Help solving the heat equation

In summary: However, you also need to consider the other initial condition of uxxx(x,0)= 0. This means that uxx(x,0)= 0, and therefore A(t) must also be equal to 0. This allows us to simplify the solution to u(x,t) = B(t)x + C(t). Then, using the initial conditions, we can solve for B(t) and C(t). In summary, the problem is asking to solve the heat equation with the given initial conditions of u(x,0)= x2 and uxxx(x,0)= 0. By using the heat equation, we can show that u(x,t) must be of the form A(t)x2 + B(t)x + C(t).
  • #1
timjones007
10
0
help solving the heat equation!

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
Yes, you can say that uxxx(x,t) is identically equal to 0 if uxxx(x,0)= 0. For the initial conditions, you can use them to determine the coefficients A(0), B(0), and C(0). Then, you can use the heat equation to solve for A(t), B(t), and C(t).Yes, since u(x,0)= x2, then ux(x,0)= 2x.
 

Related to Help solving the heat equation

What is the heat equation and why is it important?

The heat equation is a partial differential equation that describes the distribution of heat in a given region over time. It is important because it is used to solve many real-world problems involving heat transfer, such as in engineering, physics, and climate science.

How do you solve the heat equation?

The heat equation can be solved using various analytical and numerical methods, such as separation of variables, finite difference methods, and Fourier transforms. The specific method used depends on the boundary conditions and other factors of the problem.

What are the key assumptions made in solving the heat equation?

The heat equation assumes that the heat transfer is primarily driven by temperature differences and that the thermal conductivity and heat capacity of the material are constant. It also assumes that the material is homogeneous and isotropic, meaning that its properties are the same in all directions.

What are the boundary conditions and why are they important in solving the heat equation?

The boundary conditions are the specified temperature, heat flux, or other parameters at the boundaries of the region being studied. They are important because they help determine the behavior of heat flow within the region and are necessary for solving the heat equation.

What are some applications of the heat equation?

The heat equation has many practical applications, such as in predicting the temperature distribution in a building, designing cooling systems for electronic devices, and understanding the Earth's climate. It is also used in fields such as thermodynamics, materials science, and chemical engineering.

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