What do you mean by one dimension?
I mean that the heat equation can come in one dimension, two dimensions, or three dimensions.
To show you that, I will use the variable \(\displaystyle u(x,t)\) because I need \(\displaystyle y\) to be an independent variable.
One dimension:
\(\displaystyle \frac{1}{k}\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}\)
Two dimensions:
\(\displaystyle \frac{1}{k}\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\)
Three dimensions:
\(\displaystyle \frac{1}{k}\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\)
Although it is possible to call this differential equation \(\displaystyle \frac{\partial y}{\partial t} = k\frac{\partial^2 y}{\partial x^2}\) two dimensional, we usually count the dimension with respect to the space variable. The space variable \(\displaystyle x\) here is one dimensional, so we say it is a one dimensional heat equation.
There are already four variables in the equation, [imath]x[/imath], [imath]y[/imath], [imath]t[/imath], [imath]k[/imath]. Why not four dimensions?
This is not the right way to tell the dimensions of an equation. I showed you above how to find the dimension. It will be an interesting partial differential equation if \(\displaystyle k\) is a variable which I don't think so.
What is the different between a unique solution and not a unique solution? Please, solve the
Why did khansaheb ask you for boundary and initial conditions? Because without them, there can be a million solution to this differential equation. To get one and only one solution (unique solution), you have to write a complete boundary and initial conditions with a domain. What I can see in your first post is like a chicken without feathers!
Please, solve the equation and let me see your solution, so that I can compare it with my solution and know where I went wrong.
I don't mind to solve the equation, but I will not. Not because the rules of the forum forbids that, it is because you are just throwing naked questions, you don't even understand the basic idea behind them.
To prove to you that, you understand nothing, you cannot even answer this simple question!
There are already four variables in the equation, [imath]x[/imath], [imath]y[/imath], [imath]t[/imath], [imath]k[/imath].
Classify which ones of these are the dependent variables, the independent variables, and the constants.