I am currently doing numerical analysis of GM type Pulse Tube Cryocooler.
Governing equations are a system of boundary value problems and one need to use shooting method to solve it (as per one research paper).
At hot end of regenerator and at hot end of pulse tube there will be different mass flow rate and we need to give two different boundary conditions for mass flow rate at both hot ends.
My question is, how to solve the continuity equation (first order equation) by using shooting method to get exact boundary values at hot ends because shooting method normally applies to second order equations?
Let get more basic!
Every n^th order differential equation needs n condition to be constrained, i.e. for a 2nd order differential equation, you need 2 conditions, that may be one initial condition and one boundary condition , or two boundary condition , the latter is when shooting method comes forward ! to change the Boundary Value Problem (say y(a)=alpha and y(b)=beta ) to a boundary and initial condition (say y(a)=alpha and y'(a)=gamma)
A first order differential equation only needs one condition to be solved ! Initial condition (for a time dependent equation) or boundary condition (for a space differentiated equation) is enough! That is how 1-D wave propagates or 1-D heat transfer and so on.
In summary: You cannot have a first order differential equation with two boundary condition, so you may need not shooting method!
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