Collocation method for a functional equation arising in behavioral sciences

TL;DR

This paper introduces a collocation method for solving a generalized nonlocal functional equation from behavioral sciences, proving existence, uniqueness, regularity, and convergence of the numerical scheme with practical efficiency.

Contribution

It develops a new collocation scheme for a complex functional equation, establishing theoretical properties and demonstrating superior computational performance over traditional methods.

Findings

Unique solution exists under growth conditions.

The scheme converges with second-order accuracy for smooth coefficients.

Numerical experiments show the method is fast and more efficient than Picard iteration.

Abstract

We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function arguments. We show that, provided some growth conditions of the coefficients, there exists a unique solution in the natural Lipschitz space. Furthermore, we prove that the regularity of the solution is inherited from the smoothness properties of the coefficients. As a natural numerical method to solve the general case, we consider the collocation scheme of piecewise linear functions. We prove that the method converges with the error bounded by the error of projecting the Lipschitz function onto the piecewise linear polynomial space. Moreover, provided sufficient regularity of the coefficients, the scheme is of the second order measured in the supremum norm. A series of numerical experiments verify the proved claims and show that the implementation is computationally cheap and exceeds the frequently used Picard iteration by orders of magnitude in the calculation time.