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Consider the differential equation Ly=f(z), (1) where I is a differential operator on a function space defined for z € (a, b), a and 6 are suitable boundary points, f(x) is a source function, and y(z) is the required solution subject to boundary conditions imposed at one or both of the boundary points. The Green's function, G(x, t), is defined to be the solution of the equation LG(x, t) = 8(z-t), where the operator L is understood to operate only on the x-coordinate, 6(x-t) is the Dirac delta, and G(x, t) satisfies the same boundary conditions as does y(x). a) Show that y(x)=G(x, t)f(t)dt is the solution to eq. (1). b) How are the boundary conditions included in the above solution for y(x)? Which part of the above solution y(z) would change if the boundary conditions imposed at a and & changed?

Consider the differential equation Ly=f(z), (1) where I is a differential operator on a function space defined for z € (a, b), a and 6 are suitable boundary points, f(x) is a source function, and y(z) is the required solution subject to boundary conditions imposed at one or both of the boundary points. The Green's function, G(x, t), is defined to be the solution of the equation LG(x, t) = 8(z-t), where the operator L is understood to operate only on the x-coordinate, 6(x-t) is the Dirac delta, and G(x, t) satisfies the same boundary conditions as does y(x). a) Show that y(x)=G(x, t)f(t)dt is the solution to eq. (1). b) How are the boundary conditions included in the above solution for y(x)? Which part of the above solution y(z) would change if the boundary conditions imposed at a and & changed?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the differential equation Ly = f(z), (1) where i is a differential operator on a function space defined for z € [a, b], a and b are suitable boundary points, f(x) is a source function, and y(x) is the required solution subject to boundary conditions imposed at one or both of the boundary points. The Green's function, G(x,t), is defined to be the solution of the equation LG(1,t) = 6(z – t), where the operator L is understood to operate only on the r-coordinate, ő(x – t) is the Dirac delta, and G(r, t) satisfies the same boundary conditions as does y(r). a) Show that y(z) =- S, G(x,t)f(t)dt is the solution to eq. (1). b) How are the boundary conditions included in the above solution for y(x)? Which part of the above solution y(x) would change if the boundary conditions imposed at a and b changed?
Transcribed Image Text:Consider the differential equation Ly = f(z), (1) where i is a differential operator on a function space defined for z € [a, b], a and b are suitable boundary points, f(x) is a source function, and y(x) is the required solution subject to boundary conditions imposed at one or both of the boundary points. The Green's function, G(x,t), is defined to be the solution of the equation LG(1,t) = 6(z – t), where the operator L is understood to operate only on the r-coordinate, ő(x – t) is the Dirac delta, and G(r, t) satisfies the same boundary conditions as does y(r). a) Show that y(z) =- S, G(x,t)f(t)dt is the solution to eq. (1). b) How are the boundary conditions included in the above solution for y(x)? Which part of the above solution y(x) would change if the boundary conditions imposed at a and b changed?
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