Finite Element Solution of a Problem for Gravity Gyroscopic Equation in the Time Domain

Affiliation(s)

National Aviation University, Kyiv, Ukraine.

Berdakh Kara-Kalpak State University, Nukus, Uzbekistan.

National Aviation University, Kyiv, Ukraine.

Berdakh Kara-Kalpak State University, Nukus, Uzbekistan.

ABSTRACT

To solve the equation for gravity-gyroscopic waves in a rectangular domain, the distinguished algorithm for the solution of the Cauchy problem for a second-order transient equation is proposed. This algorithm is developed by using the time-varying finite element method. The space derivatives in the gravity-gyroscopic wave equation are approximated with finite differences. The stability and accuracy of the method are estimated. The procedure for the implementation of the method is developed. The calculations were performed for determining the steady-state modes of fluctuations of the solutions of the gravity-gyroscopic wave equation depending on the problem parameters.

Cite this paper

Moskalkov, M. and Utebaev, D. (2014) Finite Element Solution of a Problem for Gravity Gyroscopic Equation in the Time Domain.*Applied Mathematics*, **5**, 1120-1132. doi: 10.4236/am.2014.58105.

Moskalkov, M. and Utebaev, D. (2014) Finite Element Solution of a Problem for Gravity Gyroscopic Equation in the Time Domain.

References

[1] Whitham, G.B. (1974) Linear and Nonlinear Waves. Wiley, Hoboken.

[2] Miropolsky, Y.Z. (1981) Dynamics of Internal-Waves in the Ocean. Gidrometeoizdates, Moscow.

[3] Gabov, S.A. and Sveshnikov, A.G. (1990) Linear Problems in the Theory of Non-Steady-State Internal Waves. Nauka, Moskow.

[4] Gabov, S.A. and Pletner, Y.D. (1985) A Nachal-No-Kraevoy Problem for the Gravity-Gyros-Copic Waves. Journal of Computational Mathematics and Mathematical Physics, 25, 1689-1696.

[5] Gabov, S.A. and Pletner, Y.D. (1987) The Gravity-Gyroscopic Waves: The Angular Potential and Its Application. Journal of Computational Mathematics and Mathematical Physics, 27, 102-113.

[6] Iskendarov, B.A., Mamedov, J.Y. and Sulaimanov, S.E. (2009) Mixed Problem for the Gravity-Gyroscopic Waves in the Boussinesq Approximation in an Infinite Cylindrical Domain. Journal of Computa-tional Mathematics and Mathematical Physics, 49, 1659-1675.

[7] Samarskii, A.A. (1977) Theory of Difference Schemes. Nauka, Moskow.

[8] Samarskii, A.A. and Vabishchevich, P.N. (2007) Additive Difference Schemes for Problems in Mathematical Physics. URSS, LCI, Moscow.

[9] Moskalkov, M.N. (1975) On a Property of Schemes of High Order for One-Dimensional Wave Equation. Journal of Computational Mathematics and Mathematical Physics, 15, 254-260.

[10] Zienkiewicz, O.C. (1971) The Finite Element Method in Engineering Science. McGraw-Hill, London.

[11] Zienkiewicz, O.C. and Morgan, K. (1983) Finite Elements and Approximation. Wiley, Hoboken.

[12] Zienkiewicz, O.C. and Taylor, R.L. (2000) The Finite Element Method. Butterworth-Heinemann.

[13] Moskalkov, M.N. (1980) Scheme of the High-Accuracy Finite Element Method for Solving Non-Steady-State SecondOrder Equations. Differential Equations, 16, 1283-1292.

[14] Moskalkov, M.N. and Utebaev, D. (2005) Investigation of Difference Schemes of Finite Element Method for Second-Order Unsteady-State Equations. Journal Computation and Applied Mathematics, Kiev, No. 92, 70-76.

[15] Moskalkov, M.N. and Utebaev, D. (2012) Comparison of Some Methods for Solving the Internal Wave Propagation Problem in a Weakly Stratified Fluid. Mathematical Models and Computer Simulations, 3, 264-271.

http://dx.doi.org/10.1134/S2070048211020086

[16] Moskalkov, M.N. and Utebaev, D. (2010) Finite Element Method for the Gravity-Gyroscopic Wave Equation. Journal Computation and Applied Mathematics, 2, 97-104.

[17] Moskalkov, M.N. and Utebaev, D. (2011) Convergence of the Finite Element Scheme for the Equation of Internal Waves. Cybernetics and Systems Analysis, 47, 459-465.

http://dx.doi.org/10.1007/s10559-011-9327-1

[18] Moskalkov, M.N. and Utebaev, D. (2009) The Convergence of Finite Element Method for a Hyperbolic Equation with Generalized Solutions. Uzbek Mathematical Journal, Uzbekistan, No. 2, 119-128.

[1] Whitham, G.B. (1974) Linear and Nonlinear Waves. Wiley, Hoboken.

[2] Miropolsky, Y.Z. (1981) Dynamics of Internal-Waves in the Ocean. Gidrometeoizdates, Moscow.

[3] Gabov, S.A. and Sveshnikov, A.G. (1990) Linear Problems in the Theory of Non-Steady-State Internal Waves. Nauka, Moskow.

[4] Gabov, S.A. and Pletner, Y.D. (1985) A Nachal-No-Kraevoy Problem for the Gravity-Gyros-Copic Waves. Journal of Computational Mathematics and Mathematical Physics, 25, 1689-1696.

[5] Gabov, S.A. and Pletner, Y.D. (1987) The Gravity-Gyroscopic Waves: The Angular Potential and Its Application. Journal of Computational Mathematics and Mathematical Physics, 27, 102-113.

[6] Iskendarov, B.A., Mamedov, J.Y. and Sulaimanov, S.E. (2009) Mixed Problem for the Gravity-Gyroscopic Waves in the Boussinesq Approximation in an Infinite Cylindrical Domain. Journal of Computa-tional Mathematics and Mathematical Physics, 49, 1659-1675.

[7] Samarskii, A.A. (1977) Theory of Difference Schemes. Nauka, Moskow.

[8] Samarskii, A.A. and Vabishchevich, P.N. (2007) Additive Difference Schemes for Problems in Mathematical Physics. URSS, LCI, Moscow.

[9] Moskalkov, M.N. (1975) On a Property of Schemes of High Order for One-Dimensional Wave Equation. Journal of Computational Mathematics and Mathematical Physics, 15, 254-260.

[10] Zienkiewicz, O.C. (1971) The Finite Element Method in Engineering Science. McGraw-Hill, London.

[11] Zienkiewicz, O.C. and Morgan, K. (1983) Finite Elements and Approximation. Wiley, Hoboken.

[12] Zienkiewicz, O.C. and Taylor, R.L. (2000) The Finite Element Method. Butterworth-Heinemann.

[13] Moskalkov, M.N. (1980) Scheme of the High-Accuracy Finite Element Method for Solving Non-Steady-State SecondOrder Equations. Differential Equations, 16, 1283-1292.

[14] Moskalkov, M.N. and Utebaev, D. (2005) Investigation of Difference Schemes of Finite Element Method for Second-Order Unsteady-State Equations. Journal Computation and Applied Mathematics, Kiev, No. 92, 70-76.

[15] Moskalkov, M.N. and Utebaev, D. (2012) Comparison of Some Methods for Solving the Internal Wave Propagation Problem in a Weakly Stratified Fluid. Mathematical Models and Computer Simulations, 3, 264-271.

http://dx.doi.org/10.1134/S2070048211020086

[16] Moskalkov, M.N. and Utebaev, D. (2010) Finite Element Method for the Gravity-Gyroscopic Wave Equation. Journal Computation and Applied Mathematics, 2, 97-104.

[17] Moskalkov, M.N. and Utebaev, D. (2011) Convergence of the Finite Element Scheme for the Equation of Internal Waves. Cybernetics and Systems Analysis, 47, 459-465.

http://dx.doi.org/10.1007/s10559-011-9327-1

[18] Moskalkov, M.N. and Utebaev, D. (2009) The Convergence of Finite Element Method for a Hyperbolic Equation with Generalized Solutions. Uzbek Mathematical Journal, Uzbekistan, No. 2, 119-128.