An improved algorithm based on deviation of the error estimation for first-order integro-differential equations

Abstract

In this paper, we study efficient asymptotically correct a posteriori error estimates for the numerical approximation of first-order Fredholm–Volterra integro-differential equations. In the first step, we find the deviation of the error for Fredholm–Volterra integro-differential equations by using defect correction principle. Then we show that for m degree piecewise polynomial collocation method, our method provides order $$\mathcal {O}(h^{m+1})$$ for the deviation of the error. Also we improve the piecewise polynomial collocation method by using the deviation of the error estimation. Numerical results in the last section are included to confirm the theoretical results.