In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials
are constructed and applied to evaluate the numerical solution of the general form of Caputo
fractional order diffusion wave equations. The operational matrices of ordinary and fractional
derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then,
these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem
to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation
spectral methods are employed. To demonstrate the validity and applicability of the presented
method, we offer five significant examples, including generalized Cattaneo diffusion wave and
Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented
numerical method. The advantage of having compact support and orthogonality of these family of
wavelets trigger having sparse operational matrices, which reduces the computational time and CPU
requirements.