Handbook of linear partial differential equations for by A D Poli︠a︡nin

By A D Poli︠a︡nin

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In general, a nonhomogeneous linear elliptic equation can be written as −➃ where ➃ x [➈ ➂ ]≡ ➒ ➓ ➓ ➲ ,➔ =1 ➔ → (x) ➂ ] = ❺ (x), x [➈ ↔ ➓ 2➈ ➂ + ↔ ➔ (17) ➂ ➒ ➲ ➓ =1 (x) ➂ ➓ ➈ ↔ ➓ + ➵ (x)➈ . (18) ➳ Two-dimensional problems correspond to ➇ = 2 and three-dimensional problems, to ➇ = 3. We consider equation (17)–(18) in a domain ➨ and assume that the equation is subject to the general linear boundary condition ➥ x [➈ ] = ♣ (x) for x ➄ ➛ . (19) The solution of the stationary problem (17)–(19) can be obtained by passing in (5) to the limit as ➺ .

2-2. Representation of the problem solution in terms of the Green’s function. The solution of problem (12), (13), (4), (5) can be represented as the sum ❘ ❯ ❙ (✜ , ) = ✦ 2 ❙ ❱ 0 ✚ ✚ ✦ 1 ✦ 2 − ② ✚ (€ , ❲ ) q (✜ , € , , ❲ ) ✧ 0 (€ )❥ ❙ ◗ q ❲ ( ✜ , € , , ❲ )❦ ✚ 0 1 (❲ ✧ ❧ ) ✫ (✜ 1 , ❲ ) r ❙ 1 (✜ , ,❲ )✧ ✚ ✷ ② 1 (€ ❙ ) + ② 0 (€ ) ✐ (€ , 0)✸❫q (✜ , € , , 0) ✧ ♣ 0 ✚ ❙ € ✦ 1 ❯ + ❲ ✦ 2 + € =0 ❯ ♣ ❲ ◗ ✦ 1 + ✧ € 2 (❲ ) ✫ (✜ 2 , ❲ ) r 2 (✜ ❙ , ,❲ )✧ ❲ . (14) Here, the Green’s function q (✜ , € , , ❲ ) is determined by solving the homogeneous equation ◗ 2 ◗ ❙ ◗ ❙ q 2 + ✐ (✜ , ) ◗ q ❙ −❚ ✦ ,❯ [q ] = 0 (15) with the semihomogeneous initial conditions ❙ q =0 at =❲ , (16) ◗ ❙ = ❨ ( ✜ − € ) at = ❲ , (17) q ❯ and the homogeneous boundary conditions (9) and (10).

Green’s function: ✝ ✞ (x, y, é , ñ ) = ✟ ü æ (x, y, é − ñ + 2✠ ☎ ) + ù (x, y, é + ñ + 2✠ ☎ ) ✡ . ù (13) =− 3 ✆ . ü The unknown function and its derivative are prescribed at the left and right end, respectively: = Ï 1 (x) Þ at = 0, é ☛ è✱Ï = Þ 2 (x) =☎. at é Green’s function: ✝ ✞ (x, y, é , ñ ) = ✟ æ (−1) ü (x, y, é − ñ + 2✠ ☎ ) − ù (x, y, é + ñ + 2✠ ☎ ) ✡ . ù (14) =− æ ü 4 ✆ . Mixed boundary value problem. The derivative and the unknown function itself are prescribed at the left and right end, respectively: ☛ è✱Ï = 1 (x) Þ at é = 0, Ï = Þ 2 (x) at é =☎.

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