Fourth-Order Trapezoid Algorithm with Four Iterative Schemes for Nonlinear Integral Equations

It is well known that the classical trapezoidal rule (TR2) is only second order of convergence. However, Leonhard Euler and Colin Maclaurin had independently discovered that the error terms of TR2 can be represented as an infinite sum of terms in even powers of the mesh size. In this article, we exploit this error representation to construct a nonconventional fourth-order trapezoid rule (TR4) and combine it with different iterative schemes to solve nonlinear Fredholm integral equations. First, the problem is projected into a discrete space and approximated with TR2. Then, the leading error term of the TR2 is approximated with finite difference formula, giving rise to TR4. Next, we derive three discrete fixed point methods (Picard, Ishikawa, and Argawal) and a Newton–Raphson method to approximate the solution of the resulting nonlinear system. This led to four different algorithms for solving nonlinear Fredholm equation. The algorithms were tested and compared for accuracy and efficiency by using six example problems with known exact solutions. The results show that (i) all the methods converge with fourth order accuracy, but only the Newton scheme maintained the fourth order accuracy at all levels of grid refinement and for all the six examples considered in this work, (ii) the Ishikawa and Argawal iterative processes have approximately equal performance, (iii) whenever the Picard scheme is applicable, it is fast, but may degrade the quality of the solution at very fine mesh more than the others, and (iv) the Newton approach, though may be slower in some problems, is guaranteed to produce very reliable results and maintain fourth order of accuracy.