Yıl 2020,
Cilt: 49 Sayı: 6, 1885 - 1903, 08.12.2020
N Balasubramani
M. Guru Prem Prasad
S Natesan
Kaynakça
- [1] M. Baccouch, A superconvergent local discontinuous Galerkin method for nonlinear
two-point boundary-value problems, Numer. Algor. 79 (3), 697–718, 2018.
- [2] M. Baccouch, An adaptive local discontinuous Galerkin method for nonlinear twopoint
boundary-value problems, Numer. Algor. 2019, doi:10.1007/s11075-019-00794-8.
- [3] M.F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2 (1), 303–329,
1986.
- [4] M.F. Barnsley and A.N. Harrington, The calculus of fractal interpolation functions,
J. Approx. Theory 57 (1), 14–34, 1989.
- [5] R.E. Bellman and R.E. Kalaba, Quasilinearization and Nonlinear Boundary-Value
Problems, American Elsevier, New York, 1965.
- [6] R. Bhatia, L. Elsner, and G. Krause, Bounds for the variation of the roots of a polynomial
and the eigenvalues of a matrix, Linear Algebra Appl. 142, 195–209, 1990.
- [7] S.K. Bhatta and K.S. Sastri, A sixth order spline procedure for a class of nonlinear
boundary-value problems, Int. J. Comput. Math. 49 (3–4), 255–271, 1993.
- [8] S.K. Bhatta and K.S. Sastri, Symmetric spline procedures for boundary-value problems
with mixed boundary conditions, J. Comput. Appl. Math. 45 (3), 237–250, 1993.
- [9] A.K.B. Chand and G.P. Kapoor, Generalized cubic spline fractal interpolation functions,
SIAM J. Numer. Anal. 44 (2), 655–676, 2006.
- [10] A.K.B. Chand and P. Viswanathan, A constructive approach to cubic hermite fractal
interpolation function and its constrained aspects, BIT 53 (4), 841–865, 2013.
- [11] M.M. Chawla, A sixth order tridiagonal finite difference method for nonlinear twopoint
boundary-value problems, BIT 17 (2), 128–133, 1977.
- [12] M.M. Chawla, An eighth order tridiagonal finite difference method for nonlinear twopoint
boundary-value problems, BIT 17 (3), 281–285, 1977.
- [13] M.M. Chawla and P.N. Shivakumar, Numerov’s method for nonlinear two-point
boundary-value problems, Int. J. Comput. Math. 17 (2), 167–176, 1985.
- [14] M.M. Chawla and R. Subramanian, A new fourth-order cubic spline method for nonlinear
two-point boundary-value problems, Int. J. Comput. Math. 22 (3-4), 321–341,
1987.
- [15] M.M. Chawla and R. Subramanian, A new fourth-order cubic spline method for
second-order nonlinear two-point boundary-value problems, J. Comput. Appl. Math.
23 (1), 1–10, 1988.
- [16] M.M. Chawla, R. Subramanian, and P.N. Shivakumar, Numerov’s method for nonlinear
two-point boundary-value problems II. monotone approximations, Int. J. Comput.
Math. 26 (3-4), 219–227, 1989.
- [17] U. Erdogan and T. Ozis, A smart nonstandard finite difference scheme for second
order nonlinear boundary-value problems, J. Comput. Phys. 230 (17), 6464–6474,
2011.
- [18] P. Henrici, Discrete variable methods in ordinary differential equations, John Wiley
and Sons, New York, 1962.
- [19] M.K. Kadalbajoo and K.C. Patidar, Spline techniques for solving singularly-perturbed
nonlinear problems on nonuniform grids, J. Optim. Theory Appl. 114 (3), 573–591,
2002.
- [20] M. Lees, Discrete method for nonlinear two-point boundary-value problems, in: Numerical
Solution of Partial Differential Equations, ed. J.H. Bramble, Academic Press,
New York, 1966.
- [21] L.B. Liu, H.W. Liu, and Y. Chen, Polynomial spline approach for solving second-order
boundary-value problems with Neumann conditions, Appl. Math. Comput. 217 (16),
6872–6882, 2011.
- [22] J. Rashidinia, R. Mohammadi, and R. Jalilian, Spline solution of nonlinear singular
boundary-value problems, Int. J. Comput. Math. 85 (1), 39–52, 2008.
- [23] A.S.V. Ravikanth and V. Bhattacharya, Cubic spline for a class of nonlinear singular
boundary-value problems arising in physiology, Appl. Math. Comput. 174 (1), 768–
774, 2006.
- [24] S.B.G. Karakoç, N.M. Yagmurlu, and Y. Ucar, Numerical approximation to a solution
of the modified regularized long wave equation using quintic B-splines, Bound. Value
Probl. 2013 (1), 2013.
- [25] I.A. Tirmizi and E.H. Twizell, Higher-order finite-difference methods for nonlinear
second-order two-point boundary-value problems, Appl. Math. Lett. 15 (7), 897–902,
2002.
- [26] H. Zeybek and S.B.G. Karakoç, Application of the collocation method with B-splines
to the GEW equation, Electron. Trans. Numer. Anal. 46, 71–88, 2017.
Fractal quintic spline method for nonlinear boundary-value problems
Yıl 2020,
Cilt: 49 Sayı: 6, 1885 - 1903, 08.12.2020
N Balasubramani
M. Guru Prem Prasad
S Natesan
Öz
In this article, numerical solutions of nonlinear boundary-value problems are obtained using fractal quintic spline. Convergence analysis of the proposed method is also established. Proposed method has fourth-order convergence. Numerical examples are provided to show practical usefulness of the method and numerical results are compared with the existing numerical methods.
Kaynakça
- [1] M. Baccouch, A superconvergent local discontinuous Galerkin method for nonlinear
two-point boundary-value problems, Numer. Algor. 79 (3), 697–718, 2018.
- [2] M. Baccouch, An adaptive local discontinuous Galerkin method for nonlinear twopoint
boundary-value problems, Numer. Algor. 2019, doi:10.1007/s11075-019-00794-8.
- [3] M.F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2 (1), 303–329,
1986.
- [4] M.F. Barnsley and A.N. Harrington, The calculus of fractal interpolation functions,
J. Approx. Theory 57 (1), 14–34, 1989.
- [5] R.E. Bellman and R.E. Kalaba, Quasilinearization and Nonlinear Boundary-Value
Problems, American Elsevier, New York, 1965.
- [6] R. Bhatia, L. Elsner, and G. Krause, Bounds for the variation of the roots of a polynomial
and the eigenvalues of a matrix, Linear Algebra Appl. 142, 195–209, 1990.
- [7] S.K. Bhatta and K.S. Sastri, A sixth order spline procedure for a class of nonlinear
boundary-value problems, Int. J. Comput. Math. 49 (3–4), 255–271, 1993.
- [8] S.K. Bhatta and K.S. Sastri, Symmetric spline procedures for boundary-value problems
with mixed boundary conditions, J. Comput. Appl. Math. 45 (3), 237–250, 1993.
- [9] A.K.B. Chand and G.P. Kapoor, Generalized cubic spline fractal interpolation functions,
SIAM J. Numer. Anal. 44 (2), 655–676, 2006.
- [10] A.K.B. Chand and P. Viswanathan, A constructive approach to cubic hermite fractal
interpolation function and its constrained aspects, BIT 53 (4), 841–865, 2013.
- [11] M.M. Chawla, A sixth order tridiagonal finite difference method for nonlinear twopoint
boundary-value problems, BIT 17 (2), 128–133, 1977.
- [12] M.M. Chawla, An eighth order tridiagonal finite difference method for nonlinear twopoint
boundary-value problems, BIT 17 (3), 281–285, 1977.
- [13] M.M. Chawla and P.N. Shivakumar, Numerov’s method for nonlinear two-point
boundary-value problems, Int. J. Comput. Math. 17 (2), 167–176, 1985.
- [14] M.M. Chawla and R. Subramanian, A new fourth-order cubic spline method for nonlinear
two-point boundary-value problems, Int. J. Comput. Math. 22 (3-4), 321–341,
1987.
- [15] M.M. Chawla and R. Subramanian, A new fourth-order cubic spline method for
second-order nonlinear two-point boundary-value problems, J. Comput. Appl. Math.
23 (1), 1–10, 1988.
- [16] M.M. Chawla, R. Subramanian, and P.N. Shivakumar, Numerov’s method for nonlinear
two-point boundary-value problems II. monotone approximations, Int. J. Comput.
Math. 26 (3-4), 219–227, 1989.
- [17] U. Erdogan and T. Ozis, A smart nonstandard finite difference scheme for second
order nonlinear boundary-value problems, J. Comput. Phys. 230 (17), 6464–6474,
2011.
- [18] P. Henrici, Discrete variable methods in ordinary differential equations, John Wiley
and Sons, New York, 1962.
- [19] M.K. Kadalbajoo and K.C. Patidar, Spline techniques for solving singularly-perturbed
nonlinear problems on nonuniform grids, J. Optim. Theory Appl. 114 (3), 573–591,
2002.
- [20] M. Lees, Discrete method for nonlinear two-point boundary-value problems, in: Numerical
Solution of Partial Differential Equations, ed. J.H. Bramble, Academic Press,
New York, 1966.
- [21] L.B. Liu, H.W. Liu, and Y. Chen, Polynomial spline approach for solving second-order
boundary-value problems with Neumann conditions, Appl. Math. Comput. 217 (16),
6872–6882, 2011.
- [22] J. Rashidinia, R. Mohammadi, and R. Jalilian, Spline solution of nonlinear singular
boundary-value problems, Int. J. Comput. Math. 85 (1), 39–52, 2008.
- [23] A.S.V. Ravikanth and V. Bhattacharya, Cubic spline for a class of nonlinear singular
boundary-value problems arising in physiology, Appl. Math. Comput. 174 (1), 768–
774, 2006.
- [24] S.B.G. Karakoç, N.M. Yagmurlu, and Y. Ucar, Numerical approximation to a solution
of the modified regularized long wave equation using quintic B-splines, Bound. Value
Probl. 2013 (1), 2013.
- [25] I.A. Tirmizi and E.H. Twizell, Higher-order finite-difference methods for nonlinear
second-order two-point boundary-value problems, Appl. Math. Lett. 15 (7), 897–902,
2002.
- [26] H. Zeybek and S.B.G. Karakoç, Application of the collocation method with B-splines
to the GEW equation, Electron. Trans. Numer. Anal. 46, 71–88, 2017.