving Volterras population model. Commun. Nonlinear Sci. Numer. Simul. 16(4), 1811–1819 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Parand, K., Hemami, M.: 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. (2016). doi:10.1007/s40324-016-0103-z

  51. Parand, K., Hemami, M.: Numerical study of astrophysics equations by meshless collocation method based on compactly supported radial basis function. Int. J. Appl. Comput. Math. 1–23 (2016). doi:10.1007/s40819-016-0161-z

  52. Parand, K., Hossayni, S.A., Rad, J.: Operation matrix method based on Bernstein polynomials for the Riccati differential equation and volterra population model. Appl. Math. Model. 40(2), 993–1011 (2016)

    Article  MathSciNet  Google Scholar 

  53. Parand, K., Nikarya, M.: Solving the unsteady isothermal gas through a micro–nano porous medium via bessel function collocation method. J. Comput. Theor. Nanosci. 11(1), 131–136 (2014)

    Article  Google Scholar 

  54. Parand, K., Shahini, M., Dehghan, M.: Solution of a laminar boundary layer flow via a numerical method. Commun. Nonlinear Sci. Numer. Simul. 15(2), 360–367 (2010)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  57. Rad, J., Ghaderi, S., Parand, K.: Numerical and analytical solution of gas flow through a micro–nano porous media: a comparison. J. Comput. Theor. Nanosci. 8(10), 2033–2041 (2011)

    Article  Google Scholar 

  58. Rad, J.A., Parand, K., Abbasbandy, S.: Local weak form meshless techniques based on the radial point interpolation (rpi) method and local boundary integral equation (lbie) method to evaluate European and American options. Commun. Nonlinear Sci. Numer. Simul. 22(1), 1178–1200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Rad, J.A., Parand, K., Abbasbandy, S.: Pricing European and American options using a very fast and accurate scheme: the meshless local Petrov–Galerkin method. Proc. Natl. Acad. Sci. India Sect. A: Phys. Sci. 85(3), 337–351 (2015)

  60. Rezaei, A., Parand, K., Pirkhedri, A.: Numerical study on gas flow through a micro–nano porous media based on special functions. J. Comput. Theor. Nanosci. 8(2), 282–288 (2011)

    Article  Google Scholar 

  61. Roberts, R.C.: Unsteady flow of a gas through a porous medium. J. Appl. Mech. Trans. ASME 18, 326–326 (1951) [ASME-Am. Soc. Mech. Eng., 345 E 47th ST, New York, NY 10017]

  62. Saeedi, H.: On the linear b-spline scaling function operational matrix of fractional integration and its applications in solving fractional order differential equations. Iran. J. Sci. Technol. (Sciences) (2015)

  63. Saeedi, H., Chuev, G.N.: Triangular functions for operational matrix of nonlinear fractional volterra integral equations. J. Appl. Math. Comput. 49(1–2), 213–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  64. Saeedi, H., Samimi, F.: Hes homotopy perturbation method for nonlinear Fredholm integro-differential equations of fractional order. Int. J. Eng. Res. Appl. 2(5), 52–56 (2012)

    Google Scholar 

  65. Shakeri, F., Dehghan, M.: Numerical solution of the Klein–Gordon equation via Hes variational iteration method. Nonlinear Dyn. 51(1–2), 89–97 (2008)

    MathSciNet  MATH  Google Scholar 

  66. Taghavi, A., Fani, H., et al.: Lagrangian method for solving unsteady gas equation. World Acad. Sci. Eng. Technol. Int. J. Math. Comput. Phys. Electr. Comput. Eng. 3(11), 991–995 (2009)

    MathSciNet  Google Scholar 

  67. Taghavi, A., Parand, K., Shams, A., Sofloo, H.G.: Spectral method for solving differential equation of gas flow through a micro–nano porous media. J. Comput. Theor. Nanosci. 7(3), 542–546 (2010)

    Article  Google Scholar 

  68. Tatari, M., Dehghan, M., Razzaghi, M.: Application of the adomian decomposition method for the Fokker–Planck equation. Math. Comput. Model. 45(5), 639–650 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  69. Tchiotsop, D., Wolf, D., Louis-Dorr, V., Husson, R.: Ecg data compression using Jacobi polynomials. In: Engineering in Medicine and Biology Society, 2007. EMBS 2007. 29th Annual International Conference of the IEEE, pp. 1863–1867. IEEE (2007)

  70. Upadhyay, S., Rai, K.: Collocation method applied to unsteady flow of gas through a porous medium. Int. J. Appl. Math. Res. 3(3), 251–259 (2014)

    Article  Google Scholar 

  71. Wazwaz, A.M.: The modified decomposition method applied to unsteady flow of gas through a porous medium. Appl. Math. Comput. 118(2), 123–132 (2001)

    MathSciNet  MATH  Google Scholar 

  72. Wazwaz, A.M.: The variational iteration method for solving linear and nonlinear odes and scientific models with variable coefficients. Cent. Eur. J. Eng. 4(1), 64–71 (2014)

    Google Scholar 

  73. Yi, Y., Guo, B.: Generalized Jacobi rational spectral method on the half line. Adv. Comput. Math. 37(1), 1–37 (2012)

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Authors and Affiliations

  1. Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Kourosh Parand, Pooria Mazaheri, Mehdi Delkhosh & Amin Ghaderi

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

    Kourosh Parand

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  1. Kourosh Parand
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Parand, K., Mazaheri, P., Delkhosh, M. et al. New numerical solutions for solving Kidder equation by using the rational Jacobi functions. SeMA 74, 569–583 (2017). https://doi.org/10.1007/s40324-016-0103-z

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Keywords

  • Rational Jacobi functions
  • Kidder equation
  • Jacobi polynomials
  • Collocation method
  • Quasilinearization method
  • Singular points
  • Unbounded interval

Mathematics Subject Classification

  • 34B15
  • 34B40
  • 34L30
  • 65M70
  • 65N35