Green's function

In mathematics, a Green's function (sometimes improperly termed a Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.In quantum field theory, Green's functions take the roles of propagators.A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space, at a point s, is any solution of where δ is the Dirac delta function.However, in practice, some combination of symmetry, boundary conditions and/or other externally imposed criteria will give a unique Green's function.The Green's function as used in physics is usually defined with the opposite sign, instead.has constant coefficients with respect to x, then the Green's function can be taken to be a convolution kernel, that is,In this case, Green's function is the same as the impulse response of linear time-invariant system theory.The problem now lies in finding the Green's function G that satisfies equation 1.This can be thought of as an expansion of f according to a Dirac delta function basis (projecting f overThe primary use of Green's functions in mathematics is to solve non-homogeneous boundary value problems.Therefore if the homogeneous equation has nontrivial solutions, multiple Green's functions exist.The terminology advanced and retarded is especially useful when the variable x corresponds to time.In such cases, the solution provided by the use of the retarded Green's function depends only on the past sources and is causal whereas the solution provided by the use of the advanced Green's function depends only on the future sources and is acausal.In these problems, it is often the case that the causal solution is the physically important one.The use of advanced and retarded Green's function is especially common for the analysis of solutions of the inhomogeneous electromagnetic wave equation.Applying the operator L to each side of this equation results in the completeness relation, which was assumed.A further identity follows for differential operators that are scalar polynomials of the derivative,This process yields identities that relate integrals of Green's functions and sums of the same.While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, whenThe following table gives an overview of Green's functions of frequently appearing differential operators, where[2] Where time (t) appears in the first column, the retarded (causal) Green's function is listed.Suppose the problem is to solve for φ(x) inside the region.reduces to simply φ(x) due to the defining property of the Dirac delta function and we haveHowever, application of Gauss's theorem to the differential equation defining the Green's function yieldsSupposing that the bounding surface goes out to infinity and plugging in this expression for the Green's function finally yields the standard expression for electric potential in terms of electric charge density asFirst step: The Green's function for the linear operator at hand is defined as the solution to If, then the delta function gives zero, and the general solution isOne can ensure proper discontinuity in the first derivative by integrating the defining differential equation (i.e., Eq.Note that we only integrate the second derivative as the remaining term will be continuous by construction.
An animation that shows how Green's functions can be superposed to solve a differential equation subject to an arbitrary source.
If one knows the solution to a differential equation subject to a point source and the differential operator is linear, then one can superpose them to build the solution for a general source .
fundamental solutionmathematicsimpulse responseinhomogeneousdifferential operatorDirac's delta functionconvolutionsuperposition principlelinear ordinary differential equationdelta functionsmathematicianGeorge Greenpartial differential equationsfundamental solutionsmany-body theoryphysicsquantum field theoryaerodynamicsaeroacousticselectrodynamicsseismologystatistical field theorycorrelation functionspropagatorsdistributionsEuclidean spaceDirac delta functionkernelsymmetryboundary conditionsGreen's function numberfunctionswave equationsdiffusion equationsquantum mechanicsHamiltoniandensity of statestranslation invariantconstant coefficientsconvolution kernellinear time-invariant system theorySpectral theoryVolterra integral equationprojectionFredholm integral equationFredholm theoryboundary value problemstheoretical physicsFeynman diagramscorrelation functionSturm–Liouvillecontinuous functionDerivativeGreen's function (many-body theory)propagatorcausalinhomogeneous electromagnetic wave equationdimensional analysisvolume elementspacetimed'Alembert operatoreigenvectorseigenvaluescompleteness relationfunction spacesmethod of imagesseparation of variablesLaplace transformsfundamental theorem of algebracommutes with itselfFourier transformpartial fraction decompositionHeaviside step functionBessel functionmodified Bessel function of the first kindmodified Bessel function of the second kindPoisson equationHelmholtz operatorHankel function of the second kindspherical Hankel function of the second kindSchrödinger equationfree particleYukawa potentialScreened Poisson equationwave equationdiffusionKlein–Gordon equationtelegrapher's equationrelativistic heat conductionLaplacianGreen's identitiesdivergence theoremGauss's theoremGreen's theoremLaplace's equationPoisson's equationNeumannDirichletharmonic functionselectrostaticselectric potentialelectric chargedensitysurface integralGreen's function for the three-variable Laplace equationDirichlet boundary conditionNeumann boundary conditionTaylor seriesBessel potentialTransfer functionGreen's function in many-body theoryParametrixResolvent formalismKeldysh formalismMultiscale Green's functiontrivialhomogeneousFolland, G.B.Lakhtakia, A.BibcodeEncyclopedia of MathematicsEMS PressWeisstein, Eric W.MathWorldPlanetMathWayback MachineBrady HaranUniversity of Nottingham