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

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Discussion Overview

The discussion revolves around the possibility of solving the equation u_x = (sin(x)) * (u) on a periodic domain using Fourier transforms. Participants explore the nature of the equation, its variables, and potential methods for solving it, including the use of ordinary differential equations (ODEs).

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant inquires about the feasibility of solving the equation u_x = (sin(x)) * (u) using FFT, mentioning previous unsuccessful attempts with convolutions.
  • Another participant seeks clarification on the meaning of u_x, suggesting it could refer to either a partial derivative or the x-component of a vector.
  • A later reply clarifies that u_x refers to the partial derivative of u with respect to x and states the goal is to solve for u.
  • One participant questions whether u depends on variables other than x, suggesting that if it does, the equation is an ordinary differential equation that could be solved analytically.
  • Another participant proposes rewriting the equation as y' = y * sin(x) and suggests solving it as an ODE while assuming the constant is a function of other variables.

Areas of Agreement / Disagreement

Participants express varying interpretations of the equation and its components, leading to multiple proposed approaches for solving it. No consensus is reached on a single method or interpretation.

Contextual Notes

Participants have not fully defined the scope of variables involved in the equation, and there are unresolved assumptions regarding the nature of u and its dependence on other variables.

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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