Explanation Given: The given boundary value problem is: y ″ + 4 y = 0 ; y ( 0 ) = y ( π ) , y ′ ( 0 ) = y ′ ( π ) Procedure used: Let L be the differential operator given by the L [ y ] = A 2 y ″ + A 1 y ′ + A 0 y . The formal adjoint of L is the differential operator L + define by L + [ y ] = ( A 2 y ) ′ ′ − ( A 1 y ) ′ + A 0 y provided A 2 ′ ′ , A 1 ′ and A 0 are continuous on [ a , b ] . The Green’s formula: P ( u , v ) | a b = 0 , for u in D ( L ) , for v in D ( L + ) . Here P ( u , v ) , the bilinear concomitant associated with L , is define as P ( u , v ) = u A 1 v − u ( A 2 v ) ′ + u ′ A 2 v Let B [ u ] = 0 represent the boundary conditions required for a function to be in the domain of L and let B + [ u ] = 0 be the boundary conditions required for a function to be in the domain of its adjoint operator L + . Then, the adjoint boundary value problem for L [ u ] = 0 , B [ u ] = 0 is the boundary value problem L + [ u ] = 0 , B + [ u ] = 0 Calculation: Consider the given boundary value problem: y ″ + 4 y = 0 Here A 0 = 4 , A 1 = 0 and A 2 = 1 . Substitute in the formal adjoint formula: L + [ y ] = ( 1 ⋅ y ) ′ ′ − ( 0 ⋅ y ) ′ + 4 y = y ″ + 4 y Apply the Green’s formula: P ( u , v ) ( x ) | 0 π = 0 for all u in D ( L ) and v in D ( L + ) . Apply the formula of bilinear concomitant:

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 11.4, Problem 13E

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To find:

The adjoint boundary value problem for the given boundary value problem y″+4y=0;y(0)=y(π), y′(0)=y′(π).

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Chapter 11.4, Problem 12E

Chapter 11.4, Problem 14E

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The adjoint boundary value problem for the given boundary value problem y ″ + 4 y = 0 ; y ( 0 ) = y ( π ) , y ′ ( 0 ) = y ′ ( π ) .

Solution Summary: The author explains that the adjoint boundary value problem is ly, Y(0)=y

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To find:

The adjoint boundary value problem for the given boundary value problem y″+4y=0;y(0)=y(π), y′(0)=y′(π).

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Chapter 11 Solutions

The adjoint boundary value problem for the given boundary value problem y ″ + 4 y = 0 ; y ( 0 ) = y ( π ) , y ′ ( 0 ) = y ′ ( π ) .

Solution Summary: The author explains that the adjoint boundary value problem is ly, Y(0)=y

Videos

To find: The adjoint boundary value problem for the given boundary value problem y″+4y=0;y(0)=y(π), y′(0)=y′(π).

Want to see the full answer?

Chapter 11 Solutions

To find:

The adjoint boundary value problem for the given boundary value problem y″+4y=0;y(0)=y(π), y′(0)=y′(π).