Operator modeling homogeneous heat transport \( {\cal L_1}(u) = \Delta u = \frac{\partial^2 u(x,y;t)}{\partial x^2}+\frac{\partial^2 u(x,y;t)}{\partial y^2} \). In this case, the initial condition gives us information about the temperature distribution in the modeled area at the initial moment of the simulation. Boundary conditions in turn determine the temperature at the boundary of the domain, or the rate of heat change on the boundary of the domain (how quickly heat escapes from our area, or how quickly it flows into our area), or the heat flux depending on the temperature difference, or the boundary condition describing the phenomenon radiation. The right-hand side function \( {\cal L }\left(u\left(x,y;t\right)\right) = {\cal F }(x,y;t) \) depending on the assumption of positive or negative values, can model a heater or cooler located in the area under consideration, which locally supplies heat (increases the temperature) or lowers the temperature through the cooling mechanism (heat discharge).

Operator modeling diffusion through a material with specific properties \( {\cal L_2}(u) = \nabla \left (K \nabla u\right) = \frac{\partial }{\partial x} \left( K(x,y;t) \frac{\partial u(x,y;t) }{\partial x}\right) + \frac{\partial }{\partial y} \left( K(x,y;t) \frac{\partial u(x,y;t) }{\partial y}\right) \). Here, the initial state gives us information about the initial concentration of the substance, the diffusion of which in a given area is simulated. Diffusion is a special case of heat transport equations, and analogous kinds of boundary conditions are also possible. Diffusion coefficient \( {\cal R}^2 \times [0,T] \ni (x,y)\times t \rightarrow K(x,y;t) \in {\cal R}^{2\times 2} \) may even be a matrix-valued function. The boundary condition describes the concentration on the boundary or the speed of "escape" or "inflow" of the substance under consideration, or the diffusion flux depending on the concentration difference, or the phenomenon of radiation. The right-hand side function \( {\cal L }\left(u\left(x,y;t\right)\right) = {\cal F }(x,y;t) \) here means the source of the diffusible substance, or its loss (suction).

Flow in a heterogeneous medium \( {\cal L_3 }(u) = \nabla \left( K exp(\mu u) \right) \nabla u = \frac{\partial }{\partial x} \left( K(x,y;t) exp(\mu u(x,y;t) ) \frac{\partial u(x,y;t) }{\partial x}\right) + \frac{\partial }{\partial y} \left( K(x,y;t) exp(\mu u(x,y;t) )\frac{\partial u(x,y;t) }{\partial y}\right) \). In this case, we model the pressure scalar field in the geological area into which we pump the liquid under pressure. The right-hand side function, assuming positive values, models the position of the pump pumping the liquid under pressure, and assuming negative values, it models a suction pump, used, for example, in the oil extraction process. The initial condition here means the initial distribution of the pressure field in the modeled area. Boundary condition means the value or changes of pressure on the edge of the modeled area.

Pollution propagation \( {\cal L_4 }(u) = \beta \cdot \nabla u + \nabla \cdot (K \nabla u) = \\= \frac{\partial }{\partial x} \left( K(x,y;t) \frac{\partial u(x,y;t) }{\partial x} \right) + \frac{\partial }{\partial y} \left( K(x,y;t) \frac{\partial u(x,y;t) }{\partial y }\right) + \beta_x(x,y;t) \frac{\partial u(x,y;t)}{\partial x } + \beta_y(x,y;t) \frac{u(x,y;t) }{\partial y } \). In this example, we model the concentration of pollutants in a given area. The positive value of the right-hand side function \( {\cal L }\left(u\left(x,y;t\right)\right) = {\cal F }(x,y,z) \) means the source of pollution (e.g., chimney), the negative value of the right-hand side function means the pollution suction pump (e.g., such pumps placed in the main squares in cities that suck the pollution from the atmosphere). The initial state means the initial concentration of pollutants in the modeled area. Boundary conditions mean the concentration of pollutants on the boundary, or the speed of the blowing in of pollutants from outside the area, or the interaction of a pollutant field with the ground.

Non-stationary nonlinear flow equations for incompressible fluids, called the Navier-Stokes equations, are a vector equation in which the velocity vector field and the pressure scalar field are sought \( {\cal L_5}({\bf u}, p) = \left( \frac{\partial {\bf u}}{\partial t } - \mu \Delta {\bf u} + ({\bf u} \cdot \nabla ) {\bf u} + \nabla p = f; div p = 0 \right) \). A typical model problem in which we solve the Navier-Stokes equations is, for example, a flow in a square-shaped region, forced by a river flowing along the upper edge of the region. The non-linear term in non-stationary equations can be treated "explicite". \( {\bf u}_{t+1} -\mu \Delta {\bf u}_{t+1 } + \nabla p_{t+1 } = f - ({\bf u}_t \cdot \nabla ) {\bf u}_t \). Commonly used algorithms for solving Navier-Stokes equations are the pressure correction algorithms called SIMPLE, SIMPLEC or PISO. Modern numerical schemes used in the Navier-Stokes equations simulations can be found in paper [1].