A combined efficient method for approximate two-dimensional integral equations

In this paper, we combine the two-dimensional (2D) Haar wavelet functions (HWFs) with the block-pulse functions (BPFs) to solve the 2D linearVolterra-Fredholm integral equations (2D-L(VF)IE), so we present a new hybrid computational effcient method based on the 2D-HWFs and 2D-BPFs to approximate the solution ofthe 2D linear Volterra-Fredholm integral equations. In fact, the HWFs and theirrelations to the BPFs are employed to derive a general procedure to formoperational matrix of Haar wavelets. Theoretical erroranalysis of the proposed method is done. Finally some examples arepresented to show the effectiveness of the proposed method.