{"id":2185,"date":"2021-02-14T13:04:38","date_gmt":"2021-02-14T07:34:38","guid":{"rendered":"https:\/\/soulofmathematics.com\/?page_id=2185"},"modified":"2021-02-14T18:40:56","modified_gmt":"2021-02-14T13:10:56","slug":"greens-function","status":"publish","type":"page","link":"https:\/\/soulofmathematics.com\/index.php\/greens-function\/","title":{"rendered":"GREEN’S FUNCTION"},"content":{"rendered":"\n

Green’s functions are named after the British mathematician George Green<\/a>, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green’s functions are studied largely from the point of view of fundamental solutions instead.<\/p><\/blockquote><\/figure>\n\n\n\n

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The term is also used in physics, specifically in quantum field theory, aerodynamics, aeracoustics, 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.<\/p><\/blockquote>\n\n\n\n

<\/p>\n<\/div><\/div>\n\n\n\n

Preliminary ideas and motivation<\/h2>\n\n\n\n

The delta function<\/h4>\n\n\n\n

Definition- The \u03b4-function is defined by the following three properties,<\/p>\n\n\n\n

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where f is continuous at x = a. The last is called the shifting property of the \u03b4-function. To make proofs with the \u03b4-function more rigorous, we consider a \u03b4-sequence, that is, a sequence of functions that converge to the \u03b4-function, at least in a pointwise sense. Consider the sequence<\/p>\n\n\n\n

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Note that,<\/p>\n\n\n\n

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The 2D \u03b4-function is defined by the following three properties,<\/p>\n\n\n\n

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Finding the Green\u2019s function<\/h2>\n\n\n\n
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To find the Green\u2019s function for a 2D domain D, we first find the simplest function that satisfies \u22072v = \u03b4(r). Suppose that v (x, y) is axis-symmetric, that is, v = v (r). Then<\/p>\n\n\n\n

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For r > 0,<\/p>\n\n\n\n

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Integrating gives<\/p>\n\n\n\n

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For simplicity, we set B = 0. To find A, we integrate over a disc of radius \u03b5 centered at(x, y),D\u03b5,<\/p>\n\n\n\n

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From the Divergence Theorem, we have<\/p>\n\n\n\n

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where C\u03b5 is the boundary of D\u03b5, i.e. a circle of circumference 2\u03c0\u03b5. Combining the previous two equations gives<\/p>\n\n\n\n

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Hence<\/p>\n\n\n\n

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This is called the fundamental solution for the Green\u2019s function of the Laplacian on 2D domains. For 3D domains, the fundamental solution for the Green\u2019s function of the Laplacian is \u22121\/(4\u03c0r), where r = (x \u2212 \u03be)2<\/sup> + (y \u2212 \u03b7)2<\/sup> + (z \u2212 \u03b6)2<\/sup>. The Green\u2019s function for the Laplacian on 2D domains is defined in terms of the corresponding fundamental solution,<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n

The term \u201cregular\u201d means that h is twice continuously differentiable in(\u03be, \u03b7)on D. Finding the Green\u2019s function G is reduced to finding a C2 function h on D that satisfies<\/p>\n\n\n\n

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The definition of G in terms of h gives the BVP for G. Thus, for 2D regions D, finding the Green\u2019s function for the Laplacian reduces to finding h.<\/p>\n\n\n\n

Examples<\/h3>\n\n\n\n