Eliminating φ from η equations

Approximating φ to a given order and specified mode

This is the equation that needs to be solved for φ_i:    η'=ω_i φ_i + αUnderoverscript[∑, n = 1, arg3]Underoverscript[∑, m = 1, arg3]C_ (i, m, n) φ_m η_n

phiEquation expands the right hand side of the phi equation to order N and mode M.  The highest order alpha term in the double sum will always be dropped, so the sum only goes to α^(n - 1).

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phiEqn[i, 2, M]

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This turns off the right hand side pattern warning message.  The reason this defination of phi works is because phiEqn contains the lower orders of phi, which are immidiately evaluated.  φ is programs to save calculated orders, so one it figures an order out, it plugs it in everywhere, and the solver works.

In[3]:=

φ[isub_, N_Integer, Msub_] := (φ[i_, N, M_] = Solve[phiEqn[i, N, M], φdummy[i, N, M]]/.{{{_→r_}} →r} ; φ[isub, N, Msub])

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φ[i, 1, M]

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-(Underoverscript[∑, n_1 = 1, arg3] Underoverscript[∑, m_1 = 1, arg3] (α c[i, m_1, n_1] η[n_1][t] η[m_1]^′[t])/ω[m_1])/(α ω[i])

φt is a total order sum for a given mode

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φtotal[i_, N_, M_] := Underoverscript[∑, h = 0, arg3] α^h φ[i, h, M]

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φtotal[i_1, 1, M]

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Plugging in for Phi into η equations

We are trying to eliminate φ from the following equations.  Mathematica seems to be finicky (or I haven't figured out an alternative) with taylor series O notation; this needs to be figured out.

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etaEqn[i, N, M]

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