Research Article Received 13 June 2012 Published online 19 June 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.2817 MOS subject classification: 65N35 Bernstein series solution of linear second-order partial differential equations with mixed conditions Osman Rasit Isik a * † , Mehmet Sezer b and Zekeriya Guney c Communicated by J. E. Muñoz Rivera 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 abso- lute error can be estimated. Also, one can specify the optimal truncation limit n, which gives better result in any norm kk. 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. Keywords: numerical solution of PDEs; collocation method; Bernstein series solution; residual error estimation 1. Introduction To numerically solve PDEs, there are some well-known numerical methods such as finite difference methods, finite element methods, polynomial approximate methods, spectral methods, Galerkin, and collocation methods (see, e.g. [1, 2]). Also, various approximate methods were given in the literature such as differential transform method [3], Legendre wavelet method [4], Legendre expansion method [5], Chebyshev tau method [6, 7], Homotopy perturbation method [8], Ultraspherical tau method [9], Chebyshev expan- sion method [10, 11], and Chebyshev matrix method [12–14]. In this paper, we have developed a matrix method which is based on Bernstein polynomials and collocation points. The method was given by error estimation and error analysis. In this study, we consider the second-order linear partial differential equation on  D Œ0, R 1   Œ0, R 2  P @ 2 u @x 2 C R @ 2 u @x@y C S @ 2 u @y 2 C T @u @x C U @u @y C Vu D G (1) with the three mixed conditions:  For .˛ k , ˇ k / 2 @, a k i,j , k D 1, . ., t and  t are constants t X kD1 1 X iD0 1 X jD0 a k i,j u .i,j/ .˛ k , ˇ k / D  t ,  For .x,  k / 2 @ and b k i,j , k D 1, . ., p are constants p X kD1 1 X iD0 1 X jD0 b k i,j u .i,j/ .x,  k / D k.x/, and a Elemantary Mathematics Education Program, Faculty of Education, Mugla Sitki Kocman University, Mugla 48000, Turkey b Department of Mathematics, Faculty of Sciences and Arts, Manisa Celal Bayar University, 45000 Manisa, Turkey c Department of Mathematics, Faculty of Education, Mugla Sitki Kocman University, 48000 Mugla, Turkey *Correspondence to: Osman Rasit Isik, Elemantary Mathematics Education Program, Faculty of Education, Mugla Sitki Kocman University, Mugla 48000, Turkey. † E-mail: osmanrasitisik@hotmail.com Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014, 37 609–619 609