In this work we present a new numerical method for the solution of the distributed order time fractional\ud diffusion equation. The method is based on the approximation of the solution by a double\ud Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of\ud equations at suitable collocation points. An error analysis is provided and a comparison with other\ud methods used in the solution of this type of equation is also performed

For a class of tempered fractional terminal value problems of the Caputo type, we study the existence and uniqueness of the solution, analyze the continuous dependence on the given data, and using a shooting method we present and discuss three numerical schemes for the numerical approximation of such problems. Some numerical examples are considered in order to illustrate the theoretical results and evidence the efficiency of the numerical methods.