RK method for heat equation with dependent variables

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SUMMARY

The discussion focuses on solving the heat equation with a dependent variable, specifically addressing the challenge of preserving the k(y) term during numerical integration. The user initially struggled with the Runge-Kutta method, which inadvertently eliminated the k(y) term. Ultimately, the issue was resolved by employing a proper substitution technique, allowing for the successful integration of the equation without losing the dependent variable.

PREREQUISITES
  • Understanding of the heat equation and its variables
  • Familiarity with numerical methods, specifically Runge-Kutta integration
  • Knowledge of variable substitution techniques in differential equations
  • Basic principles of analytical versus numerical solutions
NEXT STEPS
  • Research the application of variable substitution in solving differential equations
  • Study the Runge-Kutta method in detail, focusing on its implementation for partial differential equations
  • Explore advanced techniques for preserving dependent variables in numerical simulations
  • Learn about alternative numerical methods for solving the heat equation, such as finite difference methods
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Mathematicians, physicists, and engineers involved in solving differential equations, particularly those working with numerical methods and heat transfer problems.

maistral
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Hello.

How do I solve this equation without killing the k(y) term:
1.png


I managed to derive an analytical solution for this one. I intend to run the numerical solution via Runge-Kutta but I can't stop myself from killing the k(y) term. I'm starting to think I'm doing something wrong... It goes something like this:

22.png


Even if I try and keep the k(y) term until I derive the systems of equations required to run Runge-Kutta, the integrator would then numerically kill it anyway. What should I do?
Any help please?
 
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Nevermind. Solved it by using a proper substitution.
 

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