# Solving a PDE: Can Someone Help?

• kaniello
In summary, The conversation discusses a PDE involving a function u that is only a function of time. It is a Riccati equation and the OP is seeking help to solve it. The conversation includes a proposed solution involving separating variables and integrating, but it is noted that this approach may not work for the specific problem at hand. The article from CompuChip is referenced as a helpful resource for solving the equation.
kaniello
Hallo,
I ended up with this PDE:

du/dt + A*u² =C

A and C are constants and U is only function of time.
It looks simple but...
Can someone help me?

It is a Riccati equation, and indeed it is slightly more tricky than you may initially think.

Thank you so much! That was a quick answer!

Can't you just separate variables ?

@dextercioby. That would work to solve for a function of u (e.g. f(u) = ...) , but kaniello here is solving for the function u(t) that satisfies the ODE... a much tricker problem that satisfies the riccati equation.

Hmmm

$$\frac{du}{dt} = C-Au^2 \Rightarrow \int dt = \int\frac{du}{C-Au^2} \Rightarrow...$$

??

..which would be
tanh⁻1($\sqrt{a/c}$ / (a/c).
I am currently studying the article from CompuChip and reference given there. Probably this is the solution that comes out from the Riccati equation setting q1=0, q0 and q2= const.

X89codered89X said:
@dextercioby. That would work to solve for a function of u (e.g. f(u) = ...) , but kaniello here is solving for the function u(t) that satisfies the ODE... a much tricker problem that satisfies the riccati equation.

CompuChip said:
It is a Riccati equation, and indeed it is slightly more tricky than you may initially think.

The Riccati equation is only "tricky" in the general case - for example when C in the OP's equation is replaced by something like $Ct^n$

As dextercioby said, this special case is straighforward to integrate.

## 1. How do I know which method to use to solve a PDE?

The method you use to solve a PDE depends on the type of PDE and the boundary conditions given. Some common methods include separation of variables, method of characteristics, and numerical methods such as finite differences or finite elements. It is important to understand the properties of the PDE and the given conditions before deciding on a method.

## 2. What are the steps for solving a PDE?

The general steps for solving a PDE are: 1) Identify the type of PDE and the given boundary conditions, 2) Simplify the PDE using any known properties or transformations, 3) Apply the chosen method to obtain a solution, 4) Check the solution for accuracy by plugging it back into the original PDE, and 5) Interpret the solution in the context of the problem.

## 3. Can I use a computer program to solve a PDE?

Yes, there are various software programs that can solve PDEs numerically. These programs use algorithms and numerical methods to approximate the solution. However, it is still important to have a good understanding of the PDE and the solution process in order to properly interpret the results and ensure their accuracy.

## 4. Are there any common mistakes to avoid when solving a PDE?

Some common mistakes when solving a PDE include incorrect application of the chosen method, errors in simplifying the PDE, and incorrect interpretation of the solution. It is important to carefully check each step of the solution process and understand the properties of the PDE to avoid these mistakes.

## 5. How can I check the accuracy of my solution for a PDE?

The best way to check the accuracy of a solution for a PDE is to plug it back into the original PDE and see if it satisfies the equation. Additionally, you can compare your solution to known solutions or use a computer program to verify the results. It is also important to check that the solution satisfies the given boundary conditions.

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