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Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability

Year 2017, Volume: 5 Issue: 3, 245 - 252, 01.07.2017

Abstract

The recent work on the solvability of the boundary value problem for the nonlinear analogue of the Boussinesq equation has been further extended to focus on the characteristics of the solution. Since this type of equation does not have a known analytical solution for arbitrary boundary conditions, the problem has been solved numerically. The stability of the solution and the effect of the input function on the stability have been investigated from the physics point of view. For the special case of a discontinuous function at the right hand side of the equation, the solution has been analyzed around the discontinuity points.

References

  • Soerensen, M. P., Christiansen, P. L., Lomdahl, P. S. (1984). Solitary waves on nonlinear elastic rods. I. The Journal of the Acoustical Society of America, 76(3), 871-879.
  • Karpman, V. I. (2016). Non-Linear Waves in Dispersive Media: International Series of Monographs in Natural Philosophy (Vol. 71). Elsevier, 15-18.
  • Li, X. L., Zheng, Y. (2010). The Effects of the Boussinesq Model to the Rising of the Explosion Clouds. Nuclear Electronics & Detection Technology, 11, 010.
  • Peirotti,M. B., Giavedoni,M. D., Deiber, J. A. (1987). Natural convective heat transfer in a rectangular porous-cavity with variable fluid properties-validity of the Boussinesq approximation. International Journal of Heat and Mass Transfer, 30(12), 2571-2581.
  • Madsen, P. A., Bingham, H. B., Sch¨affer, H. A. (2003, May). Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 459, No. 2033, pp. 1075-1104). The Royal Society.
  • Wang, M. (1995). Solitary wave solutions for variant Boussinesq equations. Physics Letters A, 199(3-4), 169-172.
  • Zufiria, J. A. (1987). Weakly nonlinear non-symmetric gravity waves on water of finite depth. Journal of Fluid Mechanics, 180, 371-385.
  • Seadawy, A. R., El-Kalaawy, O. H., Aldenari, R. B. (2016). Water wave solutions of Zufiria’s higher-order Boussinesq type equations and its stability. Applied Mathematics and Computation, 280, 57-71.
  • Seadawy, A. R., Amer, W., Sayed, A. (2014). Stability analysis for travelling wave solutions of the Olver and fifth-order KdV equations. Journal of Applied Mathematics, 2014.
  • Helal,M. A., Seadawy, A. R., Zekry,M. (2017). Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications. Chinese Journal of Physics, 55(2), 378-385.
  • Seadawy, A. R. (2015). Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part I. Computers & Mathematics with Applications, 70(4), 345-352.
  • Helal, M. A., Seadawy, A. R., Zekry, M. H. (2014). Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation. Applied Mathematics and Computation, 232, 1094-1103.
  • Seadawy, A. R. (2017). Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves. The European Physical Journal Plus, 132(1), 29.
  • Helal, M. A., Seadawy, A. R. (2012). Benjamin-Feir instability in nonlinear dispersive waves. Computers & Mathematics with Applications, 64(11), 3557-3568.
  • Selima, E. S., Seadawy, A. R., Yao, X. (2016). The nonlinear dispersive Davey-Stewartson system for surface waves propagation in shallow water and its stability. The European Physical Journal Plus, 131(12), 425.
  • Moutsopoulos, K. N. (2010). The analytical solution of the Boussinesq equation for flow induced by a step change of the water table elevation revisited. Transport in Porous Media, 85(3), 919-940.
  • Tolikas, P. K., Sidiropoulos, E. G., Tzimopoulos, C. D. (1984). A Simple Analytical Solution for the Boussinesq One-Dimensional Groundwater Flow Equation. Water Resources Research, 20(1), 24-28.
  • Basha, H. A., Maalouf, S. F. (2005). Theoretical and conceptual models of subsurface hillslope flows. Water Resources Research, 41(7).
  • Rupp, D. E., Selker, J. S. (2006). On the use of the Boussinesq equation for interpreting recession hydrographs from sloping aquifers. Water Resources Research, 42(12).
  • Brutsaert, W., Nieber, J. L. (1977). Regionalized drought flow hydrographs from a mature glaciated plateau. Water Resources Research, 13(3), 637-643.
  • Lonngren, K. E. (1978). Observation of solitons on nonlinear dispersive transmission lines. Solitons in Action, 25, 127-152.
  • Amirov, S., Kozhanov, A. I. (2016). Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation. Mathematical Notes, 99(1-2), 183-191.
  • Walker, J. A. (2013). Dynamical Systems and Evolution Equations: Theory and Applications (Vol. 20). Springer Science & Business Media.
  • Lyapunov, A. M. (1992). The general problem of the stability of motion. International Journal of Control, 55(3), 531-534.
  • Benjamin, T. B. (1972, May). The stability of solitary waves. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 328, No. 1573, pp. 153-183). The Royal Society.
  • Bona, J. L., Souganidis, P. E., Strauss, W. A. (1987, June). Stability and instability of solitary waves of Korteweg-de Vries type. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 411, No. 1841, pp. 395-412). The Royal Society.
  • Liu, Y. (1993). Instability of solitary waves for generalized Boussinesq equations. Journal of Dynamics and Differential Equations, 5(3), 537-558.
  • Grillakis,M., Shatah, J., Strauss,W. (1987). Stability theory of solitary waves in the presence of symmetry, I. Journal of Functional Analysis, 74(1), 160-197.
  • Shatah, J. (1983). Stable standing waves of nonlinear Klein-Gordon equations. Communications in Mathematical Physics, 91(3), 313-327.
  • Ohanian H.C. (1989). Physics (2nd Edition Expanded). WW Norton & Company. New York. pp. 393-394.
Year 2017, Volume: 5 Issue: 3, 245 - 252, 01.07.2017

Abstract

References

  • Soerensen, M. P., Christiansen, P. L., Lomdahl, P. S. (1984). Solitary waves on nonlinear elastic rods. I. The Journal of the Acoustical Society of America, 76(3), 871-879.
  • Karpman, V. I. (2016). Non-Linear Waves in Dispersive Media: International Series of Monographs in Natural Philosophy (Vol. 71). Elsevier, 15-18.
  • Li, X. L., Zheng, Y. (2010). The Effects of the Boussinesq Model to the Rising of the Explosion Clouds. Nuclear Electronics & Detection Technology, 11, 010.
  • Peirotti,M. B., Giavedoni,M. D., Deiber, J. A. (1987). Natural convective heat transfer in a rectangular porous-cavity with variable fluid properties-validity of the Boussinesq approximation. International Journal of Heat and Mass Transfer, 30(12), 2571-2581.
  • Madsen, P. A., Bingham, H. B., Sch¨affer, H. A. (2003, May). Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 459, No. 2033, pp. 1075-1104). The Royal Society.
  • Wang, M. (1995). Solitary wave solutions for variant Boussinesq equations. Physics Letters A, 199(3-4), 169-172.
  • Zufiria, J. A. (1987). Weakly nonlinear non-symmetric gravity waves on water of finite depth. Journal of Fluid Mechanics, 180, 371-385.
  • Seadawy, A. R., El-Kalaawy, O. H., Aldenari, R. B. (2016). Water wave solutions of Zufiria’s higher-order Boussinesq type equations and its stability. Applied Mathematics and Computation, 280, 57-71.
  • Seadawy, A. R., Amer, W., Sayed, A. (2014). Stability analysis for travelling wave solutions of the Olver and fifth-order KdV equations. Journal of Applied Mathematics, 2014.
  • Helal,M. A., Seadawy, A. R., Zekry,M. (2017). Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications. Chinese Journal of Physics, 55(2), 378-385.
  • Seadawy, A. R. (2015). Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part I. Computers & Mathematics with Applications, 70(4), 345-352.
  • Helal, M. A., Seadawy, A. R., Zekry, M. H. (2014). Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation. Applied Mathematics and Computation, 232, 1094-1103.
  • Seadawy, A. R. (2017). Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves. The European Physical Journal Plus, 132(1), 29.
  • Helal, M. A., Seadawy, A. R. (2012). Benjamin-Feir instability in nonlinear dispersive waves. Computers & Mathematics with Applications, 64(11), 3557-3568.
  • Selima, E. S., Seadawy, A. R., Yao, X. (2016). The nonlinear dispersive Davey-Stewartson system for surface waves propagation in shallow water and its stability. The European Physical Journal Plus, 131(12), 425.
  • Moutsopoulos, K. N. (2010). The analytical solution of the Boussinesq equation for flow induced by a step change of the water table elevation revisited. Transport in Porous Media, 85(3), 919-940.
  • Tolikas, P. K., Sidiropoulos, E. G., Tzimopoulos, C. D. (1984). A Simple Analytical Solution for the Boussinesq One-Dimensional Groundwater Flow Equation. Water Resources Research, 20(1), 24-28.
  • Basha, H. A., Maalouf, S. F. (2005). Theoretical and conceptual models of subsurface hillslope flows. Water Resources Research, 41(7).
  • Rupp, D. E., Selker, J. S. (2006). On the use of the Boussinesq equation for interpreting recession hydrographs from sloping aquifers. Water Resources Research, 42(12).
  • Brutsaert, W., Nieber, J. L. (1977). Regionalized drought flow hydrographs from a mature glaciated plateau. Water Resources Research, 13(3), 637-643.
  • Lonngren, K. E. (1978). Observation of solitons on nonlinear dispersive transmission lines. Solitons in Action, 25, 127-152.
  • Amirov, S., Kozhanov, A. I. (2016). Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation. Mathematical Notes, 99(1-2), 183-191.
  • Walker, J. A. (2013). Dynamical Systems and Evolution Equations: Theory and Applications (Vol. 20). Springer Science & Business Media.
  • Lyapunov, A. M. (1992). The general problem of the stability of motion. International Journal of Control, 55(3), 531-534.
  • Benjamin, T. B. (1972, May). The stability of solitary waves. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 328, No. 1573, pp. 153-183). The Royal Society.
  • Bona, J. L., Souganidis, P. E., Strauss, W. A. (1987, June). Stability and instability of solitary waves of Korteweg-de Vries type. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 411, No. 1841, pp. 395-412). The Royal Society.
  • Liu, Y. (1993). Instability of solitary waves for generalized Boussinesq equations. Journal of Dynamics and Differential Equations, 5(3), 537-558.
  • Grillakis,M., Shatah, J., Strauss,W. (1987). Stability theory of solitary waves in the presence of symmetry, I. Journal of Functional Analysis, 74(1), 160-197.
  • Shatah, J. (1983). Stable standing waves of nonlinear Klein-Gordon equations. Communications in Mathematical Physics, 91(3), 313-327.
  • Ohanian H.C. (1989). Physics (2nd Edition Expanded). WW Norton & Company. New York. pp. 393-394.
There are 30 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Sherif Amirov This is me

Mustafa Anutgan

Publication Date July 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 3

Cite

APA Amirov, S., & Anutgan, M. (2017). Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability. New Trends in Mathematical Sciences, 5(3), 245-252.
AMA Amirov S, Anutgan M. Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability. New Trends in Mathematical Sciences. July 2017;5(3):245-252.
Chicago Amirov, Sherif, and Mustafa Anutgan. “Boundary Value Problem for the Nonlinear Analogues of the Boussinesq Equation: Numerical Solution and Its Stability”. New Trends in Mathematical Sciences 5, no. 3 (July 2017): 245-52.
EndNote Amirov S, Anutgan M (July 1, 2017) Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability. New Trends in Mathematical Sciences 5 3 245–252.
IEEE S. Amirov and M. Anutgan, “Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability”, New Trends in Mathematical Sciences, vol. 5, no. 3, pp. 245–252, 2017.
ISNAD Amirov, Sherif - Anutgan, Mustafa. “Boundary Value Problem for the Nonlinear Analogues of the Boussinesq Equation: Numerical Solution and Its Stability”. New Trends in Mathematical Sciences 5/3 (July 2017), 245-252.
JAMA Amirov S, Anutgan M. Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability. New Trends in Mathematical Sciences. 2017;5:245–252.
MLA Amirov, Sherif and Mustafa Anutgan. “Boundary Value Problem for the Nonlinear Analogues of the Boussinesq Equation: Numerical Solution and Its Stability”. New Trends in Mathematical Sciences, vol. 5, no. 3, 2017, pp. 245-52.
Vancouver Amirov S, Anutgan M. Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability. New Trends in Mathematical Sciences. 2017;5(3):245-52.