Non-Homogeneous Boundary Conditions: How to Solve PDEs with Green's Function?

In summary, the conversation discusses solving PDEs using Green's function and addresses the issue of non-homogeneous boundary conditions. The solution involves making a substitution and using the method of Green functions to find the values of c_1 and c_2.
  • #1
Col.Buendia
8
0
Hey Guys;

I'm solving PDE's with the use of Green's function where all the boundary conditions are homogeneous.
However, how do you solve ones in which we have non-homogeneous b.c's.

In case it helps, the particular PDE I'm looking at is:

[tex]y'' = -x^2[/tex]

[tex]y(0) + y'(0) = 4, y'(1)= 2 [/tex]

Thanks for any help.
 
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  • #2
That is an ODE, the general solution is

[tex]y = c_1 + c_2x - \frac{1}{12}x^4 [/tex]

The equations are:

[tex]y'(1) = c_2 - \frac{1}{3}(1)^3 = 1[/tex]

Therefore we have c_2 = 4/3. Therefore the second equation reads:

[tex]c_1 + c_2 = 4[/tex]

Therefore we have c _1 = 8/3.

Doing this problem with the method of Green functions, we begin with the piecewise solution c_1 + c_2x, x < x' , and b_1 + b_2 x, x > x'. The first boundary condition says that c_1 + c_2 = 4, and the second one says b_2 = 1. Continuity at x' yields c_1 + (4-c_1)x' = b_1 + x'. Finally, the jump condition on the first derivative yields y'(x')_right - y'(x')_left = 1 implies 1 - c_2 = 1, and so c_2 is zero, at which point I'm stuck too.
 
  • #3
Col.Buendia said:
Hey Guys;

I'm solving PDE's with the use of Green's function where all the boundary conditions are homogeneous.
However, how do you solve ones in which we have non-homogeneous b.c's.

In case it helps, the particular PDE I'm looking at is:

[tex]y'' = -x^2[/tex]

[tex]y(0) + y'(0) = 4, y'(1)= 2 [/tex]

Thanks for any help.


Try making the substitution:
Y(x) = y(x) - 2x - 2.

Then
Y(0) = y(0) - 2 , Y'(0) = y'(0) - 2 and Y'(1) = y'(1) - 2.
These will make your bc homogeneous.
 
  • #4
Thanks, that seems to work perfectly.
 

What is a PDE?

A PDE, or partial differential equation, is a mathematical equation that involves partial derivatives of a function with respect to multiple independent variables.

What is the role of Green's function in solving PDEs?

Green's function is a mathematical tool used in solving PDEs. It is a solution to the homogeneous form of the PDE, and by using it with the method of superposition, we can find the solution to the inhomogeneous form of the PDE.

How do you find the Green's function for a given PDE?

The method for finding the Green's function for a given PDE varies depending on the type of PDE. In general, it involves solving the homogeneous form of the PDE and applying boundary conditions to find the specific Green's function for that PDE.

What is the physical interpretation of Green's function?

The physical interpretation of Green's function is that it represents the response of a system to a point source. In other words, it describes how a system will behave at a particular point in space when a localized input is applied.

How is Green's function used in real-life applications?

Green's function has many applications in physics, engineering, and other fields. It is often used to model and understand the behavior of physical systems, such as fluid flow, heat transfer, and electromagnetism. It is also used in solving practical problems, such as predicting the spread of pollutants in the environment or designing efficient electronic circuits.

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