Numerical Computation of Sturm-Liouville Problem with Robin Boundary Condition
The modelling of physical phenomena, such as the
earth’s free oscillations, the vibration of strings, the interaction of
atomic particles, or the steady state flow in a bar give rise to Sturm-
Liouville (SL) eigenvalue problems. The boundary applications of
some systems like the convection-diffusion equation, electromagnetic
and heat transfer problems requires the combination of Dirichlet and
Neumann boundary conditions. Hence, the incorporation of Robin
boundary condition in the analyses of Sturm-Liouville problem. This
paper deals with the computation of the eigenvalues and
eigenfunction of generalized Sturm-Liouville problems with Robin
boundary condition using the finite element method. Numerical
solution of classical Sturm–Liouville problem is presented. The
results show an agreement with the exact solution. High results
precision is achieved with higher number of elements.
[1] Sturm-Liouville problem, In Encyclopædia, 2015, Britannica. Retrieved from http://www.britannica.com/topic/Sturm-Liouville problem.
[2] A. Zettl, “Sturm-liouville theory”, American Mathematical Soc., vol. 121, 2010.
[3] M. D. Milkhailov, & N. L. Vulchanov, “Computational procedure for Sturm-Liouville problems”. Journal of Computational Physics, vol. 50 (3), 1983, 323-336.
[4] C. T. Fulton, & S. A. Pruess, “Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems”, Journal of Mathematical Analysis and Applications, vol. 188 (1), 1994, pp. 297-340.
[5] Q. Kong, A. Zettl, Eigenvalues of regular Sturm–Liouville problems. Journal of differential equations, vol. 131(1), 1996, pp. 1-19.
[6] S. Abbasbandy, A. Shirzadi,, “A new application of the homotopy analysis method: Solving the Sturm–Liouville problems”, Communications in Nonlinear Science and Numerical Simulation, vol. 16(1), 2011, pp. 112-126.
[7] Q. Kong, H. Wu, A. Zettl, “Dependence of the nth Sturm–Liouville eigenvalue on the problem”, Journal of differential equations, vol. 156 (2), 1999, pp. 328-354.
[8] S. Pruess, Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation. SIAM Journal on Numerical Analysis, 1973, vol. 10(1), pp. 55-68.
[9] M. Lakestani, M. Dehghan, “Numerical solution of fourth order integro differential equations using Chebyshev cardinal functions”, International Journal of Computer Mathematics, vol. 87, 2010, pp. 1389-1394.
[10] Q. M. Al-Mdallal, “On the numerical solution of fractional Sturm–Liouville problems”, International Journal of Computer Mathematics, vol. 87 (12), 2010, pp. 2837-2845.
[11] B. M. Levitan, ”Inverse Sturm-Liouville Problems”, 1987, VSP
[12] W. O. Amrein, A. M. Hinz, D. B. Pearson, “Sturm-Liouville theory: past and present”, Springer Science & Business Media., 2005.
[13] D. W. Hahn, M. N. Ozisk, “Heat Conduction, 3rd edition”. New York: Wiley. 2012, ISBN 978-0-470-90293-6.
[1] Sturm-Liouville problem, In Encyclopædia, 2015, Britannica. Retrieved from http://www.britannica.com/topic/Sturm-Liouville problem.
[2] A. Zettl, “Sturm-liouville theory”, American Mathematical Soc., vol. 121, 2010.
[3] M. D. Milkhailov, & N. L. Vulchanov, “Computational procedure for Sturm-Liouville problems”. Journal of Computational Physics, vol. 50 (3), 1983, 323-336.
[4] C. T. Fulton, & S. A. Pruess, “Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems”, Journal of Mathematical Analysis and Applications, vol. 188 (1), 1994, pp. 297-340.
[5] Q. Kong, A. Zettl, Eigenvalues of regular Sturm–Liouville problems. Journal of differential equations, vol. 131(1), 1996, pp. 1-19.
[6] S. Abbasbandy, A. Shirzadi,, “A new application of the homotopy analysis method: Solving the Sturm–Liouville problems”, Communications in Nonlinear Science and Numerical Simulation, vol. 16(1), 2011, pp. 112-126.
[7] Q. Kong, H. Wu, A. Zettl, “Dependence of the nth Sturm–Liouville eigenvalue on the problem”, Journal of differential equations, vol. 156 (2), 1999, pp. 328-354.
[8] S. Pruess, Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation. SIAM Journal on Numerical Analysis, 1973, vol. 10(1), pp. 55-68.
[9] M. Lakestani, M. Dehghan, “Numerical solution of fourth order integro differential equations using Chebyshev cardinal functions”, International Journal of Computer Mathematics, vol. 87, 2010, pp. 1389-1394.
[10] Q. M. Al-Mdallal, “On the numerical solution of fractional Sturm–Liouville problems”, International Journal of Computer Mathematics, vol. 87 (12), 2010, pp. 2837-2845.
[11] B. M. Levitan, ”Inverse Sturm-Liouville Problems”, 1987, VSP
[12] W. O. Amrein, A. M. Hinz, D. B. Pearson, “Sturm-Liouville theory: past and present”, Springer Science & Business Media., 2005.
[13] D. W. Hahn, M. N. Ozisk, “Heat Conduction, 3rd edition”. New York: Wiley. 2012, ISBN 978-0-470-90293-6.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71543", author = "Theddeus T. Akano and Omotayo A. Fakinlede", title = "Numerical Computation of Sturm-Liouville Problem with Robin Boundary Condition", abstract = "The modelling of physical phenomena, such as the
earth’s free oscillations, the vibration of strings, the interaction of
atomic particles, or the steady state flow in a bar give rise to Sturm-
Liouville (SL) eigenvalue problems. The boundary applications of
some systems like the convection-diffusion equation, electromagnetic
and heat transfer problems requires the combination of Dirichlet and
Neumann boundary conditions. Hence, the incorporation of Robin
boundary condition in the analyses of Sturm-Liouville problem. This
paper deals with the computation of the eigenvalues and
eigenfunction of generalized Sturm-Liouville problems with Robin
boundary condition using the finite element method. Numerical
solution of classical Sturm–Liouville problem is presented. The
results show an agreement with the exact solution. High results
precision is achieved with higher number of elements.", keywords = "Sturm-Liouville problem, Robin boundary condition,
finite element method, eigenvalue problems.", volume = "9", number = "11", pages = "679-5", }