Solving u_x=(sin(x))*(u) in Fourier space

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SUMMARY

The discussion centers on solving the equation u_x = (sin(x)) * u on a periodic domain using Fast Fourier Transform (FFT). The equation represents a first-order ordinary differential equation (ODE) where u_x denotes the partial derivative of u with respect to x. Participants suggest rewriting the equation as y' = y * sin(x) for easier analytical solutions. The use of FFT for this type of equation is questioned, with emphasis on the potential for convolution methods.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with Fast Fourier Transform (FFT) techniques
  • Knowledge of convolution methods in signal processing
  • Basic concepts of periodic domains in mathematical analysis
NEXT STEPS
  • Research methods for solving ordinary differential equations analytically
  • Explore the application of FFT in solving differential equations
  • Study convolution techniques in the context of signal processing
  • Investigate periodic boundary conditions and their implications in mathematical modeling
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Mathematicians, physicists, and engineers interested in solving differential equations, particularly those utilizing FFT and convolution methods in periodic domains.

vector_problems
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Does anyone know if it is possible to solve an equation of the type
u_x=(sin(x))*(u)
on a periodic domain using the fft.

I have tried methods using convolutions but have had no success

thanks in advance
 
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vector_problems said:
Does anyone know if it is possible to solve an equation of the type
u_x=(sin(x))*(u)
on a periodic domain using the fft.
What does u_x mean in the context of your problem? It could possibly mean the partial derivative of u with respect to x, or it could mean the x-component of a vector u.

Also, what are you trying to solve for?
vector_problems said:
I have tried methods using convolutions but have had no success

thanks in advance
 
sorry, should have been more clear. u_x is the partial derivative of u w.r.t. x and i am trying to solve for u,

du/dx=(sin(x))*(u)
 
Is u a function of any variables other than x? If so, your equation is an ordinary differential equation, and one that is separable and easy to solve analytically.
 
You have ux=sin(x)*u

Can't you rewrite it as: y'=y*sin(x)? Solve it as ODE and instead of writing the constant, just assume it's a function of the other variables.
 

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