How to reduce higher order partial differential equations

[Ojo Saheed]

Hello guys,
I have the system of PDE below and I want to solve it using finite difference method but I think I have to reduce it first to a system of first order PDE. The problem is that I don't know how to reduce this PDE to a first order system. I will appreciate any hints in this regard. Thanks

∇⁶w₁ + μ₁∂/∂t(∇⁴w₁) + μ₂∂²/∂t²(∇²w₁) + μ₃∇⁴w₁ + μ₄∂³w₁ /∂t³ + μ₅∂/∂t(∇²w₂) + μ₆∇²w₂ = g₁(x, y, t)

∇⁶w₂ + χ₁∂/∂t(∇⁴w₂) + χ₂∂²/∂t²(∇²w₂) + χ₃∇⁴w₂ + χ₄∂³w₂ /∂t³ + χ₅∂/∂t(∇²w₁) + χ₆∇⁴w₁ = g₂(x, y, t)

 

[the_wolfman]

The usual trick to reducing the order is to introduce a new variable, let's say [itex] u [/itex] such that [itex]\frac{\partial w}{\partial t} =u[/itex]. Then in your original equation you replace at explicit factors of [itex]\frac{\partial w}{\partial t}[/itex] with [itex] u [/itex].

You then have to also solve the equation [itex]\frac{\partial w}{\partial t} =u[/itex] simultaneously.

You can do this in general for higher order derivatives too. For instances let's introduce [itex] v [/itex] such that [itex]\frac{\partial^2 w}{\partial t^2} =\frac{\partial u}{\partial t}=v[/itex]. In this way we can reduces in high order differential equation into a system of first order equations.

However a word of caution. Numerically solving the your system of equations is not a trivial task. You really need to analyze the system of in order to find an optimal numerical method.

There are different ways of formulating finite difference operators. The different formulations are optimal in differential situations, and not all formulations work for all problems. This is especially true when dealing with a system of partial differential equations that has a 6th order spatial operator and a 3 order temporal operator.

 

[Ojo Saheed]

Thanks for the response the_wolfman. I will proceed as you suggested but could you give me some hints on how to analyse the system to determine the optimal finite difference formulation for my problem? This is my first time to encounter this type of equations and I just think reducing it to first order system would suffice to solve the problem.

 

[the_wolfman]

It sounds like you don't have much experience in numerical analysis. I'd start by solving some easier problems like the heat equation and the wave equation. Theres a plethora of literature out there on how to do so. Work through the von Neumann stability analysis. Derive the finite difference operators using a Taylor expansion etc. You need to be proficient in this techniques before tackling more challenging problems.

Also, where do your equations come from? Do they have a name? Do a literature review to see how other people solve them.

 

[Ojo Saheed]

Once again thanks. Actually, I have solved some problems regarding heat equations and system of ordinary differential equations recently using finite difference method. This particular equation is just a coupled problem of thermoelasticity involving heat and stress equations. Usually, I do solve problems involving systems of differential equations by reducing it to first order system and it works fine but I have never used this method for system of PDEs and I also encounter some difficulties in reducing this equation to first order system, hence my enquiries perhaps I could get some hints to simplify my problem better.