This tag is for questions about a Green's function which is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions.
Green's function is a function of many variables associated with integral representation of solution of a boundary problem for a differential equation.
Generally speaking, a Green's function is an integral kernel that can be used to solve differential equations from a large number of families including simpler examples such as ordinary differential equations with initial or boundary value conditions, as well as more difficult examples such as inhomogeneous partial differential equations (PDE) with boundary conditions.
In the general case of a linear boundary problem with homogeneous boundary conditions$$L\phi(x)=f(x),~~~~x\in D\tag1$$$$\Gamma_i\phi(x)=0,~~~~i=1,2,\cdots,I,~~x\in S\tag2$$where $~Γ_i φ(x)~$ are linear homogeneous functions of $~φ(x)~$ and its derivatives on the boundary $~S~$ of domain $~D~$. An inverse transformation (if it exists) of the form $$\phi(x)=\int_D G(x,\xi)dv\tag3$$uses Green's function $~G(x, ξ)~$ as a kernel for the given problem, Eq. (1) & (2).
Equation (3) describes the solution as a superposition of elementary solutions which can be interpreted as point sources or power pulses $~f(ξ) δ(x, ξ)~$ at the point $~x = ξ~$ (where $~δ(x, ξ)~$ is the Dirac delta function).
The function $~G(x, ξ)~$ of the argument $~x~$ must satisfy the homogeneous boundary condition $(2)$, and also the equation$$LG(x,\xi)=0,~~~\text{for}~~~x\ne \xi\tag4$$ and the condition$$\int_DLG(x,\xi)dv=1\tag5$$or, as generalized function, the equation$$LG(x,\xi)=\delta(x,\xi)\tag6$$If the operator $~L~$ is self-conjugate, Green's function $~G(x, ξ)~$ is symmetric, i.e., $~G(x, ξ) = G(ξ, x)~$. For a boundary problem for a linear ordinary differential equation$$L\phi\equiv a_n(x)\frac{d^n\phi}{dx^n}+\cdots+a_1\frac{d\phi}{dx}+a_0=f(x)\tag7$$the general solution on the section $~[a, b]~$ can be presented in the form$$\phi=\int_a^bG(x,\xi)f(\xi)d\xi+\sum_{k=1}^n C_k\phi_k(x).\tag8$$where {\phi_k} is the functional system of solutions of a homogeneous equation $~L(φ) = 0, ~~C_k~$ are arbitrary constants obtained from boundary conditions.
It often appears possible to determine Green's function so that a particular solution$$\int_a^bG(x,\xi)f(\xi)dv$$satisfies the given boundary conditions. Such Green's function must have a jump of $~(n – 1)^{th}~$ derivative for $~x = ξ~$
$$\left|\frac{\partial^{n-1}G}{\partial x^{n-1}}\right|_{x\to \xi^{+}}-\left|\frac{\partial^{n-1}G}{\partial x^{n-1}}\right|_{x\to \xi^{-}}=\frac{1}{a_n(\xi)}$$
Applications:
In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.
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