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Exact Solutions > Nonlinear Partial Differential Equations > Second-Order Parabolic Partial Differential Equations

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1. Nonlinear Parabolic Equations

1.1. Nonlinear Heat Equations with a Source of the Form wt = wxx + f(w)

  1. wt = wxx + aw(1 − w).    Fisher equation.
  2. wt = wxx + awbw3.    Newell--Whitehead equation.
  3. wt = wxxw(1 − w)(aw).    FitzHugh--Nagumo equation.
  4. wt = wxx + aw + bwm.
  5. wt = wxx + a + beλw.
  6. wt = wxx + aw ln w.

1.2. Nonlinear Heat Equations of the Form wt = [f(w)wx]x + g(w)

  1. wt = a(wmwx)x.   Heat equation with a power-law nonlinearity.
  2. wt = a(wmwx)x + bw.
  3. wt = a(wmwx)x + bwm+1.
  4. wt = a(wmwx)x + bw1−m.
  5. wt = a(w2nwx)x + bw1−n.
  6. wt = a(wnwx)x + bw + c1wm + c2wk.
  7. wt = a(eλwwx)x.   Heat equation with a exponential nonlinearity.
  8. wt = a(eλwwx)x + b + c1eβw + c2eσw.
  9. wt = [f(w)wx]x.   Nonlinear heat equation of general form.
  10. wt = [f(w)wx]x + g(w).   Nonlinear heat equation with a source of general form.

1.3. Other Nonlinear Parabolic Equations

  1. wt = wxx + wwx.    Burgers equation.
  2. wt + σwwx = awxx + b0 + b1w + b2w2 + b3w3.
  3. wt = xn[xnf(w)wx]x + g(w).
  4. wt = [f(w)(wx)n]x + g(w).

1.4. Nonlinear Schrodinger Equations

  1. iwt + wxx + k|w|2w = 0.    Schrodinger equation with a cubic nonlinearity.
  2. iwt + wxx + k|w|2nw = 0.    Schrodinger equation with a power-law nonlinearity.
  3. iwt + wxx + f(|w|)w = 0.    Schrodinger equation of general form.

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