2nd Order PDE Using Similarity Method

In summary, the conversation is about solving a PDE with a similarity solution method. The equation can be written as a diffusion reaction equation with a reaction term of R(U) = CU/A. The speaker also mentions a substitution method to solve the equation.
  • #1
keropi452
3
0
Hi All,

Does anybody know how to solve the following PDE? I tried a similarity solution method where eta = y/f(x) (which I can do successfully without the C * U term) but was unsuccessful.

upload_2017-5-14_2-54-19.png


Thank you very much in advance!
 
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  • #2
If it is a PDE, it will be ##A\frac{\partial U}{\partial x}=B\frac{\partial^2 U}{\partial y^2}+C\cdot U## ...
 
  • #3
True - Sorry about that. Please take the d's to mean partial differential. Thank you for that catch.
 
  • #4
If you assume that ##A\not=0## you can write your equation as ##\frac{\partial U}{\partial x}=\frac{B}{A}\frac{\partial^2 U}{\partial y^2}+\frac{C}{A}\cdot U## that is an example of diffusion reaction equation with ## R(U)=\frac{C}{A}U##, seehttps://en.wikipedia.org/wiki/Reaction–diffusion_systemwhere you call ##x=t## and ##y=x##, here the reaction term is simply ##\frac{C}{A}U##...

Ssnow
 
  • #5
Thank you very much for your response and observation. Are you possibly aware of any closed form solutions to the diffusion reaction eq with R(U) = CU/A?
 
  • #6
The substitution [itex]u = e^{Cx/A}v[/itex] results in a standard diffusion equation for [itex]v[/itex].
 

Related to 2nd Order PDE Using Similarity Method

1. What is a 2nd Order PDE?

A 2nd Order Partial Differential Equation (PDE) is a mathematical equation that involves partial derivatives of a function with respect to two or more independent variables. It is a type of differential equation that is commonly used to describe physical phenomena in fields such as engineering, physics, and mathematics.

2. What is the Similarity Method for solving 2nd Order PDEs?

The Similarity Method is a technique for solving 2nd Order PDEs by reducing them to ordinary differential equations (ODEs). This method involves transforming the PDE into a new coordinate system, where the PDE becomes an ODE. The solution to this ODE can then be used to obtain the solution to the original PDE.

3. When is the Similarity Method useful for solving 2nd Order PDEs?

The Similarity Method is particularly useful when dealing with PDEs that exhibit symmetry or invariance under certain transformations. This method is also beneficial in cases where the PDE cannot be solved using other techniques such as separation of variables or the method of characteristics.

4. What are the steps involved in solving a 2nd Order PDE using the Similarity Method?

The steps for solving a 2nd Order PDE using the Similarity Method are as follows:

  1. Transform the PDE into a new coordinate system using a similarity transformation.
  2. Identify and solve the resulting ODE in the new coordinate system.
  3. Transform the solution back to the original coordinate system to obtain the solution to the PDE.
  4. Apply any necessary boundary or initial conditions to determine the constants of integration.

5. What are some applications of the Similarity Method in real-world problems?

The Similarity Method has various applications in fields such as fluid mechanics, heat transfer, and electromagnetics. For example, it can be used to analyze the flow of a fluid over a flat plate or to determine the temperature distribution in a heated rod. It is also commonly used in the study of diffusion processes and chemical reactions in physical and biological systems.

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