Solving fourth-order boundary value problem with orthogonal wavelet basis

In the presented paper, an effective method based upon Legendre wavelet function is proposed for ‎the solution of the singular fourth-order boundary value problem with mixed boundary conditions. ‎The properties of Legendre wavelet function are first presented and then we introduce the ‎construction method of wavelet basis in W_2^5 [0,1] by orthogonal wavelet basis in W_2^1 [0,1]. The ‎ε-approximate solution was defined and then it was proved to be the optimal solution. In addition, the ‎stability and convergence for the method in W_2^5 [0,1] space are discussed. Numerical illustrations ‎are investigated to show the reliability and efficiency of the proposed method.‎