Composite non-polynomial spline solution of boundary value problems in plate deflection theory

References

• E. L. Reiss, A. J. Callegari, and D. S. Ahluwalia, Ordinary Differential Equations with Applications, 6th ed. New York, NY: Holt, Rinehart and Winston, 1976.
   Google Scholar
• S. Timoshenko, and S. K. Woinowsky, Theory of Plates and Shells, 2nd ed. New York, NY: McGraw-Hill, 1959.
   Google Scholar
• R. A. Usmani, “Solving boundary value problems in plate deflection theory,” Simulation, vol. 37, pp. 195–206, 1981. DOI: 10.1177/003754978103700605.
   Google Scholar
• R. A. Usmani, and S. A. Warsi, “Smooth spline solutions for boundary value problems in plate deflection theory,” Comput. Math. Appl., vol. 6, no. 2, pp. 205–211, 1980. DOI: 10.1016/0898-1221(80)90029-2.
   Web of Science ®Google Scholar
• L. Bougoffa, R. Rach, and A. M. Wazwaz, “On solutions of boundary value problem for fourth-order beam equations,” Math. Mod. Anal., vol. 21, no. 3, pp. 304–318, 2016.
   Web of Science ®Google Scholar
• R. Singh, and A. M. Wazwaz, “Numerical solutions of fourth-order volterra integro-differential equations by the Green’s function and decomposition method,” Math. Sci., vol. 10, no. 4, pp. 159–166, 2016. DOI: 10.1007/s40096-016-0190-0.
   Google Scholar
• S. Mukhigulashvili, and M. Manjikashvili, “On one two-point BVP for the fourth order linear ordinary differential equation,” Geo. Math. J., vol. 24, no. 2, pp. 265–275, 2017.
   Google Scholar
• J. Fialho, and F. Minhós, “Fourth order impulsive periodic boundary value problems,” Differ. Equ. Dyn. Syst., vol. 23, no. 2, pp. 117–127, 2015. DOI: 10.1007/s12591-013-0186-2.
   Web of Science ®Google Scholar
• J. R. Graef, L. Kong, and X. Liu, “Multiple anti-periodic solutions to a discrete fourth order nonlinear equation,” Differ. Equ. Dyn. Syst., 2016. DOI: 10.1007/s12591-016-0293-y.
   Web of Science ®Google Scholar
• G. A. Afrouzi, S. Moradi, and G. Caristi, “Infinitely Many solutions for impulsive nonlocal elastic beam equations,” Differ. Equ. Dyn. Syst., pp. 1–14, 2017. https://doi.org/10.1007/s12591-017-0397-z
   Google Scholar
• Y. Zou, “On the existence of positive solutions for a fourth-order boundary value problem,” J. Func. Spaces, vol. 2017, Article ID 4946198, p. 5, 2017. DOI: 10.1155/2017/4946198.
   Google Scholar
• H. Tariq, and G. Akram, “Quintic spline technique for time fractional fourth-order partial differential equation,” Numer. Meth. Partial Differ. Equ., vol. 33, no. 2, pp. 445–466, 2016.
   Web of Science ®Google Scholar
• O. Adeyeye, and Z. Omar, “Solving nonlinear fourth-order boundary value problems using a numerical approach. (m + 1) th-step block method,” Int. J. Differ. Equ., vol. 2017, Article ID 4925914, p. 9, 2017. DOI: 10.1155/2017/4925914.
   Google Scholar
• N. Papamichael, and A. J. Worsey, “A cubic spline method for the solution of a linear fourth-order two-point boundary value problem,” Technical Report TR/101, 1980.
   Google Scholar
• R. A. Usmani, and S. A. Warsi, “Smooth spline solutions for boundary value problems in plate deflection theory,” Comp Math Appl., vol. 6, no. 2, pp. 205–211, 1980. DOI: 10.1016/0898-1221(80)90029-2.
   Web of Science ®Google Scholar
• R. A. Usmani, “The use of quartic splines in the numerical solution of fourth-order boundary value problems,” J. Comput Appl. Math., vol. 44, no. 2, pp. 187–199, 1992. DOI: 10.1016/0377-0427(92)90010-U.
   Web of Science ®Google Scholar
• M. van Daele, V. G. Berghe, and H. de Meyer, “A smooth approximation for the solution of a fourth order boundary value problem based on non-polynomial splines,” J Comput. Appl. Math., vol. 51, pp. 383–394, 1994.
   Web of Science ®Google Scholar
• Zhu, Y. “Quartic spline collocation methods for fourth order two point boundary value problems,” MSc thesis, Department of Computer Science, University of Toronto, 2001.
   Google Scholar
• E. A. Al-Said, and M. A. Noor, “Quartic spline method for solving fourth order obstacle boundary value problems,” J. Comput. Appl. Math, vol. 143, no. 1, pp. 107–116, 2002. DOI: 10.1016/S0377-0427(01)00497-6.
   Web of Science ®Google Scholar
• S. S. Siddiqi, and G. Akram, “Quintic spline solutions of fourth order boundary value problems,” Mathematics NA/0306357, 2003.
   Google Scholar
• J. Rashidinia, and A. Golbabaee, “Convergence of numerical solution of a fourth-order boundary value problem,” Appl. Math. Comp., vol. 171, no. 2, pp. 1296–1305, 2005. DOI: 10.1016/j.amc.2005.01.117.
   Web of Science ®Google Scholar
• S. U. Islam, I. A. Tirmizi, and S. Ashraf, “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary value problems with engineering applications,” Appl. Math. Comp., vol. 174, pp. 1169–1180, 2006.
   Web of Science ®Google Scholar
• E. A. Al-Said, M. A. Noor, and T. M. Rassias, “Cubic splines method for solving fourth-order obstacle problems,” Appl. Math. Comp., vol. 174, no. 1, pp. 180–187, 2006. DOI: 10.1016/j.amc.2005.03.022.
   Web of Science ®Google Scholar
• G. B. Loghmani, and S. R. Alavizadeh, “Numerical solution of fourth-order problems with separated boundary conditions,” Appl. Math. Comput., vol. 191, pp. 571–581, 2007. DOI: 10.1016/j.amc.2007.02.116.
   Web of Science ®Google Scholar
• J. Rashidinia, R. Jalilian, and R. Mohammadi, “A class of methods based on non-polynomial spline functions for the solution of a special fourth- order boundary value problems with engineering applications,” Appl. Math. Comput., vol. 186, pp. 1572–1580, 2007. DOI: 10.1016/j.amc.2006.07.154.
   Web of Science ®Google Scholar
• S. S. Siddiqi, and G. Akram, “Solution of the system of fourth-order boundary value problems using non-polynomial spline technique,” Appl. Math. Comput., vol. 185, pp. 128–135, 2007. DOI: 10.1016/j.amc.2006.07.014.
   Web of Science ®Google Scholar
• M. A. Ramadan, I. F. Lashien, and W. K. Zahra, “Quintic non-polynomial spline solutions for fourth order two point boundary value problems,” Comm. Nonlin. Sci. Numer. Simul, vol. 14, no. 4, pp. 1105–1114, 2009. DOI: 10.1016/j.cnsns.2007.12.008.
   Web of Science ®Google Scholar
• P. Khandelwal, and A. Khan, “Exponential spline approach for the Solution of nonlinear fourth-order boundary value problems,” Publ. Inst. Math. Nouvelle Série Tome, vol. 104, no. 118, pp. 265–279, 2018. DOI: 10.2298/PIM1818265K.
   Google Scholar
• A. Khalid A, and M. N. Naeem, “Cubic spline solution of linear fourteenth order boundary value problems,” Ceylon J. Sci., vol. 47, no. 3, pp. 253–261, 2018. DOI: 10.4038/cjs.v47i3.7541.
   Google Scholar
• M. A. Ramadan, I. F. Lashien, and W. K. Zahra, “High order accuracy non-polynomial spline solutions for order two point boundary value problems,” Appl. Math. Comput., vol. 204, pp. 920–927, 2008. DOI: 10.1016/j.amc.2008.07.038.
   Web of Science ®Google Scholar
• J. R. Rashidinia, and R. J. Mohammadi, “Quintic spline solution of boundary value problems in the plate deflection theory,” Comp. Sci. Eng. Electr. Eng., vol. 16, pp. 53–59, 2009.
   Web of Science ®Google Scholar
• R. A. Usmani, “Discrete variable method for a boundary value problem with engineering applications,” Math. Comp., vol. 32, no. 144, pp. 1087–1096, 1978. DOI: 10.2307/2006334.
   Web of Science ®Google Scholar
• J. Rashidinia, “Direct methods for solution of a linear fourth-order two-point boundary value problem,” Int. J. Eng. Sci., vol. 13, pp. 37–48, 1978.
   Google Scholar
• R. A. Usmani, and M. J. Marsden, “Numerical solution of some ordinary differential equations occurring in plate deflection theory,” J. Eng. Math., vol. 9, no. 1, pp. 1–10, 1975. DOI: 10.1007/BF01535492.
   Web of Science ®Google Scholar
• M. M. Chawla, and C. P. Katti, “Finite difference methods for two-point boundary value problems involving high order differential equations,” BIT, vol. 19, no. 1, pp. 27–33, 1979. DOI: 10.1007/BF01931218.
   Google Scholar