Mastering PDEs: Solving the Non-Constant Coefficient d^2G/dxdy Equation

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In summary, this equation is saying that if you solve for both G and A, then you will get 0 as the result.
  • #1
toptrial
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d^2G/dxdy+(a-1)*dG/dx*dG/dy=0
where G is a function of x and y.

Moreover, what if a is not a constant, but instead a function of x and y?
 
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  • #2
Can't you assume a solution of the form G(x,y) = X(x)Y(y)
 
  • #3
I am afraid not.
In fact, I know G(x,y)=-ln([1+(a-1){X(x)+Y(y)}]^[1/(a-1)]) is the solution. I am just trying to figure out how to slove this pde.
Thanks
 
  • #4
If you assume G(x,y)=X(x)+Y(y)
will that do the trick?

You'll get:
(a-1)XY=0, then either X=0 or Y=0 or a=1, the last one is uninterseting, so you have two solutions here.

There not much to go out here, you use multiplication or addition, division and substraction are defined apostriori by them.

Edit: or any other composition of the elementary functions.

I am quite sure the way your textbook or teacher made this question, that they knew already the answer, and then found which equation it satisifies, you know the solution so take the derivative and see his process of devising the question.
 
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  • #5
For some problems like this one can sometimes use the transformation

z = x + y
w = x - y

Maybe some terms drop off and a new simpler PDE is achieved? It's like a rotation.
 
  • #6
What RedBranchKnight refers to is a special case of the http://en.wikipedia.org/wiki/Method_of_characteristics" .
 
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  • #7
loop quantum gravity said:
If you assume G(x,y)=X(x)+Y(y)
will that do the trick?

You'll get:
(a-1)XY=0, then either X=0 or Y=0 or a=1, the last one is uninterseting, so you have two solutions here.

Um, aren't all of those uninteresting cases? This is saying that any G which is a function of just one of the variables (x or y, but not both) is a solution. In that case, one of the 1st-derivatives will be zero, as will the mixed 2nd-derivative. The original equation becomes 0=0.
 
  • #8
Well, I understand now, the solution's of toptrial is the answer itself, because you don't know what is X(x), Y(y), you need to take the derivative of G(x,y), and find what are X(x) and Y(y).
Or so I think.
 
  • #10
Hi guys, how would I solve this pde numerically in matlab?

d^2G/dxdy+A*dG/dx*dG/dy=0
where G and A are both functions of x and y.
 

FAQ: Mastering PDEs: Solving the Non-Constant Coefficient d^2G/dxdy Equation

What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple variables and their partial derivatives. It is commonly used in physics, engineering, and other fields to model relationships between quantities that vary continuously.

How do I solve a PDE?

Solving a PDE involves finding a function or set of functions that satisfy the equation. This can be done using a variety of methods, such as separation of variables, Fourier transforms, or numerical techniques. The approach used will depend on the specific form of the PDE and the boundary conditions.

What are some applications of PDEs?

PDEs have many applications in science and engineering, including modeling heat transfer, fluid dynamics, and electromagnetic fields. They are also used in economics, biology, and other fields to describe dynamic systems and their behavior over time.

Can PDEs be solved analytically?

In some cases, PDEs can be solved analytically using exact or closed-form solutions. However, for more complex equations, it may be necessary to use numerical methods to approximate the solution.

Are there any software tools available for solving PDEs?

Yes, there are numerous software tools and packages available for solving PDEs, such as MATLAB, Mathematica, and COMSOL. These tools offer a variety of numerical methods and visualization options to help solve and analyze PDEs.

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