Reduction of 2nd order PDE to a first order equations system

It is not related to the given PDE. In summary, the given linear second order general form PDE can be converted into two equations using parametric values and the functions u and v. The equations are not complete as they do not incorporate the function g. The hint provided is not related to the given PDE and does not provide a solution for finding the parametric values.
  • #1
Auteng
20
1
I want to convert this linear second order general form PDE to two equations:
##ϕ_{xx}+bϕ_{xy}+cϕ_{yy}+dϕ_x+eϕ_y+fϕ=g(x,y)##

Converted equations:

##a_1 u_x+b_1 u_y+c_1 v_x+d_1 v_y=f_1##

##a_2 u_x+b_2 u_y+c_2 v_x+d_2 v_y=f_2##

I want to find parametric values of ##a_1 ...f_2##

How can I do it?

Hint:

##ϕ_{xx}+ϕ_{yy}=0##

##u=ϕ_x , v=ϕ_y##

##ϕ_{xx}=u_x,ϕ_{yy}=ϕ_y##

##u_x+v_y=0##

##u_y-v_x=0##
 
Last edited:
  • #3
1. The converted equations don't look right, as they do not incorporate the function ##g##.

2. What is the justification for the first line of the Hint: ##\phi_{xx}+\phi_{yy}=0##? It is not derivable from the given equation.
 
  • #4
No that hint is a typical example
 

What is the process of reducing a second order PDE to a first order equations system?

The process involves expressing the second order PDE as a system of first order equations by introducing new variables and rewriting the original equation in terms of these variables. This allows for the use of methods such as the method of characteristics to solve the PDE.

Why is it useful to reduce a second order PDE to a first order equations system?

Reducing a second order PDE to a first order equations system can make the problem easier to solve, as first order equations are often simpler and more well-studied. It also allows for the use of different methods and techniques to solve the PDE.

What are some common techniques used to reduce a second order PDE to a first order equations system?

Some common techniques include introducing new variables, using the chain rule, and rewriting the PDE in terms of these new variables. Another technique is to use characteristic curves to transform the PDE into a system of first order equations.

Are there any limitations or drawbacks to reducing a second order PDE to a first order equations system?

One limitation is that the process can sometimes introduce additional complexity or difficulties in solving the PDE. Additionally, not all second order PDEs can be reduced to a first order equations system, so this method may not always be applicable.

Can reducing a second order PDE to a first order equations system affect the accuracy of the solution?

In general, reducing a second order PDE to a first order equations system does not affect the accuracy of the solution. However, the choice of variables and the method used to reduce the PDE can impact the accuracy. It is important to carefully choose the variables and method to ensure an accurate solution.

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