Asadollah Torabi Giklou - Department of Mathematics, Guilan Science and Research Branch, Islamic Azad University, Rasht, Iran.
Mojtaba Ranjbar - Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.
Mahmoud Shafiee - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.
Vahid Roomi - Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.

In this article, we use the collocation method based on the radial basis functions with symmetric variable shape parameter (SVSP) to obtain numerical solutions of neutral-type functional-differential equations with proportional delays. In this method, we control the absolute errors and the condition number of the system matrix through the program prepared with Maple ۱۸.۰ by increasing the number of collocation points that have a direct effect on the defined shape parameter. Also, we present the tables of the rate of the convergence (ROC) to investigate and show the convergence rate of this method compared to the RBF method with constant shape parameter. Several examples are given to illustrate the efficiency and accuracy of the introduced method in comparison with the same method with the constant shape parameter (CSP) as well as other analytical and numerical methods. Comparison of the obtained numerical results shows the considerable superiority of the collocation method based on RBFs with SVSP in accuracy and convergence over the collocation method based on the RBFs with CSP and other analytical and numerical methods for delay differential equations (DDEs).

functional-differential equation, proportional delay, Radial basis function, variable shape parameter