×

Quartic splines solutions of third-order obstacle problems. (English) Zbl 1048.65068

Summary: We develop a new numerical method for solving a system of third-order boundary value problems associated with third-order obstacle problems using the quartic splines. Its convergence analysis is also considered. To illustrate its efficiency, we give an example, which shows that this method gives better results.

MSC:

65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
49M25 Discrete approximations in optimal control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Al-Said, E. A., Numerical solutions of third-order boundary value problems, Int. J. Comput. Math., 78, 111-122 (2001) · Zbl 0990.65085
[2] Al-Said, E. A.; Aslam Noor, M., Cubic splines method for a system of third-order boundary value problems, Appl. Math. Comput., 142, 195-204 (2003) · Zbl 1022.65082
[3] Al-Said, E. A.; Aslam Noor, M.; Khalifa, A. K., Finite difference scheme for variational inequalities, J. Opt. Theory Appl., 89, 453-459 (1996) · Zbl 0848.49007
[4] Al-Said, E. A.; Aslam Noor, M.; Rassias, Th. M., Numerical solutions of third-order obstacle problems, Int. J. Comput. Math., 69, 75-84 (1998) · Zbl 0905.65074
[5] Glowinski, R.; Lions, J. L.; Tremolieres, R., Numerical Analysis of Variational Inequalities (1981), North-Holland: North-Holland Amsterdam · Zbl 0508.65029
[6] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press London · Zbl 0457.35001
[7] Lewy, H.; Stampacchia, G., On the regularity of the solutions of the variational inequalities, Commun. Pure Appl. Math., 22, 153-188 (1969) · Zbl 0167.11501
[8] Aslam Noor, M., General variational inequalities, Appl. Math. Lett., 1, 119-122 (1988) · Zbl 0655.49005
[9] Aslam Noor, M., Variational inequalities in physical oceanography, (Rahman, M., Ocean Waves Engineering (1994), Comput. Mechanics Publ: Comput. Mechanics Publ U.K), 201-266
[10] Aslam Noor, M., Some recent advances in variational inequalities, Part I: Basic concepts, New Zealand J. Math., 26, 53-80 (1997) · Zbl 0886.49004
[11] Aslam Noor, M., Some recent advances in variational inequalities, Part II: Other concepts, New Zealand J. Math., 26, 229-255 (1997) · Zbl 0889.49006
[12] Aslam Noor, M.; Al-Said, E. A., Finite difference method for a system of third-order boundary value problems, J. Opt. Theory Appl., 122, 627-637 (2002) · Zbl 1002.49012
[13] Aslam Noor, M.; Khalifa, A. K., A numerical approach for odd-order obstacle problems, Int. J. Comput. Math., 54, 109-116 (1994) · Zbl 0828.65073
[14] Aslam Noor, M.; Inayat Noor, K.; Rassias, Th. M., Some aspects of variational inequalities, J. Comput. Appl. Math., 47, 285-312 (1993) · Zbl 0788.65074
[15] Stampacchia, G., Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258, 4413-4416 (1964) · Zbl 0124.06401
[16] Tonti, E., Variational formulation for every nonlinear problem, Int. J. Eng. Sci., 22, 1343-1371 (1984) · Zbl 0558.49022
[17] Usmani, R. A.; Sakai, M., Quartic spline solutions for two-point boundary problems involving third order differential equations, J. Math. Phys. Sci., 18, 365-380 (1984) · Zbl 0583.65052
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.