Bernstein series solution of linear second‐order partial differential equations with mixed conditions

The purpose of this study is to present a new collocation method for numerical solution of linear PDEs under the most general conditions. The method is given with a priori error estimate. By using the residual correction procedure, the absolute error can be estimated. Also, one can specify the optimal truncation limit n, which gives better result in any norm ∥ ∥ . Finally, the effectiveness of the method is illustrated in some numerical experiments. Numerical results are consistent with the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  Ibrahim Çelik,et al.  Collocation method and residual correction using Chebyshev series , 2006, Appl. Math. Comput..

[2]  David S. Watkins,et al.  Fundamentals of Matrix Computations: Watkins/Fundamentals of Matrix Computations , 2005 .

[3]  Ibrahim Çelik,et al.  Approximate calculation of eigenvalues with the method of weighted residuals-collocation method , 2005, Appl. Math. Comput..

[4]  Mehmet Sezer,et al.  Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel , 2011, Appl. Math. Comput..

[5]  Mehmet Sezer,et al.  A rational approximation based on Bernstein polynomials for high order initial and boundary values problems , 2011, Appl. Math. Comput..

[6]  Waleed M. Abd-Elhameed,et al.  Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method , 2005 .

[7]  Sen Bai,et al.  A numerical solution of second-order linear partial differential equations by differential transform , 2006, Appl. Math. Comput..

[8]  En-Bing Lin,et al.  Legendre wavelet method for numerical solutions of partial differential equations , 2010 .

[9]  Elsayed M. E. Elbarbary.  Legendre expansion method for the solution of the second-and fourth-order elliptic equations , 2002, Math. Comput. Simul..

[10]  Ayşegül Akyüz-Daşcıoğlu.  Chebyshev polynomial approximation for high‐order partial differential equations with complicated conditions , 2009 .

[11]  Paul Bracken,et al.  Solutions of differential equations in a Bernstein polynomial basis , 2007 .

[12]  Dale B. Haidvogel,et al.  The Accurate Solution of Poisson's Equation by Expansion in Chebyshev Polynomials , 1979 .

[13]  F. Oliveira.  Collocation and residual correction , 1980 .

[14]  Hojatollah Adibi,et al.  The Chebyshev Tau technique for the solution of Laplace's equation , 2007, Appl. Math. Comput..

[15]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[16]  R. Sacker.  Semigroups of maps and periodic difference equations , 2010 .

[17]  Elsayed M. E. Elbarbary,et al.  Chebyshev expansion method for solving second and fourth-order elliptic equations , 2003, Appl. Math. Comput..

[18]  Cenk Kesan.  Chebyshev polynomial solutions of second-order linear partial differential equations , 2003, Appl. Math. Comput..

[19]  Xionghua Wu,et al.  Chebyshev tau matrix method for Poisson-type equations in irregular domain , 2009 .