Heat equation: partial differential equations

In summary, the problem is to solve the heat equation ut = kuxx for -infinity < x < infinity and 0 < t < infinity, with initial conditions u(x,0)= x2 and uxxx(x,0)= 0. To find the solution, it is first shown that uxxx(x,t) must be of the form A(t)x2 + B(t)x + C(t), which is then integrated multiple times to get u(x,t). It is assumed that uxxx(x,t) is identically equal to 0 since uxxx(x,0)= 0, allowing for the desired solution form. The initial conditions can then be used to find A(t), B(t), and C(t), although only
  • #1
timjones007
10
0
solve the heat equation
ut = kuxx

-infinity < x < infinity and 0 < t < infinity

with u(x,0)= x2 and uxxx(x,0)= 0

first i showed that uxxx(x,t) solves the equation (easy part)

the next step is to conclude that u(x,t) must be of the form A(t)x2 + B(t)x + C(t).
i tried to do this by integrating uxxx(x,t) with respect to x and i got uxx(x,t) + A(t). then i solved for uxx(x,t), integrated it with respect to x, and so on until i got u(x,t) = (triple integral of uxxx(x,t) with respect to x) + A(t)x2/2 - B(t)x - C(t).

It seems like the only way to get rid of the uxxx(x,t) is to say that it is identically equal to 0 since uxxx(x,0)= 0. Then i can get the desired form of the solution. But, can you say that uxxx(x,t) is identically equal to 0?

Once I show that A(t)x2 + B(t)x + C(t) is the form of the equation, I'm supposed to use the initial conditions to find A(t), B(t), and C(t). But, the initial conditions will only tell me what A(0), B(0), and C(0) are.That's another problem

and lastly, this may have little to do with the problem but if u(x,0)= x2 then does it mean that ux(x,0)= 2x just by differentiating both sides with respect to x or are there special conditions that must be satisfied?
 
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  • #2
Are you sure you posted the problem correctly? You need two boundary conditions and one initial condition. You've posted two initial conditions and no boundary conditions. Huh?
 
  • #3
yes that's the all that's give in the problem. there's no boundary conditions
 

Related to Heat equation: partial differential equations

1. What is the heat equation and what does it describe?

The heat equation is a partial differential equation that describes the transfer of heat energy in a given region over time. It is used to model the behavior of temperature distribution in various physical systems, such as heat conduction in solids or fluids.

2. What are the main variables in the heat equation?

The main variables in the heat equation are time, temperature, and position in space. Time is denoted by t, temperature by T, and position by x, y, or z depending on the number of dimensions in the system.

3. How is the heat equation solved?

The heat equation is typically solved using numerical methods, such as finite difference or finite element methods. These methods involve discretizing the equation into smaller parts and solving them iteratively to approximate the solution.

4. What are the boundary conditions for the heat equation?

The boundary conditions for the heat equation specify the temperature at the boundaries of the system. These conditions can be either fixed temperature boundaries, where the temperature is known, or insulated boundaries, where there is no heat transfer.

5. What are some real-world applications of the heat equation?

The heat equation has numerous real-world applications, including predicting the temperature distribution in buildings, designing heat sinks for electronic devices, and analyzing the cooling of nuclear reactors. It is also used in weather forecasting and climate modeling to study the transfer of heat in the Earth's atmosphere and oceans.

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