A novel approach using a new generalization of Bernoulli wavelets for solving fractional integro-differential equations with singular kernel

In recent years, numerous fractional-order basis functions have been developed and applied for solving different classes of fractional problems. In this work, a new generalization of fractional-order Bernoulli wavelets is introduced. These new basis functions are used to provide a numerical solution for Hammerstein-type fractional integro-differential equations with a weakly singular kernel. To achieve this, the Riemann-Liouville integral operator is applied to the basis functions, and the result is computed exactly using the analytic form of Bernoulli polynomials. Through this process, key properties of the Riemann-Liouville integral and Caputo derivative are utilized to define two remainders associated with the main problem. After that, using an appropriate set of collocation points, the problem is converted to a system of algebraic equations. Due to the efficiency and high accuracy of this new technique, we extend the method for solving fractional Fredholm-Volterra integro-differential equations. Then, an upper bound of the error is discussed for the approximation of a function based on the fractional-order Bernoulli wavelets. Finally, the method is utilized for solving some illustrative examples to check its performance.

کلیدواژه ها:

Weakly singular fractional integro-differential equations ، Generalized fractional-order Bernoulli wavelets ، Caputo derivative ، Riemann-Liouville integral