 EqWorld The World of Mathematical Equations Exact Solutions > Nonlinear Partial Differential Equations > Second-Order Parabolic Partial Differential Equations ## 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.

### 1.4. 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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