Diffuusiyhtälö

Diffuusiyhtälö, also known as the diffusion equation, is a partial differential equation that describes the distribution and evolution of a quantity over time, such as heat, mass, or probability. It is widely used in various fields, including physics, chemistry, biology, and engineering.

The one-dimensional diffusion equation is typically written as:

∂u/∂t = D * ∂²u/∂x²

where u(x,t) represents the quantity being diffused, D is the diffusion coefficient, x is the spatial coordinate, and t is the time. The equation states that the rate of change of u with respect to time is proportional to the second spatial derivative of u, which measures the curvature of the quantity's distribution.

The diffusion equation can be derived from Fick's laws of diffusion, which describe the flux of a substance as being proportional to the gradient of its concentration. By combining Fick's laws with the conservation of mass, one can obtain the diffusion equation.

Solutions to the diffusion equation often exhibit Gaussian or bell-shaped profiles, reflecting the spreading and smoothing of the diffused quantity over time. The equation can be solved analytically for simple geometries and boundary conditions, or numerically for more complex cases.

The diffusion equation is closely related to other partial differential equations, such as the heat equation and the Schrödinger equation, which describe the evolution of heat and quantum wave functions, respectively. These equations share the same mathematical form and can be transformed into one another through appropriate changes of variables.