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Using Adomian decomposition method (ADM), we have theoretically investigated the nonlinear dynamics of the magnetic field diffusion and filamentation instability in a current-carrying plasma. Based on the Adomian solutions, the spatiotemporal behavior of the magnetic field is obtained, and it is shown that its profile changes from a sinusoidal shape to a saw-tooth one. Furthermore, the electron density distribution becomes highly peaked. In contrast to the numerical methods, the ADM predicts the microcurrents. In addition, the observation time of other events, like the filamentation instability as well as steepening of electron density, is tremendously increased in comparison with the time obtained by the traditional methods.