1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

368 Chapter 6 Laplace Transform b. f(t)= { 0 , 0 <t<a, 1 , a<t<b, 0 , b<t; c. f(t)= {t, 0 <t<a, a, a<t. ...

6.2 Partial Fractions and Convolutions 369 a.^1 (s^2 +ω^2 )^2 ; c. s 2 (s^2 +ω^2 )^2 ; b. s (s^2 +ω^2 )^2 ; d. s 3 (s^2 +ω^2 )^2 ...

370 Chapter 6 Laplace Transform In general we may outline our procedure as follows: Original problem L Solution of original prob ...

6.2 Partial Fractions and Convolutions 371 The rules of elementary algebra suggest thatUcan be written as a sum, cs+d (s−r 1 )(s ...

372 Chapter 6 Laplace Transform The algebraic determination of theA’s is very tedious, but notice that (s−r 1 )q(s) p(s) =A 1 +A ...

6.2 Partial Fractions and Convolutions 373 The first term in this expression is recognized as the transform ofu 0 e−at.If F(s)is ...

374 Chapter 6 Laplace Transform The transformed equation is readily solved, yielding U(s)=su^0 +u^1 s^2 −a +^1 s^2 −a F(s). Beca ...

6.2 Partial Fractions and Convolutions 375 Solve the initial value problem u′′+ 2 au′+u= 0 , u( 0 )=u 0 , u′( 0 )=u 1 in these ...

376 Chapter 6 Laplace Transform 6.3 Partial Differential Equations In applying the Laplace transform to partial differential equ ...

6.3 Partial Differential Equations 377 Example 1. ∂^2 u ∂x^2 =∂u ∂t , 0 <x< 1 , 0 <t, u( 0 ,t)= 1 , u( 1 ,t)= 1 , 0 < ...

378 Chapter 6 Laplace Transform from which we find the solution, u(x,t)= ( cos(πt)−^1 π sin(πt) ) sin(πx).
Example 3. Now we c ...

6.3 Partial Differential Equations 379 only wheresor sinh( √ s)is zero. Because sinh( √ s)=0 has no real root besides zero, we s ...

380 Chapter 6 Laplace Transform whereqandpare the obvious choices. We take √ rn=+inπin all calculations: p′(s)=sinh (√ s ) +s^1 ...

6.3 Partial Differential Equations 381 Part a. (r 0 =0.) The limit assapproaches zero ofsU(x,s)may be found by L’Hôpital’s rule ...

382 Chapter 6 Laplace Transform EXERCISES 1.Find all values ofs, real and complex, for which the following functions are zero. a ...

6.4 More Difficult Examples 383 6.4 More Difficult Examples The technique of separation of variables, once mastered, seems more ...

384 Chapter 6 Laplace Transform The only solution occurs whenξ=0, for otherwise both terms of this equa- tion have the same sign ...

6.4 More Difficult Examples 385 The transformed equation and its general solution are d^2 U dx^2 =sU,^0 <x, U(^0 ,s)=F(s), U( ...

386 Chapter 6 Laplace Transform 1 + 21 iexp [ iωt− √ ω 2 (^1 +i)x ] −α 2 iexp [ −iωt− √ ω 2 (^1 −i)x ] = 1 +αexp ( − √ ω 2 x ) s ...

6.4 More Difficult Examples 387 At the pointss=0,s=±iπ,s=±( 2 n− 1 )iπ,n= 2 , 3 ,...,U(x,s)becomes undefined. The computation of ...