Exact Solutions > Systems of Partial Differential Equations > Nonlinear Systems of Two Elliptic Equations (Reaction-Diffusion Equations)

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3. Nonlinear Systems of Two Elliptic Equations

3.1. Reaction-Diffusion Systems of the Form u_xx + u_yy = F(u, w), w_xx + w_yy = G(u, w)

Preliminary comments. Similar systems of equations are encountered in stationary problems of the theory of heat and mass transfer in reacting media, the theory of chemical reactors, combustion theory, mathematical biology, and biophysics. The functions f(φ), g(φ), and h(φ) appearing below are arbitrary functions of an argument φ = φ(u, w); the equations are arranged in order of complicating this argument.

1. u_xx + u_yy = uf(au − bw) + g(au − bw),   w_xx + w_yy = wf(au − bw) + h(au − bw).
2. u_xx + u_yy = e^λuf(λu − σw),   w_xx + w_yy = e^σwg(λu − σw).
3. u_xx + u_yy = uf(u/w),   w_xx + w_yy = wg(u/w).
4. u_xx + u_yy = uf(u/w) + u/w h(u/w),   w_xx + w_yy = wg(u/w) + h(u/w).
5. u_xx + u_yy = uⁿf(u/w),   w_xx + w_yy = wⁿg(u/w).
6. u_xx + u_yy = uf(uⁿw^m),   w_xx + w_yy = wg(uⁿw^m).
7. u_xx + u_yy = uf(u² + w²) − wg(u² + w²),   w_xx + w_yy = wf(u² + w²) + ug(u² + w²).
8. u_xx + u_yy = uf(u² − w²) + wg(u² − w²),   w_xx + w_yy = wf(u² − w²) + ug(u² − w²).

3.2. Other Systems

9. axu_x + ayu_y = u_xx + u_yy − f(u, w),   axw_x + ayw_y = w_xx + w_yy − g(u, w).