Bulletin of TICMI Vol. 2, No. 1, 216, 25 31 b Green s Function for the Light Scattering Equations Dazmir Shulaia a,b and Giorgi Makatsaria c a Georgian Technical University 77 M. Kostava St., 175, Tbilisi, Georgia I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University 2 University St., 186, Tbilisi, Georgia; dazshul@yahoo.com c Saint Andrew The First Called Georgian University of Patriarchate of Georgia 53a, I. Chavchavadze Av., 162, Tbilisi, Georgia; giorgi.makatsaria@gmail.com (Received February 22, 216; Revised May 11, 216; Accepted June 6, 216) Abstract. The aim of this paper is to construct Green s function in an infinite medium for the light scattering equation. To this end the method of spectral resolution of the solutions by the eigenfunctions of the corresponding characteristic equation is used. Keywords: Homogeneous, Characteristic equation, Regular eigenvalues, Singular eigenfunctions, Boundary condition. AMS Subject Classification: 45B5, 45E5. We consider the equation which occurs when investigating one important problem of mathematical physics, namely the equation which describes the light scattering. The considered equation has the form I(τ,, x) τ = α(x)i(τ,, x) λ + 2 Aα(x) α(x )dx I(τ,, x )d (1) τ, x (, + ), (, +1) where α(x) is a continuous, integrable, positive function, A is a normalizing multiplier A + α(x)dx = 1, Many weel known authors [1-3] investigated this equation. We seek the solution of this equation in the following form I(τ,, x) = e τ/ν φ ν (, x). Corresponding author. Email: dazshul@yahoo.com ISSN: 1512-82 print c 216 Tbilisi University Press
26 Bulletin of TICMI With this assumption, Eq. (1) becomes (να(x) )φ ν (, x) = λ 2 Aν + α(x)α(x )φ ν (, x )d dx (2) where ν is a parameter, so-called characteristic equation. It is very convenient to normalize φ ν so that Then the above becomes + α(x)φ ν (, x)ddx = 1. (να(x) )φ ν (, x) = λ 2 Aνα(x) From this point the conventional argument runs as follows φ ν (, x) = λ 2 A να(x) να(x). Inserting this result into Eq.(2) yields the condition Λ(ν) 1 λ 2 A + να 2 (x) ddx =. (3) να(x) There are, as is known [4], for λ > 1, two regular purely imaginary eigenvalues of the characteristic equation (2). Here they will be denoted by ±ν. For λ < 1 the regular eigenvalues is absent. The two roots ±ν occur. With the normalization the corresponding solutions of the initial equation are where I ±(τ,, x) = e ±τ/ν φ ±(, x). (4) φ ± (, x) = λ 2 A ν α(x) ν α(x) (5) The argument has given the usual solutions of the homogeneous light scattering equation. However, there are others. It is to see that I(τ,, x) = + e τ/ν φ ν,(ζ) (, x)u(ν, ζ)dζdν (6) τ (, ) (, +1), x (, + ).
Vol. 2, No. 1, 216 27 where φ ν,(ζ) (, x) = λ 2 Aνρ(ν)α(x)α(ζ) να(x) (7) + (δ(ζ x) λ2 ) A νρ(ν)α(x)α(ζ) να(x) d δ(να(x) ) ν (, + ), ζ. is solution of the initial equation (1). Here, ρ (ν) = 1 λ 2 A + να 2 (x)θ( ν α(x) 1) ddx, να(x) = {x (, + ) : ν α(x) < 1}, u(ν, ζ) is a continuous, integrable function satisfying H condition [6] with respect to variable ν To summarize: There are, when λ > 1, two discrete eigenfunctions given by Eqs. (5) and also the class of the so-called singular eigenfunctions given by Eqs. (7). Now consider the equation which is named the adjoint of the characteristic equation (να(x) )φ ν(, x) = λ 2 Aν + where ν is a parameter. Now, it is very convenient to normalize φ ν so that α 2 (x)φ ν(, x )d dx (8) Then + φ ν(, x)ddx = 1. (να(x) )φ ν(, x) = λ 2 Aνα2 (x) From this gives φ ν(, x) = λ 2 A να2 (x) να(x). Inserting this result into Eq.(8) we have also the condition Λ(ν) 1 λ 2 A + να 2 (x) ddx =. να(x)
28 Bulletin of TICMI Consequently, also for the adjoint characteristic equation, when λ > 1, there are, two regular eigenfunctions and a class of singular eigenfunctions φ ν,(ζ) (, x) = λ 2 Aνρ(ν)α2 (x) να(x) (9) + (δ(ζ x) λ2 A νρ(ν)α 2 ) (x) να(x) d δ(να(x) ) ν (, + ), ζ, = {x (, + ) : ν α(x) < 1} The usefulness of these functions arises from the fact that they are both biorthogonal and complete. This can be stated in the form of theorems Theorem 1 : + α(x) φ ν(, x)φ ν (, x)ddx =, ν ν (1) Proof : φ ν and φ ν satisfy the equations ( 1 1 ) φ ν(, x) = λ2 + ν α(x) Aν α(x)φ ν(, x )d dx ( 1 1 ) ν φ ν (, x) = λ2 + α(x) Aν α (x)φ ν (, x )d dx Multiplying the first of these by φ ν (, x), the second by φ ν(, x), subtracting, and integrating, we get ( 1 ν 1 ) + ν α(x) φ ν(, x)φ ν (, x)ddx =. It is seen that if g(ν, ζ, ζ) is an arbitrary integrable function then φ ν,(ζ) (, x) = φ ν,(ζ) (, x) + g(ν, ζ, ζ )φ ν,(ζ )(, x)dζ is also a singular eigenfunction of the adjoint equation. Moreover, if S(ν, ζ, x) = π 2 ν 2 λ 2 4 A2 ρ 2 (ν)α(x)α 2 (ζ )dζ
λ 2 A Vol. 2, No. 1, 216 29 νρ(ν)α(x)α(ζ ) να(ζ ) d λ 2 A νρ(ν)α 2 (ζ ) να(ζ ) d dζ + λ 2 A νρ(ν)α(x)α(ζ) να(ζ) d + λ 2 A and g(ν, ζ, x) is the solution of the equation νρ(ν)α 2 (x) να(x) d, g(ν, ζ, x) S(ν, ζ, x)g(ν, ζ, ζ )dζ = S(ν, ζ, x) ν (, + ) ζ then φ ν,(ζ)(, x) will also be singular eigenfunction of the adjoint characteristic equations and the following equality (cf.[5]) holds. + α(x) φ ν,(ζ )(, x)φ ν,(ζ) (, x)ddx = δ(ν ν)δ(ζ ζ) Theorem 2: The functions φ and φ ν,(ζ),, < ν < + are complete for ± functions ψ(, x) defined in < < +1, < x < + satisfying H condition [6] with respect to variable, integrable with respect to x, i.e. where ψ(, x) = a ±φ ±(, x) + a ± = 1 N ± + + φ ν,(ζ) (, x)u(ν, ζ)dζdν α(x) φ ±(, x)ψ(, x)ddx, u(ν, ζ) = + α(x) φ ν,(ζ) (, x)ψ(, x)ddx.
3 Bulletin of TICMI From this theorem there follows correctness of the formula α(x)δ( )δ(x x ) = 1 φ ±(, x)φ N ±(, x ) ± + + φ ν,(ζ) (, x) φ ν,(ζ) (, x )dζdν, (, 1) x, x (, + ) which can be useful when investigating the problems of the point sources. Theorem 3 : Every solution of the equation (1) can be represented in the form I(τ,, x) = c ±e ±τ/ν φ ±(, x) (11) + + e τ/ν φ ν,(ζ) (, x)v(ν, ζ)dζdν τ (, + ) (, +1), x (, + ) where c ± and v are defined uniquely by I, and Vice versa, this expansion is a formal solution of equation (1) for arbitrary c ± and v, moreover this expansion gives us a general presentation of solutions of equation (1) for arbitrary c ± and v guaranteeing convergence of the integrals in the right part of formula (11). As an illustration of the applicability of the results Green s function for the light scattering equation will be constructed. To be definite λ < 1 is assumed here. Green s function I g (τ,, x) satisfies the equation I g(τ,, x) τ = α(x)i g (τ,, x) λ + 2 Aα(x) α(x )dx I g (τ,, x )d (12) +δ(τ)δ( )δ(x x ) τ, x (, + ), (, +1) Integrating across the plane τ = shows that I g satisfies the homogeneous equation (1) for τ and the jump condition (see [1]). (I g ( +,, x) I g (,, x)) = α(x)δ( )δ(x x )
Vol. 2 No. 1, 216 31 Let us look for the solution I g which vanishes as τ. It is sufficient to expand I g in the form I g = + e τ/ν φ ν,(ζ) (, x)u(ν, ζ)dζdν, τ < or I g = e τ/ν φ ν,(ζ) (, x)u(ν, ζ)dζdν, τ > The jump condition then gives an integral equation to determine the expansion coefficients. It is + α(x)δ( )δ(x x ) = φ (ν,(ζ) (, x)u(ν, ζ)dζdν. The solution obtained using the orthogonality relations is u(ν, ζ) = φ ν,(ζ) (, x ) Hence I g can be written in the typical normal mode expansion I g = + e τ/ν φ ν,(ζ) (, x )φ ν,(ζ) (, x)dζdν τ <, (, +1) x, x (, + ) I g = e τ/ν φ ν,(ζ)(, x )φ ν,(ζ) (, x)dζdν τ >, (, +1) x, x (, + ). References [1] K.M. Case, On Wiener-Hopf equations, Ann. Phys., 2 (1957), 384 [2] S. Chavsrasekhar, Radiative Transfer, Clarendon Press, Oxford, 195 [3] M.G. Krein, Integral equations in half line whit kernel depending difference of arguments. Uspechi Mathem. nauk., 13, 5, 3 (1958) [4] I. Ershov, S. Shichov, The Methods of Solving of the Boundary Values Problems of Transport Theory (Russian), M. Atomizdat, 1977 [5] D. Shulaia, On the expansion of solutions of equations of the linear multivelocity transport theory by eigenfunctions of the characteristic equation (Russian), Doklady Akademii Nauk SSSR 31, 4 (199), 844-849 [6] N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953