Segmented Tau Approximation for a Non-Autonomous Functional Differential Equation of Mixed Type

TL;DR

This paper introduces a segmented Tau method to numerically solve a non-autonomous mixed-type functional differential equation, demonstrating promising results through numerical experiments and a novel problem generation approach.

Contribution

The paper develops a segmented Tau method for solving non-autonomous mixed-type functional differential equations and provides a new way to generate problems with known solutions.

Findings

The method yields accurate numerical solutions.

The approach is simple and effective for mixed-type problems.

Numerical experiments confirm the method's robustness.

Abstract

The segmented formulation of the Tau method is used to numerically solve the non-autonomous forward-backward functional differential equation x'(t) = a(t)x(t) + b(t)x(t-1) + c(t)x(t+1), where x is the unknown function, a, b, and c are known functions. The step by step Tau method is applied to approximate the solution of this equation by a piecewise polynomial function. A boundary value problem is posed, numerically solved, and analyzed. Also, a novel way to generate a set of non-autonomous problems with known analytical solution is provided. From it, several non-autonomous problems were constructed and resolved with the proposed method. We conclude that the good numerical results obtained in our numerical experimentation and the relative simplicity of the Tau method demonstrate that it is a promising strategy for numerically solving mixed-type problems, as presented here.