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To understand the complex problem of surface growth phenomena, we developed a model in which the regular diffusion equation is coupled to the Kardar–Parisi–Zhang (KPZ) equation. The fundamental or Gaussian solution of the regular diffusion equation is considered as an external noise or source term in the KPZ equation. The obtained system of partial differential equations is analytically solved by a self-similar Ansatz and expressed the solution as a combination of elementary and special functions. Using this solution, the effects of the physical parameters were explicitly investigated. Our recent explicit numerical method, the leapfrog-hopscotch algorithm, is also tested for this problem. It is shown that this method can be safely used with orders of magnitude larger time step sizes than the usual explicit (Euler) scheme as well if we are far from the cusp-like solutions. We pointed out that the cusps themselves cannot be properly simulated by any method that we know.

This study proposes an intelligent tracking method based on the expectation maximisation approach. To obtain the consistent tracking performance, the observer should maintain contact with the target and continuously update its tracking data. Unfortunately, a significant problem arises from the uncertainty of the target motion. The reasons behind this uncertainty include losses, time‐varying noises, external inputs, and others. Above all, a sharp manoeuvre by the arbitrary acceleration input mainly degrades the tracking performance. A variety of techniques were needed and studied to enhance performance levels. This study focuses on acceleration as the cause of the uncertainty and a consistent approximation of the acceleration, leading to highly efficient tracking performance. The EM makes it possible to separate the noise term from the acceleration inputs and the noise components. Finally, a manoeuvring submarine with three‐dimensional coordinates is provided as an example to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]