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

## 3. Nonlinear Elliptic Equations

### 3.2. Heat Equations of the Form [f(x)wx]x + [g(y)wy]y = h(w)

1. (axnwx)x + (bymwy)y = f(w).    Anisotropic heat (diffusion) equation.
2. awxx + (beμywy)y = f(w).    Anisotropic heat (diffusion) equation.
3. (aeβxwx)x + (beμywy)y = f(w).    Anisotropic heat (diffusion) equation.
4. [f(x)wx]x + [g(y)wy]y = kw ln w.    Anisotropic heat (diffusion) equation.

### 3.3. Heat Equations of the Form [f(w)wx]x + [g(w)wy]y = h(w)

1. wxx + [(αw + β)wy]y = 0.    Stationary Khokhlov--Zabolotskaya equation.
2. wxx + (aeβwwy)y = 0.    Anisotropic heat (diffusion) equation.
3. [f(w)wx]x + [g(w)wy]y = 0.    Anisotropic heat (diffusion) equation.

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