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A First Course in Partial Differential Equations: with by H. F. Weinberger

By H. F. Weinberger

Textual content offers the final houses of partial differential equations comparable to features, domain names of independence, and greatest ideas. suggestions.

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Additional resources for A First Course in Partial Differential Equations: with Complex Variables and Transform Methods (Dover Books on Mathematics)

Example text

Then the substitution ➆ = 1 ✡ ( − 0 ) leads to a similar Abel equation: 1 ➈✧➆ + ➆ ✄✁ = 0 + ✠➲➉ ☛✁ − ☎ − 2 ☎ 1 2 0 0 ➆ +✠ 0 2 − ☎ 0 2 +✠ ➆ . (6) If ☎ 0 ≡ 0, equation (1) has the trivial particular solution 0 = 0. In this case, the change of variable ➆ = 1 ✡✢ leads to an Abel equation of the form (6) with 0 = 0. 2 ❜ . Given a particular solution 0 = 0 (✝ ) of the Abel equation of the second kind ✦ ✄ ✁ = ☎ (✝ ) + ☎ (✝ ), 1 0 the substitution ➝ = ✺ (✝ ) , 2 ( − ) 0 0 where ✺ (✝ ) = exp ➈ ❳ (7) ☎ 1 0 ✝ , ▼ ➉ (8) Page 11 brings (7) to another, similar Abel equation: ➝✯➝ ✄✁ = ➳ (✝ )➝ + ➳ (✝ ).

0 ✌■❉ (8) One looks for a solution of the Cauchy problem for equation (7) with the initial condition (2) as 0 in the form of a regular expansion in powers of the small parameter: = ➼ ÷ ✌ ✖ ♥ (✝ ). ✖ ✖ =0 (9) Relation (9) is substituted in equation (7) taking into account (8). Then one expands the functions ☎ ✖ into a power series in ✌ and matches the coefficients of like powers of ✌ to obtain a system of equations for ♥ ✖ (✝ ): ♥ 0✁ = ☎ 0 (✝ , ♥ 0 ), ♥ 1✁ = ✠ (✝ , ♥ 0 ) ♥ 1 + ☎ 1 (✝ , ♥ 0 ), ☎ ✠ (✝ , ) = ◆ 0 .

Now let us take a look at an equation of the form ✌ 2 ✎✄ ✁ ✁ ✄ + ✌✢✠ (✝ ) ✄ ✁ + ☎ (✝ ) = 0 (5) on a closed interval ✆ ≤ ✝ ≤ ✟ . Assume ❝ (✝ ) ≡ [✠ (✝ )]2 − 4 ☎ (✝ ) ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (5), as ✌♣❉ 0, are expressed by 1 2 1 1 ✠ ❳ ❝ (✝ ) ✝ − ❳ ↔ ◗ 2✌ ü ▼ 2 1 1 ✠ ✄✁ = | ❝ (✝ )|−1 ❏ 4 exp ❳ ❝ (✝ ) ✝ − ❳ ↔ ◗ 2✌ ü ▼ 2 ❝ = | ❝ (✝ )|−1 ❏ 4 exp − ✄ ✁ (✝ ) ✝✥❙ ❱ 1 + ❬ (✌ )❲ , ❝ (✝ ) ▼ (✝ ) ✝✥❙ ❱ 1 + ❬ (✌ )❲ . 3-6. Equations of the general form.

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