Why seperation of variables methods (https://en.wikipedia.org/wiki/Separation_of_variables) can't be applied into Burger's Equation (https://en.wikipedia.org/wiki/Burgers%27_equation)?
Dear friend Hussein Muhammed
The Helmholtz equation is a linear partial differential equation that describes the behavior of waves in a homogeneous medium. It can be derived from the wave equation by separating variables and assuming harmonic time dependence.
However, Burger's equation is a non-linear partial differential equation that describes the behavior of waves in a medium with viscosity and convection. Due to the non-linearity of the equation, the method of separation of variables cannot be directly applied.
Separation of variables involves assuming that the solution to a partial differential equation can be expressed as a product of functions of each independent variable. This method relies on the linearity of the equation, which allows the resulting equations for each independent variable to be combined into a solution for the entire system.
In Burger's equation, the convection term introduces a non-linearity that prevents the application of separation of variables. This term makes the equation inherently nonlinear, and the solution cannot be expressed as a product of functions of each independent variable.
To solve Burger's equation, other techniques such as numerical methods or approximate analytical methods may be used. These methods involve discretizing the equation and solving it numerically or approximating the solution using techniques such as perturbation theory or asymptotic analysis.
To solve nonlinear PDEs (like Burger or Burger-Huxley etc.), you can use the "Invariant subspace Method (ISM)."
The solution obtained by ISM is expressed as a product of functions, where each function depends on only one independent variable. This factorization of the solution makes it easier to solve the PDE, as it reduces the problem to solving a system of ordinary differential equations (ODEs) instead of a complex PDE. The method involves finding a set of invariant subspaces for the given PDE. By doing this, the solution can be obtained more straightforwardly without requiring complex numerical methods or approximations.
I think the best way to address this research problem is by solving it numerically. As we know burgers' equation contains a term like the wave equation if the speed is unity and the other terms are set to equal zero. Then, we first solve the shock-wave equation using a differential geometrical operator and then inserting a FFT function to calculate the other term, eventually converting them from freq. to time domain and plot it. I am almost done with work and soon will post it.
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