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

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 $$\kappa =0.5$$.

This is a preview of subscription content, access via your institution.

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)

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)

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)

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

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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

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

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)

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)

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

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)

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

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

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)

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

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

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)

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)

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

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

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

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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

Author information

Authors

Corresponding author

Correspondence to Kourosh Parand.

Rights and permissions

Reprints and Permissions

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

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1007/s40324-016-0103-z

Keywords

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

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