Nonlinear Dynamics

, Volume 88, Issue 4, pp 2817–2829 | Cite as

Group preserving scheme and reproducing kernel method for the Poisson–Boltzmann equation for semiconductor devices

  • Ali AkgülEmail author
  • Mustafa Inc
  • Mir Sajjad Hashemi
Original Paper


This paper introduces that the nonlinear Poisson–Boltzmann equation for semiconductor devices describing potential distribution in a double-gate metal oxide semiconductor field effect transistor (DG-MOSFET) is exactly solvable. The DG-MOSFET shows one of the most advanced device structures in semiconductor technology and is a primary focus of modeling efforts in the semiconductor industry. Lie symmetry properties of this model is investigated in order to extract some exact solutions. The reproducing kernel Hilbert space method and group preserving scheme also have been applied to the nonlinear equation. Numerical results show that the present methods are very effective.


Group preserving scheme Poisson–Boltzmann equation Reproducing kernel method Lie symmetry analysis 

Mathematics Subject Classification



Compliance with ethical standards

Conflict of interests

The authors declare that they do not have any competing or conflict of interests.


  1. 1.
    Abbasbandy, S., Azarnavid, B.: Some error estimates for the reproducing kernel Hilbert spaces method. J. Comput. Appl. Math. 296, 789–797 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abbasbandy, S., Hashemi, M.S.: Group preserving scheme for the cauchy problem of the laplace equation. Eng. Anal. Bound. Elem 35, 1003–1009 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Akgül, A.: New reproducing kernel functions. Math. Probl. Eng., 10 (2015). Art. ID 158134Google Scholar
  4. 4.
    Akgül, A., Inc, M., Karatas, E., Baleanu, D.: Numerical solutions of fractional differential equations of Lane–Emden type by an accurate technique. Adv. Differ. Equ. 2015(12), 220 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bhrawy, A.H., Alzaidy, J.F., Abdelkawy, M.A., Biswas, A.: Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations. Nonlinear Dynamics 84(3), 1553–1567 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Biswas, A., Kara, Abdul H., Moraru, L., Bokhari, A.H., Zaman, F.D.: Conservation laws of coupled Klein–Gordon equations with cubic and power law nonlinearities. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 15(2), 123–129 (2014)MathSciNetGoogle Scholar
  8. 8.
    Biswas, A., Mirzazadeh, M., Eslami, M.: Soliton solution of generalized chiral nonlinear Schrödinger’s equation with time-dependent coefficients. Acta Phys. Pol. B 45(4), 849–866 (2014)CrossRefGoogle Scholar
  9. 9.
    Biswas, A., Song, M., Triki, H., Kara, A.H., Ahmed, B.S., Strong, A., Hama, A.: Solitons, shock waves, conservation laws and bifurcation analysis of Boussinesq equation with power law nonlinearity and dual dispersion. Appl. Math. Inf. Sci. 8(3), 949–957 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Biswas, A., Song, M., Zerrad, E.: Bifurcation analysis and implicit solution of Klein–Gordon equation with dual-power law nonlinearity in relativistic quantum mechanics. Int. J. Nonlinear Sci. Numer. Simul. 14(5), 317–322 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cao, J., Song, M., Biswas, A.: Topological solitons and bifurcation analysis of the PHI-four equation. Bull. Malays. Math. Sci. Soc. (2) 37(4), 1209–1219 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chang, C.-W., Liu, Chein-Shan: A backward group preserving scheme for multi-dimensional backward heat conduction problems. CMES Comput. Model. Eng. Sci. 59(3), 239–274 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chang, C.-W., Liu, C.-S.: The backward group preserving scheme for multi-dimensional nonhomogeneous and nonlinear backward wave problems. Appl. Math. Model. 38(15–16), 4027–4048 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, Y.-W., Liu, C.-S., Chang, J.-R.: Applications of the modified Trefftz method for the Laplace equation. Eng. Anal. Bound. Elem. 33(2), 137–146 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cui, M., Lin, Yingzhen: Nonlinear numerical analysis in the reproducing kernel space. Nova Science Publishers Inc., New York (2009)zbMATHGoogle Scholar
  16. 16.
    Ekici, M., Mirzazadeh, M., Eslami, M.: Solitons and other solutions to boussinesq equation with power law nonlinearity and dual dispersion. Nonlinear Dynamics 84(2), 669–676 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Geng, F., Cui, M., Zhang, Bo: Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods. Nonlinear Anal. Real World Appl. 11(2), 637–644 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hashemi, M.S.: Constructing a new geometric numerical integration method to the nonlinear heat transfer equations. Commun. Nonlinear Sci. Numer. Simul. 22(1), 990–1001 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hashemi, M.S., Baleanu, D.: Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line. J. Comput. Phys. 316, 10–20 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hashemi, M.S., Baleanu, D., Parto-Haghighi, M.: A lie group approach to solve the fractional poisson equation. Rom. J. Phys. 60, 1289–1297 (2015)Google Scholar
  21. 21.
    Hashemi, M.S., Haji-Badali, A., Vafadar, P.: Group invariant solutions and conservation laws of the fornberg-whitham equation. Z. Naturforsch. A 69(8–9), 489–496 (2014)Google Scholar
  22. 22.
    Hashemi, M.S., Nucci, M.C., Abbasbandy, S.: Group analysis of the modified generalized vakhnenko equation. Commun. Nonlinear Sci. Numer. Simul. 18, 867–877 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Inc, M., Akgül, A., Geng, Fazhan: Reproducing kernel Hilbert space method for solving Bratu’s problem. Bull. Malays. Math. Sci. Soc. 38(1), 271–287 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Inc, M., Akgül, A., Kılıçman, A.: Numerical solutions of the second-order one-dimensional telegraph equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal., 13, (2013). Art. ID 768963Google Scholar
  25. 25.
    Krishnan, E.V., Ghabshi, M.A., Mirzazadeh, M., Bhrawy, A.H., Biswas, A., Belic, M.: Optical solitons for quadratic law nonlinearity with five integration schemes. J. Comput. Theor. Nanosci. 12(11), 4809–4821 (2015)Google Scholar
  26. 26.
    Lee, H.-C., Chen, C.-K., Hung, Chen-I: A modified group-preserving scheme for solving the initial value problems of stiff ordinary differential equations. Appl. Math. Comput. 133(2–3), 445–459 (2002)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lee, H.-C., Liu, Chein-Shan: The fourth-order group preserving methods for the integrations of ordinary differential equations. CMES Comput. Model. Eng. Sci. 41(1), 1–26 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shijun, L.: Beyond Perturbation, Volume 2 of CRC Series: Modern Mechanics and Mathematics. Introduction to the homotopy analysis method. Chapman & Hall/CRC, Boca Raton (2004)Google Scholar
  29. 29.
    Liu, C.-S., Hong, H.-K., Liou, D.-Y.: Two-dimensional friction oscillator: group-preserving scheme and handy formulae. J. Sound Vib. 266(1), 49–74 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu, Chein-Shan: Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Mech. 36(7), 1047–1068 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Liu, Chein-Shan: Two-dimensional bilinear oscillator: group-preserving scheme and steady-state motion under harmonic loading. Int. J. Non-Linear Mech. 38(10), 1581–1602 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Liu, Chein-Shan: Nonstandard group-preserving schemes for very stiff ordinary differential equations. CMES Comput. Model. Eng. Sci. 9(3), 255–272 (2005)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Liu, C.-S., Chang, Chih-Wen: A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains. Eng. Anal. Bound. Elem. 36(2), 211–219 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Liu, C.-S., Chang, C.-W., Chang, J.-R.: The backward group preserving scheme for 1D backward in time advection-dispersion equation. Numer. Methods Partial Differ. Equ. 26(1), 61–80 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Liu, C.-S., Ku, Y.-L.: A combination of group preserving scheme and Runge-Kutta method for the integration of Landau-Lifshitz equation. CMES Comput. Model. Eng. Sci. 9(2), 151–177 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mirzazadeh, M.: Analytical study of solitons to nonlinear time fractional parabolic equations. Nonlinear Dynamics 85(4), 2569–2576 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Mirzazadeh, M., Biswas, A.: Optical solitons with spatio-temporal dispersion by first integral approach and functional variable method. Optik-Int. J. Light Electron Opt. 125(19), 5467–5475 (2014)CrossRefGoogle Scholar
  38. 38.
    Mirzazadeh, M., Eslami, M., Bhrawy, A.H., Biswas, A.: Integration of complex-valued kleingordon equation in\( phi\)-4 field theory. Rom. J. Phys 60(3–4), 293 (2015)Google Scholar
  39. 39.
    Morris, R.M., Kara, A.H.: Biswas, Anjan: An analysis of the Zhiber-Shabat equation including Lie point symmetries and conservation laws. Collect. Math. 67(1), 55–62 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nassar, C.J., Revelli, J.F., Bowman, R.J.: Application of the homotopy analysis method to the Poisson–Boltzmann equation for semiconductor devices. Commun. Nonlinear Sci. Numer. Simul. 16(6), 2501–2512 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Shivanian, E., Abbasbandy, S., Alhuthali, M.S.: Exact analytical solution to the Poisson–Boltzmann equation for semiconductor devices. Eur. Phys. J. Plus 129, 104 (2014)CrossRefGoogle Scholar
  42. 42.
    Song, M., Biswas, Anjan: Topological defects and bifurcation analysis of the DS equation with power law nonlinearity. Appl. Math. Inf. Sci. 9(4), 1719–1724 (2015)MathSciNetGoogle Scholar
  43. 43.
    Üreyen, A.E.: An estimate of the oscillation of harmonic reproducing kernels with applications. J. Math. Anal. Appl. 434(1), 538–553 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Wang, G.-W., Xu, T.Z., Zedan, H.A., Abazari, R., Triki, H., Biswas, A.: Solitary waves, shock waves and other solutions to Nizhniki–Novikov–Veselov equation. Appl. Comput. Math. 14(3), 260–283 (2015)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Wang, G., Kara, A.H., Fakhar, K., Vega-Guzman, Jose, Biswas, Anjan: Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation. Chaos Solitons Fractals 86, 8–15 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Xie, D., Jiang, Yi: A nonlocal modified Poisson–Boltzmann equation and finite element solver for computing electrostatics of biomolecules. J. Comput. Phys. 322, 1–20 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Xu, M.Q., Lin, Y.-Z.: Simplified reproducing kernel method for fractional differential equations with delay. Appl. Math. Lett. 52, 156–161 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Zhou, Q., Zhong, Y., Mirzazadeh, M., Bhrawy, A.H., Zerrad, E., Biswas, Anjan: Thirring combo-solitons with cubic nonlinearity and spatio-temporal dispersion. Waves Random Complex Media 26(2), 204–210 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Art and Science FacultySiirt UniversitySiirtTurkey
  2. 2.Department of Mathematics, Science FacultyFırat UniversityElazığTurkey
  3. 3.Department of Mathematics, Basic Science FacultyUniversity of BonabBonabIran

Personalised recommendations