Shifted Legendre approximation with the residual correction to solve pantograph-delay type differential equations

In this paper, a Legendre collocation method based on the residual correction technique is proposed to solve the multi-pantograph and generalized pantograph equations with initial conditions. By using the residual function of the operator equation, an error differential equation is constructed and thus the approximate solution obtained by Legendre collocation method is corrected. Also, we give an upper bound of the absolute errors for the corrected shifted Legendre solution. Finally, we illustrate the method by solving the problems with initial conditions. The obtained results are compared by the known results; the error estimation and the upper bounds of the absolute errors are performed for the approximate solutions in examples.