Recently, there has been a great interest to apply innovative methods to solve the fractional diffusion equations, due to its great importance in modeling turbulent flow, chaotic dynamics of the classical fusty system, groundwater contaminant transfer and other applications in physics, biology, chemistry and many engineering applications. This paper aims to present an analytical and approximation method to get the solution of the space-time fractional diffusion equation by using Caputo fractional derivative. This suggested method is based on a combination of the double and triple Laplace transforms with the Adomain decomposition method. The cardinal idea of the LT is that it converts a differential equation into an algebraic equation, which can be solved more easily. The double and triple LT are considered as an extensive form of the original version. The presented methodology is tested on illustrative examples and the results show that it is a simple, efficient, and reliable method.