Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet

In the current study, new functions called generalized fractional-order Bernoulli wavelet functions (GFBWFs) based on the Bernoulli wavelets are defined to obtain the numerical solution of fractional-order pantograph differential equations in a large interval. For the concept of fractional derivative we will use Caputo sense by using Riemann–Liouville fractional integral operator. First, the generalized fractional-order Bernoulli wavelets are constructed. Then, these functions and their properties are employed to derive the GFBWFs operational matrices of fractional integration and pantograph. The operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations. Finally, some examples are included for demonstrating the validity and applicability of our method.