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 w_t = w_xx + f(w)

1. w_t = w_xx + aw(1 − w).    Fisher equation.
2. w_t = w_xx + aw − bw³.    Newell--Whitehead equation.
3. w_t = w_xx − w(1 − w)(a − w).    FitzHugh--Nagumo equation.
4. w_t = w_xx + aw + bw^m.
5. w_t = w_xx + a + be^λw.
6. w_t = w_xx + aw ln w.

1.2. Nonlinear Heat Equations of the Form w_t = [f(w)w_x]_x + g(w)

1. w_t = a(w^mw_x)_x.   Heat equation with a power-law nonlinearity.
2. w_t = a(w^mw_x)_x + bw.
3. w_t = a(w^mw_x)_x + bw^m+1.
4. w_t = a(w^mw_x)_x + bw^1−m.
5. w_t = a(w²ⁿw_x)_x + bw¹⁻ⁿ.
6. w_t = a(wⁿw_x)_x + bw + c₁w^m + c₂w^k.
7. w_t = a(e^λww_x)_x.   Heat equation with a exponential nonlinearity.
8. w_t = a(e^λww_x)_x + b + c₁e^βw + c₂e^σw.
9. w_t = [f(w)w_x]_x.   Nonlinear heat equation of general form.
10. w_t = [f(w)w_x]_x + g(w).   Nonlinear heat equation with a source of general form.

1.3. Other Nonlinear Parabolic Equations

1. w_t = w_xx + ww_x.    Burgers equation.
2. w_t + σww_x = aw_xx + b₀ + b₁w + b₂w² + b₃w³.
3. w_t = x⁻ⁿ[xⁿf(w)w_x]_x + g(w).
4. w_t = [f(w)(w_x)ⁿ]_x + g(w).

1.4. Nonlinear Schrodinger Equations

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