Explain example 1.11.5 G(r.) = -log[(x - 5) + (y-n)]. (1.11.52) This is the Green's function for

Explain example 1.11.5

  

 

  

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G(r.) = -log[(x - 5)² + (y-n)²]. (1.11.52) This is the Green's function for the two-dimensional Poisson equation V² = -f(x, y). Thus, the solution of the Poisson equation is d§, u(x, y) = G(r. E) ƒ (E) dE. where G(r.) is given by (1.11.52). (1.11.53) Example 1.11.5 (Green's function for the Three-Dimensional Helmholtz Equation). We consider the three-dimensional wave equation [u-c²V²u] =q(r.1)., (1.11.54) where q(r,t) is a source. If q(r.t) =q(r)exp(-iar) represents a source oscillating with a single frequency , then, as expected, at least after an initial transient period, the entire motion reduces to a wave motion with the same frequency w so that we can write u(r,t)= u(r)exp(-iat). Consequently, the wave equation (1.11.54) reduces to the three-dimensional Helmholtz equation -(V²+k²)u(r) = f(r), (1.11.55) where k = and f(r)=c2q(r). The function u(r) satisfies this equation in some domain DCR with boundary JD, and it also satisfies some prescribed boundary conditions. We also assume that u(r) satisfies the Sommerfeld radiation condition which simply states that the solution behaves like outgoing waves generated by the source. In the limit as w→0, so that k→0 and f(r) can be interpreted as a heat source, the equation (1.11.55) results into a three- dimensional Poisson equation. The solution u(r) would represent the steady temperature distribution in D due to the heat source f(r). However, in general, u(r) can be interpreted as a function of physical interest. We construct a Green's function G(r.) for equation (1.11.55) so that G(r.) satisfies the equation -(V²+k²) G=8(x) 8(y) 8(z). (1.11.56) Using the spherical polar coordinates, the three-dimensional Laplacian can be expressed in terms of radial coordinater only so that (1.11.56) assumes the form 89 G(r.) = -log[(x - 5)² + (y-n)²]. (1.11.52) This is the Green's function for the two-dimensional Poisson equation V² = -f(x, y). Thus, the solution of the Poisson equation is d§, u(x, y) = G(r. E) ƒ (E) dE. where G(r.) is given by (1.11.52). (1.11.53) Example 1.11.5 (Green's function for the Three-Dimensional Helmholtz Equation). We consider the three-dimensional wave equation [u-c²V²u] =q(r.1)., (1.11.54) where q(r,t) is a source. If q(r.t) =q(r)exp(-iar) represents a source oscillating with a single frequency , then, as expected, at least after an initial transient period, the entire motion reduces to a wave motion with the same frequency w so that we can write u(r,t)= u(r)exp(-iat). Consequently, the wave equation (1.11.54) reduces to the three-dimensional Helmholtz equation -(V²+k²)u(r) = f(r), (1.11.55) where k = and f(r)=c2q(r). The function u(r) satisfies this equation in some domain DCR with boundary JD, and it also satisfies some prescribed boundary conditions. We also assume that u(r) satisfies the Sommerfeld radiation condition which simply states that the solution behaves like outgoing waves generated by the source. In the limit as w→0, so that k→0 and f(r) can be interpreted as a heat source, the equation (1.11.55) results into a three- dimensional Poisson equation. The solution u(r) would represent the steady temperature distribution in D due to the heat source f(r). However, in general, u(r) can be interpreted as a function of physical interest. We construct a Green's function G(r.) for equation (1.11.55) so that G(r.) satisfies the equation -(V²+k²) G=8(x) 8(y) 8(z). (1.11.56) Using the spherical polar coordinates, the three-dimensional Laplacian can be expressed in terms of radial coordinater only so that (1.11.56) assumes the form 89