New algorithms for Taylor coefficients of indefinite integrals and their applications

New recurrence algorithms for calculating the Taylor coefficients of indefinite integral are introduced and proved in this paper. These algorithms are simple in form, only involve the analytic operations: addition and multiplication, and can be directly implemented in any symbolic language. Also, no specific restriction on the kernel is required. Several types of Volterra equations such as integral, integro-differential and system of integro-differential equations with nondegenerate kernel are then solved to demonstrate the accuracy and efficiency of the proposed scheme.