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5. Systems of Partial Differential Equations of General Form

Notation: L is an arbitrary linear differential operator in coordinates x₁, ..., x_n.

5.1. Linear Systems of PDEs

1. u_t = L[u] + f₁(t)u + g₁(t)w,   w_t = L[w] + f₂(t)u + g₂(t)w.
2. u_tt = L[u] + a₁u + b₁w,   w_tt = L[w] + a₂u + b₂w.

5.2. Nonlinear Systems of Two PDEs Containing the First Derivatives in t

1. u_t = L[u] + uf(t, au − cw) + g(t, au − cw),   w_t = L[w] + wf(t, au − cw) + h(t, au − cw).
2. u_t = L₁[u] + uf(u/w),   w_t = L₂[w] + wg(u/w).
3. u_t = L[u] + uf(t, u/w),   w_t = L[w] + wg(t, u/w).
4. u_t = L[u] + uf(u/w) + g(u/w),   w_t = L[w] + wf(u/w) + h(u/w).
5. u_t = L[u] + uf(t, u/w) + u/w h(t, u/w),   w_t = L[w] + wg(t, u/w) + h(t, u/w).
6. u_t = L[u] + uf(t, u/w) ln u + ug(t, u/w),   w_t = L[w] + wf(t, u/w) ln w + wh(t, u/w).

5.3. Nonlinear Systems of Two PDEs Containing the Second Derivatives in t

1. u_tt = L[u] + uf(t, au − bw) + g(t, au − bw),   w_tt = L[w] + wf(t, au − bw) + h(t, au − bw).
2. u_tt = L₁[u] + uf(u/w),   w_tt = L₂[w] + wg(u/w).
3. u_tt = L[u] + uf(t, u/w),   w_tt = L[w] + wg(t, u/w).
4. u_tt = L[u] + uf(u/w) + g(u/w),   w_tt = L[w] + wf(u/w) + h(u/w).
5. u_tt = L[u] + au ln u + uf(t, u/w),   w_tt = L[w] + aw ln w + wg(t, u/w).

5.4. Nonlinear Systems of Many PDEs Containing the First Derivatives in t

1. (u_m)_t = L[u_m] + u_mf(t, u₁ − b₁u_n, ..., u_{n − 1} − b_{n − 1}u_n)   + g_m(t, u₁ − b₁u_n, ..., u_{n − 1} − b_{n − 1}u_n),
   m = 1, ..., n.
2. (u_m)_t = L[u_m] + u_mf_m(t, u₁/u_n, ..., u_{n − 1}/u_n) + u_m/u_n g(t, u₁/u_n, ..., u_{n − 1}/u_n),   m = 1, ..., n − 1,
   (u_n)_t = L[u_n] + u_nf_n(t, u₁/u_n, ..., u_{n − 1}/u_n) + g(t, u₁/u_n, ..., u_{n − 1}/u_n).