Understanding relationship between heat equation & Green's function

In summary: That makes sense. So, the heat equation can be represented as an operator and the Green's function is the solution to the non-homogeneous problem with zero initial conditions. This allows us to solve the heat equation with any initial condition by convolving the Green's function with the initial condition function. In summary, the Green's function for the heat equation is the solution to a non-homogeneous problem with zero initial conditions, allowing us to solve the heat equation with any initial condition by convolving the Green's function with the initial condition function.
  • #1
lriuui0x0
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TL;DR Summary
Trying to understand heat equation general solution through Green's function
Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is:

$$
u(x, t) = \int \phi(x-y, t)f(y)dy
$$

where ##\phi(x, t)## is the heat kernel.

The integral looks a lot similar to using Green's function to solve differential equation. The fact that ##\phi(x, 0) = \delta(x)## also signals something related to Green's function. This wikipedia page talks about Green's function related to heat equation as well.

However after searching on the internet, I don't get how do I understand the Green's function in the context of heat equation. As I understand, Green's function is related to a particular ordinary linear differential operator. What's the differential operator for heat equation?
 
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  • #2
lriuui0x0 said:
Summary:: Trying to understand heat equation general solution through Green's function

Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is:

$$
u(x, t) = \int \phi(x-y, t)f(y)dy
$$

where ##\phi(x, t)## is the heat kernel.

The integral looks a lot similar to using Green's function to solve differential equation. The fact that ##\phi(x, 0) = \delta(x)## also signals something related to Green's function. This wikipedia page talks about Green's function related to heat equation as well.

However after searching on the internet, I don't get how do I understand the Green's function in the context of heat equation. As I understand, Green's function is related to a particular ordinary linear differential operator. What's the differential operator for heat equation?
Have you had a look at
http://web.pdx.edu/~daescu/mth428_528/Green_functions.pdf
 
  • #3
  • #6
lriuui0x0 said:
Summary:: Trying to understand heat equation general solution through Green's function

Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is:

$$
u(x, t) = \int \phi(x-y, t)f(y)dy
$$

where ##\phi(x, t)## is the heat kernel.

The integral looks a lot similar to using Green's function to solve differential equation. The fact that ##\phi(x, 0) = \delta(x)## also signals something related to Green's function. This wikipedia page talks about Green's function related to heat equation as well.

However after searching on the internet, I don't get how do I understand the Green's function in the context of heat equation. As I understand, Green's function is related to a particular ordinary linear differential operator. What's the differential operator for heat equation?
First, as others have noted, Green's functions are used for partial differential equations all the time, not just ordinary differential equations.

Your particular problem is ##u_t - u_{xx} = 0## with ##u(x,t=0) = \delta(x)##, for ##-\infty<x<\infty##.

This is equivalent to the problem ##u_t - u_{xx} = \delta(x) \delta(t) ## with ##u(x,t<0) = 0##, for ##-\infty<x<\infty##. The solution to this problem is the Green's function for the operator ##L(u) = u_t - u_{xx}##.

So in this case the homogeneous initial value problem is equivalent to a non-homogeneous problem with zero initial conditions.

Was that the connection you were looking for?

jason
 
  • #7
jasonRF said:
First, as others have noted, Green's functions are used for partial differential equations all the time, not just ordinary differential equations.

Your particular problem is ##u_t - u_{xx} = 0## with ##u(x,t=0) = \delta(x)##, for ##-\infty<x<\infty##.

This is equivalent to the problem ##u_t - u_{xx} = \delta(x) \delta(t) ## with ##u(x,t<0) = 0##, for ##-\infty<x<\infty##. The solution to this problem is the Green's function for the operator ##L(u) = u_t - u_{xx}##.

So in this case the homogeneous initial value problem is equivalent to a non-homogeneous problem with zero initial conditions.

Was that the connection you were looking for?

jason
Thanks!
 

Related to Understanding relationship between heat equation & Green's function

1. What is the heat equation and why is it important?

The heat equation is a partial differential equation that describes how heat diffuses through a given material over time. It is important because it allows us to predict and understand the behavior of heat in various systems, such as in engineering, physics, and biology.

2. What is Green's function and how does it relate to the heat equation?

Green's function is a mathematical tool used to solve differential equations. In the context of the heat equation, it represents the response of a system to a point source of heat. It helps us to understand the behavior of heat in a given system and to find solutions to the heat equation.

3. How do we use Green's function to solve the heat equation?

To solve the heat equation using Green's function, we first need to find the Green's function for the specific system we are studying. This involves solving the heat equation for a point source of heat. Once we have the Green's function, we can use it in combination with the initial conditions of the system to find a solution to the heat equation.

4. Can Green's function be used for other types of differential equations?

Yes, Green's function can be used to solve a variety of differential equations, not just the heat equation. It is a general mathematical tool that can be applied to many different types of problems in physics, engineering, and other fields.

5. Is there a physical interpretation of Green's function in the context of the heat equation?

Yes, Green's function can be interpreted as the temperature distribution in a system due to a point source of heat. It represents the response of the system to a localized heat input and can help us to understand how heat spreads and dissipates in a given material.

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