A computational approach for distributed-order time-fractional wave-diffusion equations using generalized fractional Bernoulli wavelet functions

This work presents a novel numerical technique for addressing a category of distributed-order fractional partial differential equations, with particular emphasis on time-fractional wave-diffusion equations. The approach involves the development of an extended form of Bernoulli wavelet functions and the derivation of an exact expression for their Riemann-Liouville integral. By employing the Gauss-Legendre quadrature rule, a carefully selected set of collocation points, and approximations for the unknown function and its derivatives, the original problem is converted into a system of algebraic equations. An error analysis is performed to evaluate the precision of bivariate function approximation using fractional-order Bernoulli wavelets. To validate the method's efficacy, some illustrative examples are solved, with the numerical outcomes demonstrating its accuracy and reliability.