معادله گرما

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

${\displaystyle \mathrm{\Phi }(x,t)=\frac{1}{\sqrt{4\pi kt}}\exp \left(-\frac{x^2}{4kt}\right).}$

${\displaystyle u(x,t)=\int \mathrm{\Phi }(x-y,t)g(y)dy.}$