Solving Non-Linear Differential Equation with Fourier Transforms

In summary, the conversation discusses solving a separable ODE under certain assumptions using Fourier Transforms. However, the integral involved is too complicated to be practical, unless specific values are given for the constants.
  • #1
Gengar
13
0
Hiya. I have to solve this bad boy under the assumptions that f, f' and f'' tend to 0 as |x| tends to infinity:

1/2(f')^2 = f^3 + (c/2)f^2 + af + b

where a,b,c are constants. My thoughts are use Fourier Transforms to use the assumptions given, but not sure how to do them on these terms. Thanks!
 
Physics news on Phys.org
  • #2
Hi !

This is a separable ODE which (in theory) can be solved by direct integration :
df / sqrt(2 f^3 + c f^2 + 2af + 2b) = dx
But the integral involves very complicated elliptic functions, so that it will be of no use in practice.
In particular cases, i.e. for some particular values of a, b, c, it might reduce to simpler functions.
 

Related to Solving Non-Linear Differential Equation with Fourier Transforms

1. What is a non-linear differential equation?

A non-linear differential equation is an equation that contains one or more non-linear terms, meaning that the dependent variable and its derivatives are raised to powers other than one. This makes it more complicated to solve compared to linear differential equations.

2. How does the Fourier transform help in solving non-linear differential equations?

The Fourier transform is a mathematical tool that breaks down a function into its constituent frequencies, making it easier to analyze and solve. By transforming a non-linear differential equation into the frequency domain, it becomes a simpler algebraic equation, which can be solved more easily.

3. Can all non-linear differential equations be solved using Fourier transforms?

No, not all non-linear differential equations can be solved using Fourier transforms. The equation needs to have certain properties, such as being linearly separable, for the Fourier transform method to be applicable.

4. Are there other methods for solving non-linear differential equations?

Yes, there are other methods for solving non-linear differential equations, such as numerical methods, perturbation methods, and power series methods. However, the Fourier transform method is particularly useful for problems with periodic boundary conditions.

5. What are some practical applications of solving non-linear differential equations with Fourier transforms?

Solving non-linear differential equations with Fourier transforms has various applications in physics, engineering, and mathematics. It is commonly used in the analysis of electromagnetic waves, heat transfer, and vibration problems. It also has applications in signal processing and image reconstruction.

Similar threads

A Polar Fourier transform of derivatives
    • Differential Equations
Replies
4
Views
2K
Why is the Heat equation solved with separation of variables but not with Fourier transformations?
    • Differential Equations
Replies
4
Views
621
I Explanation of all "its linearly independent derivatives"
    • Differential Equations
Replies
3
Views
1K
A Partial differential equation containing the Inverse Laplacian Operator
    • Differential Equations
Replies
3
Views
2K
I Why is this differential equation non-linear?
    • Differential Equations
Replies
12
Views
2K
I Determining linear indepedence/dependence of a set of functions
    • Differential Equations
Replies
11
Views
583
B Why the Fourier series doesn't work to solve any differential equation?
    • Differential Equations
Replies
11
Views
4K
I Solve Int. Eq.: Exponential Growth Diff. Eq.
    • Differential Equations
Replies
7
Views
1K
I Question about solving linear first order non-homogeneous ODEs
    • Differential Equations
Replies
3
Views
862
I Can all differential equations be turned into algebraic equations via the FT?
    • Differential Equations
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top