Solving heat equation BACK in time

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SUMMARY

The discussion focuses on solving the one-dimensional heat partial differential equation (PDE) backward in time, specifically ∂u/∂t = -∇²u = -∂²u/∂x², with the goal of determining the initial temperature profile u(x,t=0) that results in a uniform temperature u(x,t1)=c after a diffusion time t1. The boundary conditions are convective, defined by ∂u(L,t)/∂x = -h*u(L,t). The user encounters challenges in implementing a numerical solution using finite difference methods in MATLAB, recognizing the problem as ill-posed and seeking alternative strategies for resolution.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with finite difference methods for numerical analysis
  • Proficiency in MATLAB programming
  • Knowledge of boundary value problems and stability analysis
NEXT STEPS
  • Research techniques for solving ill-posed problems in numerical analysis
  • Explore MATLAB's built-in functions for PDE solving, such as pdepe
  • Learn about regularization methods to stabilize solutions
  • Investigate vector field visualization techniques for better understanding of diffusion
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Mathematicians, numerical analysts, and engineers working on heat transfer problems, particularly those interested in backward time analysis and numerical methods for PDEs.

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I want to solve the one-dimensional heat PDE backward in time ∂u/∂t = -∇2u = -∂2u/∂x2 , x element of [0,L]

Basically, I want to find what the initial temperature profile u(x,t=0) should be such that after some time t1 of diffusion, I am left with the bar at a uniform temperature u(x,t1)=c for c>0 and boundary conditions are convective, i.e. ∂u(L,t)/∂x = -h*u(L,t)

I am having trouble doing this numerically using finite difference in MATLAB, and I realize this is an ill-posed problem. But it seems to be pretty simple so it should be possible. Is there some trick to solving this, or would I have to resort to a brute-force method of guess/check?

I'd appreciate some insight. Thanks!
 
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It could help to draw a vector field and see if there is a way back. But I assume that it is not unique.
 

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