Numerical simulation of variable-order fractal-fractional delay differential equations with nonsingular derivative

This work develops a new Legendre delay operational matrix based on Legendre polynomial features that are integrated with regard to the Legendre fractional derivative operational matrix in order to solve the issues. The motivation behind solving the Atangana-Baleanu The variable-order fractal-fractional delay differential equations rely on the properties of the kernel in the Atangana-Baleanu fractal-fractional derivative operator. Atangana-Baleanu fractal-fractional derivative by the variable-order exponential kernel gives more precise results to the derivative. The Legendre operational matrix of the fractional derivative error bound is also shown here. The variable-order fractal-fractional delay differential equations with Atangana-Baleanu derivatives are reduced to a set of algebraic equations using a collocation strategy based on these operational matrices. The numerical findings show that the proposed approach is a useful mathematical tool for calculating numerical solutions to variable-order fractal-fractional delay differential equations with an Atangana-Baleanu derivative compared to earlier techniques. At last, the numerical examples are employed to show the performance and efficiency of the method.