Abstract
A spectral method based on operational matrices of Bernstein polynomials using collocation method is elaborated and employed for solving nonlinear ordinary and partial differential equations with multi-point boundary conditions. First, properties of Bernstein polynomial, operational matrices of integration, differentiation and product are introduced and then utilized to reduce the given differential equation to the solution of a system of algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent integro-differential equation which implies all boundary condition. Approximate solution is achieved by expanding the desired function in terms of a Bernstein basis and employing operational matrices. Unknown coefficients are determined by collocation. The method is compared with modified Adomian decomposition method, Birkhoff-type interpolation method, reproducing kernel Hilbert space method, fixed point method, finite-difference Keller-box method, multilevel augmentation method and shooting method. Illustrative examples are included to demonstrate the high precision, validity and good performance of the new scheme even for solving nonlinear singular differential equations.
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The author would like to sincerely thank professor Mehdi Dehghan (Amirkabir University of Technology, Tehran, Iran) for his helpful discussions and significant suggestions which helped the author to improve the contents of this article. Also, author is very grateful to one of the referees for carefully reading this paper and for the comments which have enhanced the quality of paper.
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Communicated by Jan Hesthaven.
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Behroozifar, M. Spectral method for solving high order nonlinear boundary value problems via operational matrices. Bit Numer Math 55, 901–925 (2015). https://doi.org/10.1007/s10543-015-0544-2
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DOI: https://doi.org/10.1007/s10543-015-0544-2
Keywords
- Nonlinear differential equations
- Multi-point boundary value problem
- Bernstein basis
- Operational matrix
- Collocation spectral method