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

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1. Linear Parabolic Equations

1. w_t = aw_xx.    Heat equation (linear heat equation).
2. w_t = aw_xx + Φ(x, t).    Nonhomogeneous heat equation.
3. w_t = aw_xx + bw_x + cw + Φ(x, t).    Convective heat equation with a source.
4. w_t = a(w_rr + r⁻¹w_r).    Heat equation with axial symmetry.
5. w_t = a(w_rr + r⁻¹w_r) + Φ(r, t).    Heat equation with axial symmetry (with a source).
6. w_t = a(w_rr + 2r⁻¹w_r).    Heat equation with central symmetry.
7. w_t = a(w_rr + 2r⁻¹w_r) + Φ(r, t).    Heat equation with central symmetry (with a source).
8. iw_t = −kw_xx + U(x)w.    Schrodinger equation (linear Schrodinger equation).