Galerkin solution of a singular integral equation with constant coefficients

Galerkin methods are used to approximate the singular integral equation a φ ( x ) + b π ∫ − 1 1 φ ( t ) t − x d t + λ ∫ − 1 1 k ( t , x ) φ ( t ) d t = f ( x ) , − 1 < x < 1 , with solution φ having weak singularity at the endpoint −1, where a , b ≠ 0 are constants. In this case φ is decomposed as φ ( x ) = ( 1 − x ) α ( 1 + x ) β u ( x ) , where β = − α , 0 < α < 1 . Jacobi polynomials are used in the discussions. Under the conditions f ∈ H μ [ − 1 , 1 ] and k ( t , x ) ∈ H μ , μ [ − 1 , 1 ] × [ − 1 , 1 ] , 0 < μ < 1 , the error estimate under a weighted L 2 norm is O ( n − μ ) . Under the strengthened conditions f ″ ∈ H μ [ − 1 , 1 ] and ∂ 2 k ∂ x 2 ( t , x ) ∈ H μ , μ [ − 1 , 1 ] × [ − 1 , 1 ] , 2 α − ϱ < μ < 1 , the error estimate under maximum norm is proved to be O ( n 2 α − ϱ − μ + ϵ ) , where ϱ = min { α , 1 2 } , ϵ > 0 is a small enough constant.