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

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3. Nonlinear Elliptic Equations

3.1. Nonlinear Heat Equations of the Form w_xx + w_yy = f(w)

1. w_xx + w_yy = aw + bwⁿ.
2. w_xx + w_yy = awⁿ + bw²ⁿ⁻¹.
3. w_xx + w_yy = ae^βw.
4. w_xx + w_yy = ae^βw + be^2βw.
5. w_xx + w_yy = aw ln(βw).
6. w_xx + w_yy = a sin(βw).
7. w_xx + w_yy = f(w).

3.2. Heat Equations of the Form [f(x)w_x]_x + [g(y)w_y]_y = h(w)

1. (axⁿw_x)_x + (by^mw_y)_y = f(w).    Anisotropic heat (diffusion) equation.
2. aw_xx + (be^μyw_y)_y = f(w).    Anisotropic heat (diffusion) equation.
3. (ae^βxw_x)_x + (be^μyw_y)_y = f(w).    Anisotropic heat (diffusion) equation.
4. [f(x)w_x]_x + [g(y)w_y]_y = kw ln w.    Anisotropic heat (diffusion) equation.

3.3. Heat Equations of the Form [f(w)w_x]_x + [g(w)w_y]_y = h(w)