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Purpose - Initial value problems for the one-dimensional third-order dispersion equations are investigated using the reliable Adomian decomposition method (ADM). Design/methodology/approach - The solutions are obtained in the form of rapidly convergent power series with elegantly computable terms. Findings - It was found that the technique is reliable, powerful and promising. It is easier to implement than the separation of variables method. Modifications of the ADM and the noise terms phenomenon are successfully applied for speeding up the convergence of non-homogeneous equations. Research limitations/implications - The method is restricted to initial value problems in which the space variable fills the whole real axis. Modifications are required to deal with initial boundary value problems. Further, the input initial condition is required to be an infinitely differentiable function and obviously, the convergence radius of the decomposition series depends on the input data. Practical implications - The method was mainly illustrated for linear partial differential equations occuring in water resources research, but the natural extension of the ADM to solving nonlinear problems is extremely useful in nonlinear studies and soliton theory. Originality/value - The study undertaken in this paper provides a reliable approach for solving both linear and nonlinear dispersion equations and new explicit or recursively-based exact solutions are found. [ABSTRACT FROM AUTHOR]