A boundary element method formulation based on the Caputo derivative for the solution of the diffusion-wave equation

A boundary element method formulation is developed and validated through the solution of problems governed by the diffusion-wave equation, for which the order of the time derivative, say α, ranges in the interval (1, 2). This fractional time derivative is defined as an integro-differential operator in the Caputo sense. For α = 2, one recovers the classical wave equation. From the weighted residual method point of view, with the Laplace equation fundamental solution playing the role of the weighting function, a D-BEM type formulation is generated, in which the integrand of the domain integral contains the fractional time derivative. The Houbolt scheme is adopted to represent the second order time derivative of the variable of interest, say u, which appears in the Caputo operator. In all the examples, the proposed formulation provided accurate results for α as small as 1.05. Indeed, the analyses were carried out for α = 1.05, 1.2, 1.5, 1.8, 2.0, with the aim of verifying the influence of the order of the time derivative in the results.