An $\epsilon$-approximate solution of BVPs based on improved multiscale orthonormal basis

In the present paper, we construct a set of multiscale orthonormal basis based on Legendre polynomials. Using this orthonormal basis, a new algorithm is designed for solving the second-order boundary value problems. This algorithm is to find numerical solution by seeking $\epsilon$-approximate solution. Moreover, we prove that the order of convergence depends on the boundedness of $u^{(m)}(x)$. In addition, third numerical examples are provided to validate the efciency and accuracy of the proposed method. Numerical results reveal that the present method yields extremely accurate approximation to the exact solution. Meanwhile, compared with the other algorithms, the results obtained demonstrate that our algorithm is remarkably effective and convenient.