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Example
Created on 2024-06-20Asked by Emma Brown (Solvelet student)
Find the Green's function for the differential operator L[y]=y′′−y L[y] = y'' - y with boundary conditions y(0)=0 y(0) = 0 and y(1)=0 y(1) = 0 .

Solution

To find the Green's function for the differential operator L[y]=y′′−y L[y] = y'' - y with boundary conditions y(0)=0 y(0) = 0 and y(1)=0 y(1) = 0 : The Green's function G(x,ξ) G(x, \xi) satisfies: L[G(x,ξ)]=δ(x−ξ) L[G(x, \xi)] = \delta(x - \xi) where δ \delta is the Dirac delta function. For 0≤x<ξ 0 \leq x < \xi : G(x,ξ)=Aex+Be−x G(x, \xi) = A e^x + B e^{-x} Applying G(0,ξ)=0 G(0, \xi) = 0 : A+B=0 A + B = 0 B=−A B = -A G(x,ξ)=A(ex−e−x) G(x, \xi) = A (e^x - e^{-x}) For ξ<x≤1 \xi < x \leq 1 : G(x,ξ)=Cex+De−x G(x, \xi) = C e^x + D e^{-x} Applying G(1,ξ)=0 G(1, \xi) = 0 : Ce+De−1=0 C e + D e^{-1} = 0 C=−De−2 C = -D e^{-2} G(x,ξ)=D(e−2ex−e−x) G(x, \xi) = D (e^{-2} e^x - e^{-x}) Applying continuity at x=ξ x = \xi : A(eξ−e−ξ)=D(e−2eξ−e−ξ) A (e^\xi - e^{-\xi}) = D (e^{-2} e^\xi - e^{-\xi}) A=De−2 A = D e^{-2} G(x,ξ)=e−2(ex−e−x) for 0≤x<ξ G(x, \xi) = e^{-2} (e^x - e^{-x}) \text{ for } 0 \leq x < \xi G(x,ξ)=e−2(eξ−e−ξ) for ξ<x≤1 G(x, \xi) = e^{-2} (e^\xi - e^{-\xi}) \text{ for } \xi < x \leq 1 Solved on Solvelet with Basic AI Model
Some of the related questions asked by Samuel Mitchell on Solvelet
1. Find the Green's function for the one-dimensional heat equation ∂u∂t−k∂2u∂x2=0\frac{\partial u}{\partial t} - k\frac{\partial^2 u}{\partial x^2} = 0 with Dirichlet boundary conditions u(0,t)=u(L,t)=0u(0, t) = u(L, t) = 0,2. Use the Green's function to solve the one-dimensional wave equation ∂2u∂t2−c2∂2u∂x2=0\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0 subject to the initial conditions u(x,0)=f(x)u(x, 0) = f(x) and ∂u∂t(x,0)=g(x)\frac{\partial u}{\partial t}(x, 0) = g(x).,
DefinitionMathematically they are used to solve inhomogeneous differntial equations that has some particular boundary conditions. They're are the impulse response of a linear system and they used for convert solution of the differential equation to an integral representation. Green's functions are used as a mathematical representation of the operation of a linear differential operator, or more generally by a linear operator. They are used in physics, engineering, and other fields to solve certain properties functions of the Green's function can be used to represent the behavior of certain operating systems or material properties. Eg: There is a differential operator L, an inhomogeneous term f(x), then solution u(x) can be written as u(x)=∫G(x,x′)f(x′)dx′, here G(x,x′) is the Greens function for L
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