Question

1. $\frac{partial u}{partial t}=aleft(\frac{partial^{2} u}{partial x^{2}}+\frac{partial^{2} u}{partial y^{2}}

ight)$
first i notice that the expression has a clear

Explanation:

Step1: Identify the equation type

This is a two - dimensional heat equation (a type of partial differential equation).

Step2: Analyze the components

The left - hand side $\frac{\partial u}{\partial t}$ represents the rate of change of the function $u$ with respect to time $t$. The right - hand side $a(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}})$ involves second - order spatial derivatives with respect to $x$ and $y$, where $a$ is a positive constant related to the physical properties of the medium.

Answer:

This is a two - dimensional heat equation.