×

Solutions of differential equations in a Bernstein polynomial basis. (English) Zbl 1118.65087

Summary: An algorithm for approximating solutions to differential equations in a modified new Bernstein polynomial basis is introduced. The algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the Galerkin method to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of Bernstein polynomials and the procedure is much simpler compared to the piecewise B-spline method for solving differential equations.
A recursive definition of the Bernstein polynomials and their derivatives are also presented. The current procedure is implemented to solve three linear equations and one nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M.I. Bhatti, K.D. Coleman, W.F. Perger, Static polarizabilities of hydrogen in B splines basis, Phys. Rev. A 68 (2003) 044503-1.; M.I. Bhatti, K.D. Coleman, W.F. Perger, Static polarizabilities of hydrogen in B splines basis, Phys. Rev. A 68 (2003) 044503-1.
[2] Bottcher, C.; Strayer, M. R., Relativistic theory of fermions and classical field on a collocation lattice, Ann. Phys. (NY), 175, 64 (1987)
[3] Fischer, C.; Guo, W., Spline algorithms for the Hartree-Fock equation for the helium ground state, J. Comput. Phys., 90, 486 (1990) · Zbl 0714.65103
[4] Fischer, C.; Idrees, M., Spline algorithms for continuum functions, Comput. Phys., 3, 53 (1998)
[5] Fletcher, C. W.A., Computational Galerkin Methods (1984), Springer: Springer New York · Zbl 0533.65069
[6] Gelbaum, B. R., Modern Real and Complex Analysis (1995), Wiley: Wiley New York
[7] Johnson, W. R.; Blundell, S. A.; Saperstein, J., Finite basis sets for the Dirac equation constructed from B-splines, Phys. Rev. A, 37, 307 (1988)
[8] Johnson, W. R.; Idrees, M.; Saperstein, J., Second-order energies and third-order matrix elements of alkali atoms, Phys. Rev., 35, 3218 (1987)
[9] Mathematica, Wolfram Research Inc. 2004.; Mathematica, Wolfram Research Inc. 2004.
[10] Perger, W. F.; Xia, M.; Flurchick, K.; Bhatti, M. I., Comput. Sci. Eng., 3, 38 (2001)
[11] Qiu, Y.; Fischer, C. F., Integration by cell algorithm for Slater integrals in a spline basis, J. Comput. Phys., 156, 257 (1999) · Zbl 0967.81062
[12] Saperstein, J.; Johnson, W. R., The use of basis splines in theoretical atomic physics, J. Phys. B, 29, 5213 (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.