Efficient mapped Jacobi spectral method for integral equations with two-sided singularities

In this paper, we develop a mapped Jacobi spectral Galerkin method for solving the multi-term Fredholm integral equations (MFIEs) with two-sided weakly singularities. We introduce a new family of mapped Jacobi functions (MJFs) and establish the corresponding spectral approximation results on these MJFs in weighted Sobolev spaces involving the mapped Jacobi weight function. These MJFs serve as the basis functions in our algorithm design and are tailored to the two-sided end-points singularities of the solution by using suitable mapping. Moreover, we derive the error estimates of the proposed method for MFIEs. Finally, the numerical examples are provided to demonstrate the accuracy and efficiency of the proposed method.