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Zachary F
Zachary F
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I'm studying partial differential equations and I am struggling to understand the heat equation $u_t =ku_{xx}$. I understand the physical interpretation of $u_{t}$, but the meaning of $u_{x}$ eludes me.

This question arises from the following problem:

Consider the 1-D heat equation in a rod of length $L$ with diffusion constant $k$. Suppose the left endpoint is fixed at 100°, while the right endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant $K=1$) with an outside medium which is 500°. The initial temperature distribution in the rod is given by $f(x)$.

While setting up the PDE, I interpreted the boundary condition at the right end as $u_{t}(L,t)=u(L,t)-500$ but the answer in the back of the text book has $u_{x}(L,t)=u(L,t)-500$$-u_{x}(L,t)=u(L,t)-500$. Why is that?

I'm studying partial differential equations and I am struggling to understand the heat equation $u_t =ku_{xx}$. I understand the physical interpretation of $u_{t}$, but the meaning of $u_{x}$ eludes me.

This question arises from the following problem:

Consider the 1-D heat equation in a rod of length $L$ with diffusion constant $k$. Suppose the left endpoint is fixed at 100°, while the right endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant $K=1$) with an outside medium which is 500°. The initial temperature distribution in the rod is given by $f(x)$.

While setting up the PDE, I interpreted the boundary condition at the right end as $u_{t}(L,t)=u(L,t)-500$ but the answer in the back of the text book has $u_{x}(L,t)=u(L,t)-500$. Why is that?

I'm studying partial differential equations and I am struggling to understand the heat equation $u_t =ku_{xx}$. I understand the physical interpretation of $u_{t}$, but the meaning of $u_{x}$ eludes me.

This question arises from the following problem:

Consider the 1-D heat equation in a rod of length $L$ with diffusion constant $k$. Suppose the left endpoint is fixed at 100°, while the right endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant $K=1$) with an outside medium which is 500°. The initial temperature distribution in the rod is given by $f(x)$.

While setting up the PDE, I interpreted the boundary condition at the right end as $u_{t}(L,t)=u(L,t)-500$ but the answer in the back of the text book has $-u_{x}(L,t)=u(L,t)-500$. Why is that?

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Zachary F
Zachary F
  • 145
  • 2
  • 10

If $u(x,t)$ describes heat in a 1-D rod, what is $u_{x}$?

I'm studying partial differential equations and I am struggling to understand the heat equation $u_t =ku_{xx}$. I understand the physical interpretation of $u_{t}$, but the meaning of $u_{x}$ eludes me.

This question arises from the following problem:

Consider the 1-D heat equation in a rod of length $L$ with diffusion constant $k$. Suppose the left endpoint is fixed at 100°, while the right endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant $K=1$) with an outside medium which is 500°. The initial temperature distribution in the rod is given by $f(x)$.

While setting up the PDE, I interpreted the boundary condition at the right end as $u_{t}(L,t)=u(L,t)-500$ but the answer in the back of the text book has $u_{x}(L,t)=u(L,t)-500$. Why is that?