Numerical analysis of pantograph differential equation of the stretched type associated with fractal-fractional derivatives via fractional order Legendre wavelets

In the present paper, a method of using fractional order Legendre wavelets is proposed for solving the pantograph differential equation of the stretched type involved with Caputo fractal-fractional and Atangana-Baleanu fractal-fractional derivatives. The suggested approach is based on the fractal-fractional integral operational matrix of fractional order Legendre wavelets and collocation method. The purpose of this article is to analyze the behavior of the pantograph differential equation of the stretched type under the fractal-fractional operators. Two illustrative numerical examples are taken and the results achieved for these examples with different fractional order and fractal order predict the applicability and efficiency of the suggested method using graphs and tables. (C) 2020 Elsevier Ltd. All rights reserved.