Cubic spline approximation based on half-step discretization for 2D quasilinear elliptic equations

References

• D. M. Young and J. H. Dauwalder, “Discrete representation of partial differential equations,” in Errors in Digital Computation, L. Rall, Ed. New York: Academic Press, 1965, p. 181.
   Google Scholar
• R. E. Lynch and J. R. Rice, “The Hodie method and its performance for solving elliptic partial differential equations,” in Recent Advances in Numerical Analysis, C. De Boor and G. H. Golub, Eds. New York: Academic Press, 1978, pp. 143–175.
   Google Scholar
• R. E. Lynch and J. R. Rice, “High accuracy finite difference approximation to solutions of elliptic partial differential equations,” Proc. Natl. Acad. Sci. U. S. A., vol. 75, pp. 2541–2572, 1978. DOI: 10.1073/pnas.75.6.2541.
   Google Scholar
• R. F. Boisvert, “Families of high order accurate discretization of some elliptic problems,” SIAM J. Sci. Stat. Comput, vol. 2, no. 3, pp. 268–285, 1981. DOI: 10.1137/0902022.
   Web of Science ®Google Scholar
• I. Yavneh, “Analysis of fourth-order compact scheme for convection diffusion equation,” J. Comput. Phys., vol. 133, no. 2, pp. 361–373, 1997. DOI: 10.1006/jcph.1997.5659.
   Web of Science ®Google Scholar
• M. K. Jain, R. K. Jain, and R. K. Mohanty, “A fourth order difference method for elliptic equations with nonlinear first derivative terms,” Numer. Methods Partial Differ. Eq., vol. 5, no. 2, pp. 87–95, 1989. DOI: 10.1002/num.1690050203.
   Google Scholar
• M. K. Jain, R. K. Jain, and R. K. Mohanty, “Fourth order difference methods for the system of 2D nonlinear elliptic partial differential equations,” Numer. Methods Partial Differ. Eq., vol. 7, no. 3, pp. 227–244, 1991. DOI: 10.1002/num.1690070303.
   Google Scholar
• R. K. Mohanty, “Order h4 difference methods for a class of singular two space elliptic boundary value problems,” J. Comp. Appl. Math., vol. 81, no. 2, pp. 229–247, 1997. DOI: 10.1016/S0377-0427(97)00058-7.
   Google Scholar
• R. K. Mohanty and S. Singh, “A new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems,” Appl. Math. Comput., vol. 175, no. 2, pp. 1400–1414, 2006. DOI: 10.1016/j.amc.2005.08.023.
   Web of Science ®Google Scholar
• R. K. Mohanty, S. Karaa, and U. Arora, “Fourth order nine point unequal mesh discretization for the solution of 2-D nonlinear elliptic partial differential equations,” Neural Parallel Sci. Comput., vol. 14, pp. 453–470, 2006.
   Google Scholar
• J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications. New York: Academic Press, 1967.
   Google Scholar
• W. G. Bickley, “Piecewise cubic interpolation and two point boundary value problems,” Comput. J., vol. 11, no. 2, pp. 206–208, 1968. DOI: 10.1093/comjnl/11.2.206.
   Web of Science ®Google Scholar
• D. J. Fyfe, “The use of cubic splines in the solution of two point boundary value problems,” Comput. J., vol. 12, no. 2, pp. 188–192, 1969. DOI: 10.1093/comjnl/12.2.188.
   Web of Science ®Google Scholar
• E. L. Albasiny and W. D. Hoskins, “Cubic spline solution of boundary value problem,” Comput. J., vol. 12, no. 2, pp. 151–153, 1969. DOI: 10.1093/comjnl/12.2.151.
   Web of Science ®Google Scholar
• E. L. Albasiny and W. D. Hoskins, “Increased accuracy cubic spline solutions to two point boundary value problems,” IMA J. Appl. Math., vol. 9, no. 1, pp. 47–55, 1972. DOI: 10.1093/imamat/9.1.47.
   Google Scholar
• P. M. Prenter, Splines and Variational Methods. New York: Wiley, 1975.
   Google Scholar
• R. P. Tewarson, “On the use splines for the numerical solution of non-linear two-point boundary value problems,” BIT, vol. 20, no. 2, pp. 223–232, 1980. DOI: 10.1007/BF01933195.
   Google Scholar
• M. K. Jain and T. Aziz, “Cubic spline solution of two point boundary value problems with significant first derivatives,” Comput. Meth. Appl. Mech. Engg., vol. 39, no. 1, pp. 83–91, 1983. DOI: 10.1016/0045-7825(83)90075-0.
   Web of Science ®Google Scholar
• E. A. Al-Said, “Spline methods for solving a system of second order boundary value problems,” Int. J. Comput. Math., vol. 70, no. 4, pp. 717–727, 1999. DOI: 10.1080/00207169908804784.
   Web of Science ®Google Scholar
• E. A. Al-Said, “The use of cubic splines in the numerical solution of a system of second-order boundary value problem,” Comput. Math. Appl., vol. 42, no. 6–7, pp. 861–869, 2001. DOI: 10.1016/S0898-1221(01)00204-8.
   Web of Science ®Google Scholar
• R. K. Mohanty and D. J. Evans, “A fourth-order accurate cubic spline alternating group explicit method for nonlinear singular two-point boundary value problems,” Int. J. Comput. Math., vol. 80, no. 4, pp. 479–492, 2003. DOI: 10.1080/0020716021000009219.
   Web of Science ®Google Scholar
• R. K. Mohanty, P. L. Sachdev, and N. Jha, “An O(h4) accurate cubic spline TAGE method for singular two point boundary value problems,” Appl. Math. Comput., vol. 158, pp. 853–868, 2004.
   Web of Science ®Google Scholar
• A. Khan, “Parametric cubic spline solution of two point boundary value problems,” Appl. Math. Comput., vol. 154, no. 1, pp. 175–182, 2004. DOI: 10.1016/S0096-3003(03)00701-X.
   Web of Science ®Google Scholar
• A. Khan, I. Khan, and T. Aziz, “A survey on parametric spline function approximation,” Appl. Math. Comput., vol. 171, no. 2, pp. 983–1003, 2005. DOI: 10.1016/j.amc.2005.01.112.
   Web of Science ®Google Scholar
• J. Rashidinia, R. Mohammadi, and M. Ghasemi, “Cubic spline solution of singularly perturbed boundary value problems with significant first derivatives,” Appl. Math. Comput., vol. 190, no. 2, pp. 1762–1766, 2007. DOI: 10.1016/j.amc.2007.02.050.
   Web of Science ®Google Scholar
• J. Rashidinia, R. Mohammadi, and R. Jalilian, “Spline solution of nonlinear singular boundary value problems,” Int. J. Comput. Math., vol. 85, no. 1, pp. 39–52, 2008. DOI: 10.1080/00207160701293048.
   Web of Science ®Google Scholar
• E. N. Houstis, J. R. Rice, and E. A. Vavalis, “Convergence of O(h4) cubic spline collocation methods for elliptic partial differential equations,” SIAM J. Numer. Anal., vol. 25, no. 1, pp. 54–74, 1988. DOI: 10.1137/0725006.
   Google Scholar
• A. Hadjidimos, E. N. Houstis, J. R. Rice, and E. A. Vavalis, “Iterative line cubic spline collocation methods for elliptic partial differential equations in several dimensions,” SIAM J. Sci. Comput., vol. 14, no. 3, pp. 715–734, 1993. DOI: 10.1137/0914045.
   Web of Science ®Google Scholar
• R. K. Mohanty and S. Dey, “A new finite difference discretization of order four for (∂u/∂n) for two dimensional quasi-linear elliptic boundary value problems,” Int. J. Comput. Math., vol. 76, no. 4, pp. 505–516, 2001. DOI: 10.1080/00207160108805043.
   Web of Science ®Google Scholar
• R. K. Mohanty and U. Arora, “A new discretization method of order four for the numerical solution of one space dimensional second order quasi-linear hyperbolic equation,” Int. J. Math. Edu. Sci. Technol., vol. 33, no. 6, pp. 829–838, 2002. DOI: 10.1080/00207390210162465.
   Google Scholar
• R. K. Mohanty and M. K. Jain, “High accuracy cubic spline alternating group explicit methods for 1D quasi-linear parabolic equations,” Int. J. Comput. Math., vol. 86, no. 9, pp. 1556–1571, 2009. DOI: 10.1080/00207160801923049.
   Web of Science ®Google Scholar
• R. K. Mohanty and N. Setia, “A new fourth order compact off-step discretization for the system of 2D non-linear elliptic partial differential equations,” E. Asian J. Appl. Math., vol. 2, no. 1, pp. 59–82, 2012. DOI: 10.4208/eajam.291211.080212a.
   Web of Science ®Google Scholar
• R. K. Mohanty and N. Setia, “A new compact high order off-step discretization for the system of 2D quasi-linear elliptic partial differential equations,” Adv. Differ. Equ., vol. 2013, no. 1, pp. 223, 2013. DOI: 10.1186/1687-1847-2013-223.
   Google Scholar
• R. K. Mohanty, M. K. Jain, and D. Dhall, “High accuracy cubic spline approximation for two dimensional quasi-linear elliptic boundary value problems,” appl. Math. Modell., vol. 37, no. 1–2, pp. 155–171, 2013. DOI: 10.1016/j.apm.2012.02.020.
   Web of Science ®Google Scholar
• R. K. Mohanty and R. Kumar, “A new fast algorithm based on half-step discretization for one space dimensional quasilinear hyperbolic equations,” Appl. Math. Comput., vol. 244, pp. 624–641, 2014. DOI: 10.1016/j.amc.2014.07.020.
   Web of Science ®Google Scholar
• R. K. Mohanty and R. Kumar, “A novel numerical algorithm of Numerov type for 2D quasilinear elliptic boundary value problems,” Int. J. Comput. Methods Eng. Sci. Mech., vol. 15, no. 6, pp. 473–489, 2014. DOI: 10.1080/15502287.2014.934488.
   Google Scholar
• R. K. Mohanty and G. Khurana, “A new high accuracy cubic spline method based on half-step discretization for the system of 1D non-linear wave equations,” EC, vol. 36, no. 3, pp. 930–957, 2019. DOI: 10.1108/EC-04-2018-0194.
   Google Scholar
• R. K. Mohanty, G. Manchanda, and A. Khan, “Compact half-step approximation in exponential form for 2D second order quasi-linear elliptic partial differential equations,” J. Difference Equ. Appl., vol. 25, no. 5, pp. 716–749, 2019. DOI: 10.1080/10236198.2019.1624737.
   Web of Science ®Google Scholar
• R. K. Mohanty and S. Sharma, “A new two-level implicit scheme based on cubic spline approximations for the 1D time-dependent quasilinear biharmonic problems,” Eng. Comput. DOI: 10.1007/s00366-019-00778-1.
   Web of Science ®Google Scholar
• R. K. Mohanty and S. Sharma, “A new high accuracy method based on off-step cubic polynomial approximations for the solution of coupled Burgers’ equations and Burgers-Huxley equation,” Eng. Comput. DOI: 10.1007/s00366-020-00982-4.
   Web of Science ®Google Scholar
• I. Priyadarshini and R. K. Mohanty, “High resolution compact numerical method for the system of 2D quasilinear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications,” Eng. Comput. DOI: 10.1007/s00366-020-01150-4.
   Google Scholar
• R. C. Mittal and A. Tripathi, “A collocation method for numerical solutions of coupled Burgers’ equations,” Int. J. Comput. Methods Eng. Sci. Mech., vol. 15, no. 5, pp. 457–471, 2014. DOI: 10.1080/15502287.2014.929194.
   Google Scholar
• R. C. Mittal and A. Tripathi, “Numerical solutions of symmetric regularized long wave equations using collocation of cubic B-splines finite element,” Int. J. Comput. Methods Eng. Sci. Mech., vol. 16, no. 2, pp. 142–150, 2015. DOI: 10.1080/15502287.2015.1011812.
   Web of Science ®Google Scholar
• R. C. Mittal and A. Tripathi, “Numerical solutions of two-dimensional Burgers’ equations using modified bi-cubic B-spline finite elements,” Eng. Comput. vol. 32, no. 5, pp. 1275–1306, 2015. DOI: 10.1108/EC-04-2014-0067.
   Google Scholar
• R. C. Mittal and A. Tripathi, “Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines,” Int. J. Comput. Math., vol. 92, no. 5, pp. 1053–1077, 2015. DOI: 10.1080/00207160.2014.920834.
   Web of Science ®Google Scholar
• R. C. Mittal and A. Tripathi, “Numerical solutions of two-dimensional unsteady convection–diffusion problems using modified bi-cubic B-spline finite elements,” Int. J. Comput. Math., vol. 94, no. 1, pp. 1–21, 2017. DOI: 10.1080/00207160.2015.1085976.
   Web of Science ®Google Scholar
• C. T. Kelly, Iterative Methods for Linear and Non-Linear Equations. Philadelphia: SIAM Publications, 1995.
   Google Scholar
• R. S. Varga, Matrix Iterative Analysis. New York: Springer, 2000.
   Google Scholar
• Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. Philadelphia: SIAM Publisher, 2003.
   Google Scholar
• L. A. Hageman and D. M. Young, Applied Iterative Methods. New York: Dover Publication, 2004.
   Google Scholar