General solution of heat equation?

In summary, a solution to the heat equation is given by $$K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$$ and it can be proven that $$u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy$$ is also a solution to the equation. Additionally, it can be shown that $$\lim_{t\to 0^+} u(x,t) = f(x)$$ by using the properties of the Gaussian function and the limiting case of linear combinations of it. This proof can be found in textbooks on PDE.
  • #1
101
25
We know

$$
K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})
$$

is a solution to the heat equation:

$$
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
$$

I would like to ask how to prove:

$$
u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy
$$

is also the solution to the equation, and also:

$$
\lim_{t\to 0^+} u(x,t) = f(x)
$$
 
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  • #2
The first part is a lot easier than you might think, you just use the fact that K satisfies the heat equation, that is that $$\frac{\partial K}{\partial t}=\frac{\partial^2K}{\partial x^2}$$ and you go ahead to prove that ##u(x,t)## as it is defined also satisfies the heat equation by working on both sides of the heat equation for this specific ##u##.
All you use is that in the expression for## u##, because we integrate with respect to## y ##which is independent variable with respect to ##x ##and## t##, the partial derivative with respect to t (or x) passes under the integral sign, that is for example $$\frac{\partial u}{\partial t}=\frac{\partial \int_{-\infty}^{\infty}K(x-y,t)f(y)dy}{\partial t}=\int_{-\infty}^{\infty}\frac{\partial K(x-y,t)}{\partial t} f(y) dy$$ and similar for the first and second partial derivative with respect to x.
 
  • #3
the downstairs limit of the integral must be zero. Actually the whole question is contained in textbooks in PDE
 
  • Informative
Likes Delta2
  • #4
Thanks! I found some reference on this. Basically the integral is a solution follows from the limiting case of linear combination of ##K(x-y,t)##, and the limit follows from the limit of Gaussian function is delta function.
 

1. What is the general solution of the heat equation?

The general solution of the heat equation is a mathematical expression that describes the distribution of heat in a given system over time. It takes into account factors such as the initial temperature distribution, the thermal conductivity of the material, and the boundary conditions.

2. How is the general solution of the heat equation derived?

The general solution of the heat equation is derived using mathematical techniques such as separation of variables, Fourier series, and Laplace transforms. These methods involve breaking down the equation into simpler parts and solving them individually, then combining the solutions to obtain the general solution.

3. What are the boundary conditions in the general solution of the heat equation?

The boundary conditions in the general solution of the heat equation refer to the conditions at the boundaries of the system, such as the temperature at the edges of a material or the rate of heat transfer at the surface. These conditions are important in determining the behavior of the system and are often specified in the problem statement.

4. Can the general solution of the heat equation be applied to all systems?

The general solution of the heat equation can be applied to a wide range of systems, including solids, liquids, and gases. However, it is important to note that the assumptions and simplifications made in deriving the equation may not always hold true for every system. Therefore, it is necessary to carefully consider the applicability of the general solution in each case.

5. How is the general solution of the heat equation used in practical applications?

The general solution of the heat equation is used in many practical applications, such as in designing heating and cooling systems, analyzing the temperature distribution in materials during manufacturing processes, and predicting the behavior of weather systems. It is also used in fields such as thermodynamics, fluid mechanics, and materials science to understand the transfer of heat in various systems.

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