Solving a class of distributed-order time fractional wave-diffusion differential equations using the generalized fractional-order Bernoulli wavelets

In this research, we propose a new numerical method for solving a class of distributed-order fractional partial differential equations, specifically focusing on distributed-order time fractional wave-diffusion equations. The method begins by introducing a novel generalization of Bernoulli wavelets and deriving an exact result for the Riemann–Liouville integral of these new basis functions. Utilizing the Gauss–Legendre quadrature formula and a strategically chosen set of collocation points, along with approximations for the unknown function and its derivatives, we transform the problem into a system of algebraic equations. An error analysis is then conducted for the approximation of a bivariate function using fractional-order Bernoulli wavelets. Finally, three examples are solved to demonstrate the method’s applicability and accuracy, with the numerical results confirming its effectiveness. These findings demonstrate that the parameters of the basis functions can be adjusted to suit the given problem, thereby enhancing the accuracy of the method.