# उष्मा समीकरण

यहाँ जाएँ: भ्रमण, खोज

उष्मा समीकरण (heat equation) महत्वपूर्ण आंशिक अवकल समीकरण है जो किसी वस्तु के किसी क्षेत्र में समय के साथ ताप की स्थिति बताता है। तीन स्पेस चरों (x,y,z) एवं समय t के किसी फलन u(x,y,z,t) के लिये उष्मा समीकरण निम्नवत है:

$\frac{\partial u}{\partial t} -\alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0$

also written

$\frac{\partial u}{\partial t} - \alpha \Delta u=0$

or sometimes

$\frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$

where $\alpha$ is a constant and $\Delta\$ or $\nabla^2\$ denotes the Laplacian operator. For the mathematical treatment it is sufficient to consider the case α=1.

The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker–Planck equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes.