Fractional calculus extends classical calculus by allowing derivatives and integrals of non-integer order and has important applications in physics, engineering, fluid mechanics, and other fields. Fractional partial differential equations (FPDEs) are widely used to model complex phenomena more accurately than integer-order models. Various analytical and numerical methods have been developed to solve these problems, among which the Laplace Adomian Decomposition Method (LADM) is notable for its simplicity and accuracy. In this study, diffusion equations and FPDEs are solved using LADM with the Caputo operator for fractional derivatives. Results are obtained for both integer and fractional orders and show strong agreement with exact solutions. It is observed that fractional-order solutions converge to integer-order solutions as the order approaches unity. The results also indicate that fractional-order models provide better accuracy compared to classical models. Overall, LADM proves to be an effective and reliable method for solving both linear and nonlinear FPDEs.