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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.