Heat/Diffusion Equation on the whole real line

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In summary, the problem involves solving a partial differential equation with the given boundary conditions. The general solution for the whole line is known, but there is a way to solve it without computing any integrals. By rewriting the argument of the exponential in the integral, the integral can be simplified to \sqrt{\pi}, avoiding the need for explicit calculation.
  • #1
eventhorizonof

Homework Statement



boundary conditions:

u(x,0) = exp[-((x-14)^2)/4

Homework Equations



u_t = u_xx

x [tex]\in[/tex] R


The Attempt at a Solution



the problem says there is a way for this partial differential equation to be solved without computing any integrals. i know the general solution for the whole line is:

u(x,t) = 1 / sqrt(4kt) [tex]\int[/tex] exp[-((x-y)^2)/4kt] * PHI(y) dy


where PHI(y) is the boundary conditions.

is there some kind of substitution or trick to be used here to not have to compute any integral?

Thanks.
 
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  • #2
I'm under the impression that you are mixing a few things on the notation. You use k in the resulting integral, but that is nowhere defined. I know that sometimes the partial differential equation is written as:

[tex]a^2 \cdot \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} [/tex]

or:

[tex]k \cdot \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} [/tex]

In your case a² or k is equal to 1. However you still have it in the integral. There is also something wrong with a pi. I assume you mean that the solution has to be:

[tex]u(x,t)=\frac{1}{2a\sqrt{\pi t}} \cdot \int_{-\infty}^{\infty} f(y) e^{-\frac{(x-y)^2}{4a^2t}} dy[/tex]

or in your case:

[tex]u(x,t)=\frac{1}{\sqrt{4 \pi t}} \cdot \int_{-\infty}^{\infty} f(y) e^{-\frac{(x-y)^2}{4t}} dy[/tex]

Now as for the question, You can't do without the integral. However you can avoid having to calculate it explicitly by changing the argument of the exponential. This means that you need to write it as:

[tex]u(x,t)=\frac{1}{\sqrt{4 \pi t}} \cdot \int_{-\infty}^{\infty} e^{-\frac{(y-14)^2}{4}} e^{-\frac{(x-y)^2}{4t}} dy[/tex]

And now transform the argument of the exponential to something like:

[tex]A+B\cdot (y-C)^2[/tex]

The part with A can be put outside the integral and you need to use the following substitution:

[tex]\sqrt{B}(y-C)=p[/tex]

Giving you a final integral as:

[tex]\int_{-\infty}^{\infty}e^{-p^2}dp[/tex]

Which is equal to [tex]\sqrt{\pi}[/tex]. So you do not need to calculate it, just rewriting it in another form.
 

Related to Heat/Diffusion Equation on the whole real line

1. What is the Heat/Diffusion Equation on the whole real line?

The Heat/Diffusion Equation on the whole real line is a partial differential equation that describes how heat or other quantities such as mass or energy are distributed over time in a given region. It is typically used in physics, chemistry, and engineering to model the flow of heat or other substances through a medium.

2. What are the applications of the Heat/Diffusion Equation?

The Heat/Diffusion Equation has a wide range of applications, including predicting the temperature distribution in a material, analyzing heat transfer in buildings or other structures, and understanding the spread of pollutants in the environment. It is also used in fields such as finance to model the diffusion of stock prices.

3. How is the Heat/Diffusion Equation solved on the whole real line?

The Heat/Diffusion Equation can be solved using various methods, including separation of variables, Fourier series, and numerical techniques such as finite difference or finite element methods. The specific approach depends on the boundary conditions and initial conditions of the problem.

4. What are the limitations of the Heat/Diffusion Equation on the whole real line?

The Heat/Diffusion Equation assumes that heat or other quantities are evenly distributed and that there are no external sources or sinks. In reality, these assumptions may not hold true, especially in complex systems. Additionally, the Heat/Diffusion Equation does not account for convective heat transfer, which may be significant in certain applications.

5. How does the Heat/Diffusion Equation relate to other partial differential equations?

The Heat/Diffusion Equation is closely related to other partial differential equations, such as the wave equation and the Laplace equation. In fact, the Heat/Diffusion Equation can be derived from the wave equation by assuming that the wave speed is zero. It can also be obtained from the Laplace equation by introducing a time variable and considering the evolution of a system over time.

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