Segmented Tau Approximation for a Non-Autonomous Functional Differential Equation of Mixed Type
This work provides a numerical method for solving a specific class of non-autonomous mixed-type functional differential equations, which is an incremental contribution to numerical analysis.
The paper applies the segmented Tau method to numerically solve a non-autonomous functional differential equation of mixed type, achieving good numerical results on constructed test problems with known analytical solutions.
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.