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PurposeThe purpose of this paper is to explain the choice of Alpert multi‐wavelet as basis functions to discrete Fredholm integral equation of the second kind by using Petrov‐Galerkin method.Design/methodology/approachIn this process, two kinds of matrices are obtained from inner product between basis of test space and trial space; some of them are diagonal with positive elements and some others are invertible. These matrices depend on type of selection of test and trial space basis.FindingsIn this process, solution of Fredholm integral equation of the second kind is found by solving the generated system of linear equations.Originality/valueIn previous work, convergence of Petrov‐Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree. In point of computation, because of appearance of diagonal and invertible matrices, a small dimension system with a more accurate solution is obtained. The numerical examples illustrate these facts.

PurposeThe purpose of this paper is to use Alpert wavelet basis and modify the integrand function approximation coefficients to solve Fredholm‐Hammerstein integral equations.Design/methodology/approachL2[0, 1] was considered as solution space and the solution was projected to the subspaces of L2[0, 1] with finite dimension so that basis elements of these subspaces were orthonormal.FindingsIn this process, solution of Fredholm‐Hammerstein integral equation is found by solving the generated system of nonlinear equations.Originality/valueComparing the method with others shows that this system has less computation. In fact, decreasing of computations result from the modification.