PDE problem, Solve using Method of Characteristics

In summary, the conversation is about a problem with solving a partial differential equation on an exam. The solution involves a hyperbolic function in the form of u(x,y)= f(x+y) + g (x+y). The person is asking for help with solving the initial value problem for this equation, and someone suggests differentiating and integrating to find the functions f and g. The person then asks for clarification on the methodology and the conversation ends with confirmation that their approach is correct.
  • #1
Red_CCF
532
0
Hi

This problem occurred on my final and I could not figure it out.

Homework Statement



The problem was a partial differential equation (I forgot the exact equation) but the solution was a hyperbolic function in the form of u(x,y)= f(x+y) + g (x+y), it was part b that gave me the problem.

Part b asked to solve the initial value problem give u(0,y)=y and u_x_(0,y)=y^2 (I'm not 100% sure what they were, but they were functions of y). I just want to know how I would solve an initial value problem for these equations as we never covered it in class.

Thanks.
 
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  • #2
So can anyone help me out?

Thanks
 
  • #3
So I ordered my exam, it turns out the actual question was:

Solve u_xx+u_xy-6u_yy = 0. The solution which I solved correctly was a hyperbolic function u (x,y)= f(2x+y)+g(3x-y).

The IVP is u(0,y) = y^2 and u_x_(0,y)=3y.

Thanks.
 
  • #4
So found the characteristics and now you apply the boundary conditions. What is your problem exactly?

Mat
 
  • #5
hunt_mat said:
So found the characteristics and now you apply the boundary conditions. What is your problem exactly?

Mat

Hi

I just don't know how to solve the IVP u(0,y) = y^2 and u_x_(0,y)=3y when u (x,y)= f(2x+y)+g(3x-y).

I found a solved example online (http://www.mathsman.co.uk/DEnotes[2].pdf, the IVP is solved beginning page 48) but I'm very confused on their methodology, so if someone can explain this or do it some different way I would really appreciate it.

Thanks
 
  • #6
Just differentiate!
[tex]
\frac{\partial u}{\partial x}(0,y)=2f'(y)+3g'(-y)=3y
[/tex]
Integrating w.r.t. y shows that:
[tex]
2f(y)-3g(-y)=\frac{3y^{2}}{2}
[/tex]
Likewise setting x=0 in the equations shows that:
[tex]
u(0,y)=f(y)+g(-y)=y^{2}
[/tex]
To find f and g just add and subtract.
 
  • #7
hunt_mat said:
Just differentiate!
[tex]
\frac{\partial u}{\partial x}(0,y)=2f'(y)+3g'(-y)=3y
[/tex]
Integrating w.r.t. y shows that:
[tex]
2f(y)-3g(-y)=\frac{3y^{2}}{2}
[/tex]
Likewise setting x=0 in the equations shows that:
[tex]
u(0,y)=f(y)+g(-y)=y^{2}
[/tex]
To find f and g just add and subtract.

Hi

Just one question, when we take derivative of f and g, what is it with respect to? I substituted w = 2x+y and v = 3x-y, so u_x = df/dw*dw/dx + dg/dv*dv/dx and when I integrated w.r.t. y I let dw=dy and dv = -dy which gave me the same equation you had, is this correct?
 
  • #8
Sounds good.
 

1. What is a PDE problem?

A PDE (partial differential equation) problem involves finding a function that satisfies a given differential equation involving multiple independent variables. These types of problems are commonly used in physics, engineering, and other scientific fields to model complex systems.

2. What is the Method of Characteristics?

The Method of Characteristics is a mathematical technique used to solve specific types of PDE problems. It involves transforming the PDE into a system of ordinary differential equations and then solving for the characteristics, which are curves along which the solution must be constant.

3. When is the Method of Characteristics used?

The Method of Characteristics is typically used to solve linear, first-order PDEs with constant coefficients. It is also useful for solving boundary value problems and initial value problems.

4. What are the steps involved in solving a PDE problem using the Method of Characteristics?

The steps involved in solving a PDE problem using the Method of Characteristics are as follows:
1. Transform the PDE into a system of ordinary differential equations.
2. Identify the characteristics of the PDE and solve for them.
3. Use the characteristics to find a general solution to the PDE.
4. Apply any initial or boundary conditions to determine the constants in the solution.
5. Simplify and express the final solution in terms of the original variables.

5. What are some limitations of the Method of Characteristics?

The Method of Characteristics is limited to solving specific types of PDEs, particularly linear, first-order PDEs with constant coefficients. It may also be difficult to find the characteristics in some cases, making it challenging to apply this method. Additionally, the solution may become complex and difficult to interpret for more complex PDE problems.

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