Haar wavelet Method for Solving Initial and Boundary Value Problems of Bratu-type
In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution.
[1] U.M. Ascher, R. Matheij, R.D. Russell, Numerical solution of boundary
value problems for ordinary differential equations, SIAM, Philadelphia,
PA, 1995.
[2] J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation
of two branches of a function with application to the onedimensional
Bratu equation, Applied Mathematics and Computation 142
(2003) 189-200.
[3] J.P. Boyd, An analytical and numerical study of the two-dimensional
Bratu equation, Journal of Scientific Computing 1 (2) (1986) 183-206.
[4] R. Buckmire, Investigations of nonstandard Mickens-type finitedifference
schemes for singular boundary value problems in cylindrical
or spherical coordinates, Numerical Methods for partial Differential
equations 19 (3) (2003) 380-398.
[5] R. Buckmire, Application of Mickens finite-difference scheme to the
cylindrical Bratu Gelfand problem, doi:10.1002/num.10093.
[6] C.F.Chen, C.H.Hsiao, Haar wavelet method for solving lumped and
distributed- parameter systems, IEE Proc.Pt.D 144(1) (1997) 87-94.
[7] D.A. Frank-Kamenetski, Diffusion and Heat Exchange in Chemical
Kinetics, Princeton University Press, Princeton, NJ, 1955.
[8] I.H.A.H. Hassan, V.S. Erturk, Applying differential transformation
method to the One-dimensional planar Bratu problem, International
Journal of Contemporary Mathematical Sciences 2 (2007) 1493-1504.
[9] Hikmet Caglar, Nazan Caglar, Mehmet zer, Antonios Valaristo, Amalia
N. Miliou, Antonios N. Anagnostopoulos, Dynamics of the solution of
Bratu-s Equation, Nonlinear Analysis, (Press).
[10] C.H.Hsiao, Haar wavelet approach to linear stiff systems, Mathematics
and Computers in simultion ,Vol 64, 2004, pp.561-567.
[11] J. Jacobson, K. Schmitt, The Liouville-Bratu-Gelfand problem for radial
operators, Journal of Differential Equations 184 (2002) 283-298.
[12] U.Lepik, Numerical solution of evolution equations by the Haar wavelet
method, Applied Mathematics and Computation 185 (2007) 695-704.
[13] U.Lepik, Numerical solution of differential equations using Haar
wavelets, Mathematics and Computers in Simulation 68 (2005) 127-143.
[14] S. Li, S.J. Liao, An analytic approach to solve multiple solutions of a
strongly nonlinear problem, Applied Mathematics and Computation 169
(2005) 854-865.
[15] A.S. Mounim, B.M. de Dormale, From the fitting techniques to accurate
schemes for the Liouville Bratu Gelfand problem, Numerical Methods
for Partial Differential Equations, doi: 10.1002/num.20116.
[16] M.I. Syam, A. Hamdan, An efficient method for solving Bratu equations,
Applied Mathematics and Computation 176 (2006) 704-713.
[17] A.M. Wazwaz, A new method for solving singular initial value problems
in the second order differential equations, Applied Mathematics and
Computation 128 (2002) 47-57.
[18] A.M. Wazwaz, Adomian decomposition method for a reliable treatment
of the Bratu-type equations, Applied Mathematics and Computation 166
(2005) 652-663.
[1] U.M. Ascher, R. Matheij, R.D. Russell, Numerical solution of boundary
value problems for ordinary differential equations, SIAM, Philadelphia,
PA, 1995.
[2] J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation
of two branches of a function with application to the onedimensional
Bratu equation, Applied Mathematics and Computation 142
(2003) 189-200.
[3] J.P. Boyd, An analytical and numerical study of the two-dimensional
Bratu equation, Journal of Scientific Computing 1 (2) (1986) 183-206.
[4] R. Buckmire, Investigations of nonstandard Mickens-type finitedifference
schemes for singular boundary value problems in cylindrical
or spherical coordinates, Numerical Methods for partial Differential
equations 19 (3) (2003) 380-398.
[5] R. Buckmire, Application of Mickens finite-difference scheme to the
cylindrical Bratu Gelfand problem, doi:10.1002/num.10093.
[6] C.F.Chen, C.H.Hsiao, Haar wavelet method for solving lumped and
distributed- parameter systems, IEE Proc.Pt.D 144(1) (1997) 87-94.
[7] D.A. Frank-Kamenetski, Diffusion and Heat Exchange in Chemical
Kinetics, Princeton University Press, Princeton, NJ, 1955.
[8] I.H.A.H. Hassan, V.S. Erturk, Applying differential transformation
method to the One-dimensional planar Bratu problem, International
Journal of Contemporary Mathematical Sciences 2 (2007) 1493-1504.
[9] Hikmet Caglar, Nazan Caglar, Mehmet zer, Antonios Valaristo, Amalia
N. Miliou, Antonios N. Anagnostopoulos, Dynamics of the solution of
Bratu-s Equation, Nonlinear Analysis, (Press).
[10] C.H.Hsiao, Haar wavelet approach to linear stiff systems, Mathematics
and Computers in simultion ,Vol 64, 2004, pp.561-567.
[11] J. Jacobson, K. Schmitt, The Liouville-Bratu-Gelfand problem for radial
operators, Journal of Differential Equations 184 (2002) 283-298.
[12] U.Lepik, Numerical solution of evolution equations by the Haar wavelet
method, Applied Mathematics and Computation 185 (2007) 695-704.
[13] U.Lepik, Numerical solution of differential equations using Haar
wavelets, Mathematics and Computers in Simulation 68 (2005) 127-143.
[14] S. Li, S.J. Liao, An analytic approach to solve multiple solutions of a
strongly nonlinear problem, Applied Mathematics and Computation 169
(2005) 854-865.
[15] A.S. Mounim, B.M. de Dormale, From the fitting techniques to accurate
schemes for the Liouville Bratu Gelfand problem, Numerical Methods
for Partial Differential Equations, doi: 10.1002/num.20116.
[16] M.I. Syam, A. Hamdan, An efficient method for solving Bratu equations,
Applied Mathematics and Computation 176 (2006) 704-713.
[17] A.M. Wazwaz, A new method for solving singular initial value problems
in the second order differential equations, Applied Mathematics and
Computation 128 (2002) 47-57.
[18] A.M. Wazwaz, Adomian decomposition method for a reliable treatment
of the Bratu-type equations, Applied Mathematics and Computation 166
(2005) 652-663.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53700", author = "S.G.Venkatesh and S.K.Ayyaswamy and G.Hariharan", title = "Haar wavelet Method for Solving Initial and Boundary Value Problems of Bratu-type", abstract = "In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution.
", keywords = "Haar wavelet method, Bratu's problem, boundary value problems, initial value problems, adomain decomposition method.", volume = "4", number = "7", pages = "852-4", }
