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GREEN"S FUNCTIONS in Java Generation PDF417 in Java GREEN"S FUNCTIONS




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5.9. GREEN"S FUNCTIONS use javabean pdf417 development toinclude pdf417 for java Barcodes FAQs It follows that u satisfies javabean PDF417 the boundary conditions (BCs) (5.34), (5.35) iff aI, a2 are constants satisfying the system of equations.

adaXI(a) - ;3x~(a)]. + a2[aX2(a) -. ;3x~(a)]. (5.39) (5.40).

=A =B + ;3u~(a), ad/"XI(b) + ox~ Java PDF-417 2d barcode (b)] + a2bx2(b) + ox~(b)]. - auo(a). -/,uo(a) - ou~(a).. Since (5.38) holds, the sys tem (5.39), (5.

40) has a unique solution aI, a2. This implies that the BVP (5.33)-(5.

35) has a unique solution. 0 In the next example we give a BVP of the type (5.30)-(5.

32) that does not have just the trivial solution.. Example 5.99 Find all solutions of the BVP (p(t)x")". x"(a). = 0,. x"(b). = O.. This BVP is equivalent to a barcode pdf417 for Java BVP of the form (5.30)-(5.32), where == 0, a = /" = 0, ;3 =1= 0, and 0 =1= o.

A general solution of this differential equation is. q(t). x(t) =. c21t p(s) ds. The boundary conditions lead to the equations x"(a) = C2 p ta) = 0, x"(b) = C2 p (b) = O. Thus C2 = 0 and there is no restriction on CI. Hence for any constant CI, x(t) = CI is a solution of our BVP. In particular, our given BVP has nontrivial solutions.

6 In the next theorem we give a neccessary and sufficient condition for some boundary value problems of the form (5.30)-(5.32) to have only the trivial solution.

The proof of this theorem is Exercise 5.52. Note that Theorem 5.

100 gives the last statement in Example 5.99 as a special case..

Theorem 5.100 Let p := Then the BVP (b 1 pes) ds ;3/,. + pea) + p(b)". (p(t)x")" = 0, ax(a) - ;3x"(a) = 0, /,x(b) has only the trivial solution iff p =1= o. + ox" (b). The function G( , ) in the PDF417 for Java following theorem is called the Green"s function for the BVP (5.30)-(5.32).

. 5. THE SELF-ADJOINT SECOND-ORDER DIFFERENTIAL EQUATION Theorem 5.101 (Green"s Func PDF 417 for Java tion for General Two-Point BVP) Assume the homogeneous BVP (5.30)~(5.

32) has only the trivial solution. For each fixed s E [a, b], let u( , s) be the solution of the BVP. Lu au(a,s)-(3u"(a,s) /,u(b, s). (5.41) (5.42) (5.

43). + au" (b, s). -/,x(b, s) - ax" (b, s),. where x( , ) is the Cauchy function for Lx G(t,s):= {U(t,S), vet,s),. = O.. Define (5.44). if a ::::; t::::; s::::; b if a ::::; s::::; t::::; b,. where vet, s) := u(t, s) + barcode pdf417 for Java x(t, s), for t, s [a, b]; then x(t) :=. [a, b]. Assume h is continuous on G(t, s)h(s) ds,. for t E [a, b], defines the unique solution of the nonhomogeneous BVP Lx = h(t), (5.31), (5.32).

Furthermore, for each fixed s E [a,b], v( ,s) is a solution of Lx = 0 satisfying the boundary condition (5.32). Proof The existence and uniqueness of u(t, s) is guaranteed by Theorem 5.

98. Since v(t,s):= u(t,s) +x(t,s), we have for each fixed s that v( ,s) is a solution of Lx = O. Using the boundary condition (5.

43), it is easy to see that for each fixed s, v( , s) satisfies the boundary condition (5.32). Let G(t, s) be as in the statement of this theorem and consider x(t).

ib it it it ib ib G(t, s)h(s) ds G(t, s)h(s) ds vet, s)h(s) ds [u(t, s). Ib + Ib G(t, s)h(s) ds u(t, s)h(s) ds ds + x(t, s)]h(s). u(t, s)h(s) ds u(t, s)h(s) ds u(t, s)h(s) ds x(t, s)h(s) ds + z(t),. where, by the variation of constants formula (Theorem 5.22), z(t) .x(t, s )h(s) ds defines the solution of the IVP.

Lz = h(t),. z(a) = 0,. z"(a) = O. GREEN"S FUNCTIONS Hence Lx(t). ib ib ib ib Lu(t, s)h(s) ds Lu(t, s)h(s) ds + Lz(t). + h(t). h(t).. Thus x is a solution of Lx h(t). Note that ax(a) - /3x"(a). [au (a, s) - /3u"(a, s)) h(s) ds [au(a, s) - /3u"(a, s)) h(s) ds + az(a). - /3z"(a). since for each fixed s E [a , b], u(-, s) satisfies the boundary condition (5.42). Therefore, x satisfies the boundary condition (5.

31). It remains to show that x satisfies the boundary condition (5.32).

Earlier in this proof we had that. x(t) =. It follows that x(t). ib ib ib ib 1 u(t, s)h(s) ds x(t, s)h(s) ds. [vet, s) - x(t, s)]h(s) ds vet, s)h(s) ds vet, s)h(s) ds vet, s)h(s) ds x(t, s)h(s) ds x(t, s)h(s) ds x(t, s)h(s) ds + wet),. where, by the variation of pdf417 2d barcode for Java constants formula (Theorem 5.22), wet) .x(t, s)h(s) ds defines the solution of the IVP.

Lw = h(t),. Hence web) = 0,. w"(b) =. ")"x(b). + 8x" (b). ib ib [,v(b, s) [,v(b, s). + 8v"(b, s)) h(s) ds + ")"w(b) + 8w"(b) + 8v"(b, s)] h(s) ds since for each fixed s E [a PDF417 for Java , b], v( , s) satisfies the boundary condition (5.32). Hence x satisfies the boundary condition (5.

32). 0.
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