Finite-difference method for a system of third-order boundary-value problems. (English) Zbl 1002.49012
Summary: We develop a two-stage numerical method for computing the approximate solutions of third-order boundary-value problems associated with odd-order obstacle problems. We show that the present method is of order two. A numerical example is presented to illustrate the applicability of the new method. A comparison is also given with previously known results.
MSC:
| 49J40 | Variational inequalities |
Keywords:
convergence analysis; finite-difference techniques; variational inequalities; odd-order obstacle problems; boundary-value problemsReferences:
| [1] | STAMPACCHIA, G., Formes Bilineaires Coercitiûes sur les Ensembles Conûexes, Comptes Rendus de l’Academie des Sciences, Paris, Vol. 258, pp. 4413–4416, 1964. · Zbl 0124.06401 |
| [2] | NOOR, M. A., General Variational Inequalities, Applied Mathematics Letters, Vol. 1, pp. 119–122, 1988. · Zbl 0655.49005 · doi:10.1016/0893-9659(88)90054-7 |
| [3] | NOOR, M. A., Variational Inequalities in Physical Oceanography, Ocean Wave Engineering, Edited by M. Rahman, Computational Mechanics Publications, Southampton, England, pp. 201–226, 1994. |
| [4] | NOOR, M. A., Some Recent Adûances in Variational Inequalities, Part 1: Basic Concepts, New Zealand Journal of Mathematics, Vol. 26, pp. 53–80, 1997. · Zbl 0886.49004 |
| [5] | NOOR, M. A., Some Recent Adûances in Variational Inequalities, Part 2: Other Concepts, New Zealand Journal of Mathematics, Vol. 26, pp. 229–255, 1997. · Zbl 0889.49006 |
| [6] | NOOR, M. A., and KHALIFA, A. K., A Numerical Approach for Odd-Order Obstacle Problems, International Journal of Computer Mathematics, Vol. 54, pp. 109–116, 1994. · Zbl 0828.65073 · doi:10.1080/00207169408804343 |
| [7] | NOOR, M. A., NOOR, K. I., and RASSIAS, T. M., Some Aspects of Variational Inequalities, Journal of Computational and Applied Mathematics, Vol. 47, pp. 285–312, 1993. · Zbl 0788.65074 · doi:10.1016/0377-0427(93)90058-J |
| [8] | GIANNESSI, F., and MAUGERI, A., Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, NY, 1995. · Zbl 0834.00044 |
| [9] | LEWY, H., and STAMPACCHIA, G., On the Regularity of the Solutions of Variational Inequalities, Communication in Pure and Applied Mathematics, Vol. 22, pp. 153–188, 1969. · Zbl 0167.11501 · doi:10.1002/cpa.3160220203 |
| [10] | AL-SAID, E. A., NOOR, M. A., and KHALIFA, A. K., Finite-Difference Scheme for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 89, pp. 453–459, 1996. · Zbl 0848.49007 · doi:10.1007/BF02192538 |
| [11] | AL-SAID, E. A., NOOR, M. A., and RASSIAS, T. M., Numerical Solutions of Third-Order Obstacle Problems, International Journal of Computer Mathematics, Vol. 69, pp. 75–84, 1998. · Zbl 0905.65074 · doi:10.1080/00207169808804710 |
| [12] | BAIOCCHI, C., and CAPELO, A., Variational and Quasi-Variational Inequalities, John Wiley and Sons, New York, NY, 1994. · Zbl 1308.49002 |
| [13] | COTTLE, R. W., GIANNESSI, F., and LIONS, J. L., Variational Inequalities and Complementarity Problems: Theory and Applications, John Wiley and Sons, New York, NY, 1980. |
| [14] | CRANK, J., Free and Moûing Boundary Problems, Clarendon Press, Oxford, England, 1984. · Zbl 0547.35001 |
| [15] | DUNBAR, S. R., Geometric Analysis of Nonlinear Boundary-Value Problems from Physical Oceanography, SIAM Journal on Mathematical Analysis, Vol. 24, pp. 444–465, 1993. · Zbl 0770.34021 · doi:10.1137/0524028 |
| [16] | FILIPPOV, V. M., Variational Principles for Nonpotential Operators, American Mathematical Society, Providence, Rhode Island, Vol. 77, 1989. · Zbl 0682.35006 |
| [17] | BOERSMA, J., and MOLENAAR, J., Geometry of the Shoulder of a Packaging Machine, SIAM Review, Vol. 37, pp. 406–422, 1995. · Zbl 0838.53001 · doi:10.1137/1037084 |
| [18] | TUCK, E. O., and SCHWARTZ, L. W., A Numerical and Asymptotic Study of Some Third-Order Ordinary Differential Equations Releûant to Draining and Coating Flows, SIAM Review, Vol. 32, pp. 453–469, 1990. · Zbl 0705.76062 · doi:10.1137/1032079 |
| [19] | NOOR, M. A., New Approximation Schemes for General Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 251, pp. 217–229, 2000. · Zbl 0964.49007 · doi:10.1006/jmaa.2000.7042 |
| [20] | FRÖBERG, C., Numerical Mathematics; Theory and Computer Applications, Benjamin/Cummings, Reading, Massachusetts, 1985. · Zbl 0574.65001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
