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

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2. Nonlinear Systems of Two Parabolic Equations

2.1. Reaction-Diffusion Systems of the Form ut = auxx + F(u, w),  wt = bwxx + G(u, w)

Preliminary comments. Similar systems of equations are frequent in the theory of heat and mass transfer of reacting media, the theory of chemical reactors, combustion theory, mathematical biology, and biophysics. Systems of this form are invariant under translations in independent variables (and under the change of x to − x) and admit traveling-wave solutions, u = u(kx − λt) and w = w(kx − λt). These solutions and also degenerate solutions where one of the sought functions is zero are not considered further on. 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. ut = auxx + u exp(kw/u)f(u),   wt = awxx + exp(kw/u)[wf(u) + g(u)].
  2. ut = auxx + uf(bu − cw) + g(bu − cw),   wt = awxx + wf(bu − cw) + h(bu − cw).
  3. ut = auxx + eλuf(λu − σw),   wt = bwxx + eσwg(λu − σw).
  4. ut = auxx + uf(u/w),   wt = awxx + wg(u/w).
  5. ut = auxx + uf(u/w),   wt = bwxx + wg(u/w).
  6. ut = auxx + uf(u/w) + g(u/w),   wt = awxx + wf(u/w) + h(u/w).
  7. ut = auxx + uf(u/w) + u/w h(u/w),   wt = awxx + wg(u/w) + h(u/w).
  8. ut = auxx + u3f(u/w),   wt = awxx + u3g(u/w).
  9. ut = uxx + au − u3f(u/w),   wt = wxx + aw − u3g(u/w).
  10. ut = auxx + unf(u/w),   wt = bwxx + wng(u/w).
  11. ut = auxx + uf(u/w) ln u + ug(u/w),   wt = awxx + wf(u/w) ln w + wh(u/w).
  12. ut = auxx + uf(w/u) − wg(w/u) + u(u2 + w2)−1/2h(w/u),
    wt = awxx + wf(w/u) + ug(w/u) + w(u2 + w2)−1/2h(w/u).
  13. ut = auxx + uf(w/u) + wg(w/u) + u(u2 − w2)−1/2h(w/u),
    wt = awxx + wf(w/u) + ug(w/u) + w(u2 − w2)−1/2h(w/u).
  14. ut = auxx + uf(unwm),   wt = bwxx + wg(unwm).
  15. ut = auxx + u1 + knf(unwm),   wt = bwxx + w1 − kmg(unwm).
  16. ut = auxx + cu ln u + uf(unwm),   wt = bwxx + cw ln w + wg(unwm).
  17. ut = auxx + uf(u2 + w2) − wg(u2 + w2),   wt = awxx + wf(u2 + w2) + ug(u2 + w2).
  18. ut = auxx + uf(u2 − w2) + wg(u2 − w2),   wt = awxx + wf(u2 − w2) + ug(u2 − w2).
  19. ut = auxx + uf(u2 + w2) − wg(u2 + w2) − w arctan(w/u)h(u2 + w2),
    wt = awxx + wf(u2 + w2) + ug(u2 + w2) + u arctan(w/u)h(u2 + w2).
  20. ut = auxx + uf(u2 − w2) + wg(u2 − w2) + w artanh(w/u)h(u2 − w2),
    wt = awxx + wf(u2 − w2) + ug(u2 − w2) + u artanh(w/u)h(u2 − w2).
  21. ut = auxx + uk + 1f(φ),   wt = awxx + uk + 1[f(φ) ln u + g(φ)],   φ = u exp(−w/u).
  22. ut = auxx + uf(u2 + w2) − wg(w/u),   wt = awxx + ug(w/u) + wf(u2 + w2).
  23. ut = auxx + uf(u2 − w2) + wg(w/u),   wt = awxx + ug(w/u) + wf(u2 − w2).
  24. ut = auxx + uf(u2 + w2, w/u) − wg(w/u),   wt = awxx + wf(u2 + w2, w/u) + ug(w/u).
  25. ut = auxx + uf(u2 − w2, w/u) + wg(w/u),   wt = awxx + wf(u2 − w2, w/u) + ug(w/u).
  26. ut = auxx + F(u, w),   wt = awxx + bF(u, w).
  27. ut = auxx + uf(bu − cw) + g(bu − cw) + cΦ(u, w),   wt = awxx + wf(bu − cw) + h(bu − cw) + bΦ(u, w).

2.2. Reaction-Diffusion Systems of the Form ut = ax− n(xnux)x + F(u, w),  wt = bx− n(xnwx)x + G(u, w)

Preliminary comments. Similar systems of equations are frequent in the theory of heat and mass transfer of reacting media, the theory of chemical reactors, combustion theory, mathematical biology, and biophysics. The values n = 1 and n = 2 correspond to equations with axial and central symmetry, respectively. 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. ut = ax−n(xnux)x + uf(bu − cw) + g(bu − cw),   wt = ax−n(xnwx)x + wf(bu − cw) + h(bu − cw).
  2. ut = ax−n(xnux)x + eλuf(λu − σw),   wt = bx−n(xnwx)x + eσwg(λu − σw).
  3. ut = ax−n(xnux)x + uf(u/w),   wt = ax−n(xnwx)x + wg(u/w).
  4. ut = ax−n(xnux)x + uf(u/w),   wt = bx−n(xnwx)x + wg(u/w).
  5. ut = ax−n(xnux)x + uf(u/w) + u/w h(u/w),   wt = ax−n(xnwx)x + wg(u/w) + h(u/w).
  6. ut = ax−n(xnux)x + ukf(u/w),   wt = bx−n(xnwx)x + wkg(u/w).
  7. ut = ax−n(xnux)x + uf(u/w) ln u + ug(u/w),   wt = ax−n(xnwx)x + wf(u/w) ln w + wh(u/w).
  8. ut = ax−n(xnux)x + uf(x, ukwm),   wt = bx−n(xnwx)x + wg(x, ukwm).
  9. ut = ax−n(xnux)x + u1 + knf(unwm),   wt = bx−n(xnwx)x + w1 − kmg(unwm).
  10. ut = ax−n(xnux)x + cu ln u + uf(x, ukwm),   wt = bx−n(xnwx)x + cw ln w + wg(x, ukwm).
  11. ut = ax−n(xnux)x + uf(u2 + w2) − wg(u2 + w2),   wt = ax−n(xnwx)x + wf(u2 + w2) + ug(u2 + w2).
  12. ut = ax−n(xnux)x + uf(u2 − w2) + wg(u2 − w2),   wt = ax−n(xnwx)x + wf(u2 − w2) + ug(u2 − w2).
  13. ut = ax−n(xnux)x + uf(u2 + w2) − wg(w/u),   wt = ax−n(xnwx)x + wf(u2 + w2) + ug(w/u).
  14. ut = ax−n(xnux)x + uf(u2 − w2) + wg(w/u),   wt = ax−n(xnwx)x + wf(u2 − w2) + ug(w/u).

2.3. Other Systems

  1. ut = [f(t, u/w)ux]x + ug(t, u/w),   wt = [f(t, u/w)wx]x + wh(t, u/w).

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