Azubuike Weli, Chinedu Nwaigwe
Computational Analysis of Two Numerical Solvers for Functional Fredholm Equations
We construct two numerical methods for solving nonlinear functional Fredholm integral equations and compare their accuracy and computational costs. First, the nonlinear integral is approximated with composite trapezoid rule on the mesh. Then Picard and Newton iterations are designed to linearize the nonlinear system and approximate the solution. The in-built python function, scipy.optimize.fsolve, is used to handle the nonlinear system arising from the Newton approach. Several numerical experiments are performed to test the performance of the two methods. The results show that (i) both schemes are convergent and have the correct order of accuracy which is two, (ii) the methods have comparable accuracy, (iii) the Picard scheme is far more efficient than the Newton method, and (iv) the efficiency of the Picard scheme over the Newton method increases as the size of the problem increases. In addition to the above results, the Picard scheme is easier to program. We conclude that if the second order trapezoid rule is used to approximate the integral in a Fredholm equation, then the Picard scheme should be preferred over the Newton scheme. Further study is recommended to ascertain if this conclusion still holds when trapezoid rule of different order or a different quadrature rule is used.
Fredholm Integral equations, Banach Fixed-Point Theorem, Discrete Picard Iteration, Newton iteration, Trapezoid rule, Collocation Method, Experimental Order of Convergence
Cite this paper
Azubuike Weli, Chinedu Nwaigwe. (2023) Computational Analysis of Two Numerical Solvers for Functional Fredholm Equations. International Journal of Mathematical and Computational Methods, 8, 1-8