- Email: [email protected]
The construction of complete systems in the theory of diffraction
200 there was an area of highest variation of all required parameters and the greatest difference (up to 20%) in the data obtained for algorithms 1-3. In the remaining part of the calculated region this difference did not exceed 10% (the value of the deviation relatestotheparameters in the vicinity of the front edge of the plate). Fig.2 shows the pressure distribution over the upper and lower surfaces of the plate (the dot-dash line corresponds to algorithm 1, the solid line to algorithm 2 and the dashed line to 3). Here, when the agreement is comparatively good for data obtained by algorithms 2 and 3, a difference with number 1 was observed. Analysis showed that this difference was due to the gas-dynamic stage of algorithm 1, which greatly smooths the derived functions near the plate, in a direction perpendicular to it. Algorithms 1 and 2 were also compared for the following parameter values: v,=o.3, T,=S, Re=loo (the remaining parameters were the same as in the previous example). Fig.3 shows the isobars and Fig.4 shows the temperature distribution. Here, the difference in data obtained using the algorithms amounted to 7%. REFERENCES 1. KRIVTSOV V.M., NAUMOVA I.N., CHARAKHAN A.A. et al., Comparison of axisymmetric flows of a radiating gas. Zh. vychisl. Mat. mat. Fiz., 17, 4, 1077-1081, 1977. 2. ROUCH P., Computer hydrodynamics M.: Mir, 1980. 3. MacCORMACK R.W., A numerical method for solving the equations of compressible viscous flow. AIAA J, 20, 1275-1281, 1982. 4. ZHMAKIN A.I. and FURSENKO A.A., A monotonic difference scheme for through calculation. Zh. vychisl. Mat. mat. Fiz., 20, 4, 1021-1031, 1980. 5. KRIVTSOV V.M., A numerical scheme for solving the Navier-Stokes equations. Zh. vychisl. Mat. mat. Fiz., 26, 6, 914-923, 1986. 6. MARCHUK G.I., Methods of Computational mathematics. M.: Nauka, 1977.
Translated by C.A.M.
U.S.S.R.
Comput.Maths.Math.Phys.,
Vo1.27,No.3,pp.200-203,1987
Printed in Great Britain
0041-5553/87 $lO.CC+O.OO 01988 Pergamon Press plc
THE CONSTRUCTION OF COMPLETE SYSTEMS IN THE THEORY OF DIFFRACTION*
YU.A. EREMIN
A general scheme for constructing complete systems of functions in diffraction theory is described. The technique which has been developed enables one to obtain complete systems solely by specifying the geometry of the set of locations of the secondary sources. In recent times the method of secondary sources has become more and more widely used in the investigation of diffraction problems. At the same time, the use of functional systems, which are the simplest from the point of view of constructing algorithms based on them, to represent the fields of the secondary sources makes its application attractive due to the simplicity and reliability of the computational algorithms /l, 2/. In order that the method of secondary sources should be mathematically well founded, it is necessary to convince oneself of the completeness of the functional system which is used. A scheme for establishing this completeness is not always obvious and, in a number of cases, it is necessary to expend considerable effort in its implementation. A method for the approximate solution of certain boundary value problems in mathematical physics was proposed in /3/. In the treatment of diffraction problems the fundamental solution of the Helmholtz equation for free space is the basis for constructing systems of secondary sources. In the process of establishing the completeness of a system of secondary sources /3/, the fact that a functional relationship is obtained from the closure condition which is equivalent to the initial diffraction problem in the absence of an excitation source is the fundamental point. In the present paper we present a general scheme for establishing the completeness of the method of secondary sources which enables one to obtain a complete system of functions by specifying merely the structure of the set of locations of the secondary sources as the initial information. The use of the method which has been developed in the case of vector diffraction problems enables one to construct new complete systems of vector functions. The scheme developed is transferred to the treatment of the question of the construction of complete systems of secondary sources in the method of fundamental solutions (m.f.s.). *Zh.vychisl.Mat.mat.Fiz.,27,6,945-949,1987
201 Functional systems, the completeness of which has previously been established, are constructed for the interesting practical cases of problems of diffraction by a local scatterer in the presence of "regular" boundaries of separation of media with different characteristics. Let us consider the completeness of functional systems in LI. Then, in order to prove completeness, it sufficeatodemonstrate the closure of the system /3/. We shall study an external problem of diffraction, in the region D&i' of the field of a point source located at a point Mo=D., on a local body D,=R'\D. with a surface S=A(2.Y)and the boundary condition that the total field vanishes on S. In this case, there exists a unique classical solution of the problem u which satisfies the Sommerfeld radiation condition. By applying Green's second formula in D. to the solution of the problem and to the fundamental solution of the Helmholtz equation $(M,P), we obtain ME D,. s Ip(M,P)~uda~=$@f,M,), 6 Denoting the normal derivative on the surface S,au by l(P), we have, when there is no source of excitation. s $(M,P)f(P)dap-0,ME D,. *
(1)
Eq.(l) is a functional relationship which is equivalent to the external Dirichlet problem in the sense that its unique solvability and the validity of (1) everywhere in D, for I"&(S) leads to f-0. We shall subsequently refer to relationship (1) as the basic relationship. Let us consider the function V(M)= .F rp(M,P)f(P)dox., M=Dt. 8
(2)
In order to use (1) and (2) to construct complete systems, we first consider the closure of the functional systems rp(M,M,)with the following structure of the sets CM",,"=,: a) located everywhere densely in a volume C&CD, b) everywhere dense on a non-resonant surface Z=A('~"~located as a whole in D, c) everywhere dense on a curve of class A which, as a whole, is located in D, (see /4/). It is readily seen that, by virtue of the choice of structure of the set (M,)r-1,the conditions for the closure of these functional systems ensure that an analytical function of a real variable V(M) vanishes in the domain D,. It follows from this that these systems are complete in L%(S). Remark 1. In the case of a boundary condition ~u(s=O, it can be shown, by using the properties of the normal derivative of a double layer potential with a density from L2(S), SE AU.", (see /S/j, that functional systems of the form (c%(M.Mn)) are complete. Problem 1. To determine the form of a complete system of functions from the condition that the analytical function (2) together with all of its derivatives vanishes at a point MCFD,. Let us choose a neighbourhood do=& for the point MO and take I& as the origin of a spherical system of coordinates (r,8,cp). By virtue of the analytic nature of V, we transform in do by expanding rp in spherical harmonics, to the Fourier coefficients of the function V with respect to the variable 0 and, as a result, we obtain _ m=o,i,..., P(r,Cl)y&!"i"(kr)P?%m(eose),
(3)
where
a.“=const+o and M,,m is a system of metaharmonic functions. Let us fix an arbitrary mS0 and use the condition that the function Vm vanishes at the point MO together with all of its derivatives which, in the equivalent formulation, has the form lim[P(r,B)/rl]-0, 1=0,1,..., ve. r-0 From the latter relatinship, by virtue of the behaviour of the functions j.(z) when ~0 (see /l/), we obtain f%"=O, n--m,m+1,... (4) Relationship (4) is the condition for the closure of a system of metaharmonic functions. Hence, by selecting a single point MO as the set of locations of the secondary sources, we obtain the completeness of the system of metaharmonic Vekua functions. Problem 2.
To determine the form of a complete system of functions if (M,),~,ED+ are
202 located in a rectilinear interval d. and has a point of accumulation Ml~sDi. We note that a rectilinear interval is not a curve of class A (see /4/). Let us choose the neighbourhood of the point of accumulation dv=Di. We direct the oz axis of a cylindrical system of coordinates (p,cp, Z) along the interval d, and introduce the Fourier coefficients Ym(r,O) in do into the treatment. Let us show that the following theorem is valid. Theorem. Let wm((a)= lim[P(r, 8)/p], p-0
, ,.... m-01
Then, if {z,)L =d,*=dand. and has a point of accumulation ZO,then Wm(2")=0, n=l,2, ...=+Vm=O in do. (5) Proof. Let us use the representation (3), transform from p+O to sin&+0 and employ the expression for the associated Legendre polynomials in terms of the hypergeometric function. As a result, we obtain
By virtue of the analytical nature of W"'(z) on do'and the condition of where r,m=const+O. the theorem, we obtain (4). The theorem is proved. In order to elucidate the actual form of a functional system for which relationship (5) is the closure condition, we note that, apart from factors, the real and imaginary parts of can be combined into a complex function of the form P s S,(M,P)f(P)exp(lmcp,)da~ B where &are the function introduced by E.N. Vasil'yev /6/. By using the relationship from /6/, it is readily shown that liml&,W,P)lp~~l= PM-0
qmk~"(kRp~)Pmm(cog8~), qm=const
Hence, condition (5) implies the closure of a functional system of the form G,'"(M)= @(M,4=
k~'(kRas.,)P,m(cose~,)e~~.
(6)
Here, M=(p, cp, z),Rf =p2+(z--a,)Z. Sin 8z,=p/&.w For fixed m, system (6) constitutes a system of "lower" multipoles,nwhere each function is a combination of elementary functions. Remark 2. Complete functional systems of vector functions can be constructed in a completely analogous manner. In order to do this, it is sufficient to carry out the following constructions: a) for the boundary condition [II, E]ls=O, using Green's vector formula, we obtain the fundamental relationship of type (1) b) for the Cartesian components of an analytical vector function of type (2) which satisfy the Helmholtz equation, we carry out all the reasoning outlined above. As a result, we obtain the following complete systems of vector functions: Vxvx(M,m(P)ei),vx(MP(P)e>), I,m=O,I,..., m
vxvx(Gnm(P)eJ,
vx(G?Im(P)eJ, n=f, 2,. . ,
m=O,i,...,
where e, is a basis of unit vectors of a Cartesian coordinate system, i=i,?.3.We note that the completeness of the system of functions VxVx(G,"e,) was established in /2/ on the basis of other considerations. However, use of the scheme in /2/ to establish completeness does not make it possible to establish the completeness of a system of functions in the method of fundamental solutions. As an example, let us construct complete systems of functions of the method of fundamental solutions /?/ in the case of structure of D, which indicate that there is an ideally conducting plane or dielectric laminar medium in D ,,assuming that the boundaries of separqtion of the media with different parameters donotintersect the surface of the local body S. Let the ozaxis be perpendicular to the ideally conducting plane lI=D, and the point of accumulationof _ the set {M,),,,be Mo=DI. Problem 3. To construct a complete system for the method of fundamental solutions when the set (M,)cD, is located in an interval of the oz-axis of a cylindrical system of coordinates. Let us assume that the bounary condition on 11 has the form u(,,=O.In this case the fundamental relationship takes the form of (1) with rl, replaced by g, that is, by Green's function for an ideally conducting half-space. By applying the technique used in the solution of problem 2, we obtain a complete system of the form x""(M)=Gm(M,z.)-C~(M,z;), n=l,Z, ...I m=O, I,.... (7) where G" are the functions from (6) and zn*is the conjugate point on 0: with respect to Il. Let (M") be the same as in problem 3 and DZ be perpendicular to the boundary of
203 separation of the layers. Problem 4. TO construct a complete system of the method of fundamental solutions when there is a laminar medium in D, In this case Green's function for a laminar medium S (see /7/j is used in the fundamental relationship (1). Its Fourier coefficients can be represented in the form
gmW,P)= s
J,(hp)J,(hpp)u(z,zp,a)adh,
0
where u(z, ZP,h) is a function which ensures that the joining conditions on the boundaries of separation of media with different parameters are satisfied. By using the condition of the theorem, we have, apart from a constant factor,
lb k”‘~~,~)/p~“‘~PP-+O
Hence,
~J,(hp)h’“v(z, 0
zp, h)h dh.
in this case, the complete system of functions takes the form
U.~(M)=(G~(M,Z.)-G~(I,Z,‘)+
jJ,,,(hp)h~kdlie~” 0
The occurrence of am in theintegral (8) results from the magnitude of the singularity of the functions Gm when the arguments are identical. We note that the completeness of the functional systems (7) and (8) has not been previously established, although a numerical algorithm for investigating diffraction problems based on the method of fundamental solutions has been previouslv used successfullv in the solution of problems of diffraction bv dielectric solids of rotation located in a homogeneous half-space /8/. The author thanks A.G. Sveshnikovforhis help and interest. REFERENCES 1. EREMIN YU.A. and SVESHNIKOV A.G., Foundation of the method of non-orthogonal series and the investigation of certain inverse diffraction problems, Zh. vychisl. Mat. mat. Fiz., 23, 3, 738-742, 1983. 2. EREMIN YU.A., LEBEDEV O.A. and SVESHNIICOV A.G., The use of multipole sources in the method of non-orthogonal series in diffraction problems, Zh. vychisl. Mat. mat. Fir., 25, 3, 474-418, 1985. 3. KUPRADZE V.D., On the approximate solution of problems of mathematical physics, Uspekhi Matem. Nauk, 22, 2, 58-107, 1967. 4. KRAVTSOV V.V., Approximation of functions of many variablts by an antenna potential, Zh. vychisl. Mat. mat. Fiz., 18, 3, 672-680, 1978. 5. SVESHNIKOV A.G., EREMIN YU.A. and CORLOV N-V., The foundation of realization of the method of non-orthogonal series in problems of diffraction by dielectric bodies, Vestnik Moskovsk. Gos. Univ (MGU), Ser. 15, Vychisl. Matem. i Kibernetika, 3, 13-20, 1983. 6. VASIL'YEV E.N., GORELIKOV A.I. and FALULIN A.A., Green's tensor function in rotating coordinates, in: Collection of Scientific-MethodologicalArticles on Applied Electrodynamics, Vyssh. Shkola, Moscow, Issue 3, 3-25, 1980. 7. EREMIN YU.A., ZAKHAROV E.V. and NESMEYANOVA N.I., The method of fundamental solutions in problems of the diffraction of electromagneticwaves by bodies of rotation, in: Computational Methods and Programming, Izd. Moskovsk. Gos. Univ. (MGU), XxX11, 28-43, 1980. 8. SVESHNIKOV A.G., EREMIN YU.A. and ORLOV N.V., Investigationof some mathematical models in the theory of diffraction by the method of non-orthogonal series, Radiotekhn. i elektronika, 30, 4, 697-704, 1985.
Translated by E.L.S.
Translated by C.A.M.
U.S.S.R.
Comput.Maths.Math.Phys.,
Vo1.27,No.3,pp.200-203,1987
Printed in Great Britain
0041-5553/87 $lO.CC+O.OO 01988 Pergamon Press plc
THE CONSTRUCTION OF COMPLETE SYSTEMS IN THE THEORY OF DIFFRACTION*
YU.A. EREMIN
A general scheme for constructing complete systems of functions in diffraction theory is described. The technique which has been developed enables one to obtain complete systems solely by specifying the geometry of the set of locations of the secondary sources. In recent times the method of secondary sources has become more and more widely used in the investigation of diffraction problems. At the same time, the use of functional systems, which are the simplest from the point of view of constructing algorithms based on them, to represent the fields of the secondary sources makes its application attractive due to the simplicity and reliability of the computational algorithms /l, 2/. In order that the method of secondary sources should be mathematically well founded, it is necessary to convince oneself of the completeness of the functional system which is used. A scheme for establishing this completeness is not always obvious and, in a number of cases, it is necessary to expend considerable effort in its implementation. A method for the approximate solution of certain boundary value problems in mathematical physics was proposed in /3/. In the treatment of diffraction problems the fundamental solution of the Helmholtz equation for free space is the basis for constructing systems of secondary sources. In the process of establishing the completeness of a system of secondary sources /3/, the fact that a functional relationship is obtained from the closure condition which is equivalent to the initial diffraction problem in the absence of an excitation source is the fundamental point. In the present paper we present a general scheme for establishing the completeness of the method of secondary sources which enables one to obtain a complete system of functions by specifying merely the structure of the set of locations of the secondary sources as the initial information. The use of the method which has been developed in the case of vector diffraction problems enables one to construct new complete systems of vector functions. The scheme developed is transferred to the treatment of the question of the construction of complete systems of secondary sources in the method of fundamental solutions (m.f.s.). *Zh.vychisl.Mat.mat.Fiz.,27,6,945-949,1987
201 Functional systems, the completeness of which has previously been established, are constructed for the interesting practical cases of problems of diffraction by a local scatterer in the presence of "regular" boundaries of separation of media with different characteristics. Let us consider the completeness of functional systems in LI. Then, in order to prove completeness, it sufficeatodemonstrate the closure of the system /3/. We shall study an external problem of diffraction, in the region D&i' of the field of a point source located at a point Mo=D., on a local body D,=R'\D. with a surface S=A(2.Y)and the boundary condition that the total field vanishes on S. In this case, there exists a unique classical solution of the problem u which satisfies the Sommerfeld radiation condition. By applying Green's second formula in D. to the solution of the problem and to the fundamental solution of the Helmholtz equation $(M,P), we obtain ME D,. s Ip(M,P)~uda~=$@f,M,), 6 Denoting the normal derivative on the surface S,au by l(P), we have, when there is no source of excitation. s $(M,P)f(P)dap-0,ME D,. *
(1)
Eq.(l) is a functional relationship which is equivalent to the external Dirichlet problem in the sense that its unique solvability and the validity of (1) everywhere in D, for I"&(S) leads to f-0. We shall subsequently refer to relationship (1) as the basic relationship. Let us consider the function V(M)= .F rp(M,P)f(P)dox., M=Dt. 8
(2)
In order to use (1) and (2) to construct complete systems, we first consider the closure of the functional systems rp(M,M,)with the following structure of the sets CM",,"=,: a) located everywhere densely in a volume C&CD, b) everywhere dense on a non-resonant surface Z=A('~"~located as a whole in D, c) everywhere dense on a curve of class A which, as a whole, is located in D, (see /4/). It is readily seen that, by virtue of the choice of structure of the set (M,)r-1,the conditions for the closure of these functional systems ensure that an analytical function of a real variable V(M) vanishes in the domain D,. It follows from this that these systems are complete in L%(S). Remark 1. In the case of a boundary condition ~u(s=O, it can be shown, by using the properties of the normal derivative of a double layer potential with a density from L2(S), SE AU.", (see /S/j, that functional systems of the form (c%(M.Mn)) are complete. Problem 1. To determine the form of a complete system of functions from the condition that the analytical function (2) together with all of its derivatives vanishes at a point MCFD,. Let us choose a neighbourhood do=& for the point MO and take I& as the origin of a spherical system of coordinates (r,8,cp). By virtue of the analytic nature of V, we transform in do by expanding rp in spherical harmonics, to the Fourier coefficients of the function V with respect to the variable 0 and, as a result, we obtain _ m=o,i,..., P(r,Cl)y&!"i"(kr)P?%m(eose),
(3)
where
a.“=const+o and M,,m is a system of metaharmonic functions. Let us fix an arbitrary mS0 and use the condition that the function Vm vanishes at the point MO together with all of its derivatives which, in the equivalent formulation, has the form lim[P(r,B)/rl]-0, 1=0,1,..., ve. r-0 From the latter relatinship, by virtue of the behaviour of the functions j.(z) when ~0 (see /l/), we obtain f%"=O, n--m,m+1,... (4) Relationship (4) is the condition for the closure of a system of metaharmonic functions. Hence, by selecting a single point MO as the set of locations of the secondary sources, we obtain the completeness of the system of metaharmonic Vekua functions. Problem 2.
To determine the form of a complete system of functions if (M,),~,ED+ are
202 located in a rectilinear interval d. and has a point of accumulation Ml~sDi. We note that a rectilinear interval is not a curve of class A (see /4/). Let us choose the neighbourhood of the point of accumulation dv=Di. We direct the oz axis of a cylindrical system of coordinates (p,cp, Z) along the interval d, and introduce the Fourier coefficients Ym(r,O) in do into the treatment. Let us show that the following theorem is valid. Theorem. Let wm((a)= lim[P(r, 8)/p], p-0
, ,.... m-01
Then, if {z,)L =d,*=dand. and has a point of accumulation ZO,then Wm(2")=0, n=l,2, ...=+Vm=O in do. (5) Proof. Let us use the representation (3), transform from p+O to sin&+0 and employ the expression for the associated Legendre polynomials in terms of the hypergeometric function. As a result, we obtain
By virtue of the analytical nature of W"'(z) on do'and the condition of where r,m=const+O. the theorem, we obtain (4). The theorem is proved. In order to elucidate the actual form of a functional system for which relationship (5) is the closure condition, we note that, apart from factors, the real and imaginary parts of can be combined into a complex function of the form P s S,(M,P)f(P)exp(lmcp,)da~ B where &are the function introduced by E.N. Vasil'yev /6/. By using the relationship from /6/, it is readily shown that liml&,W,P)lp~~l= PM-0
qmk~"(kRp~)Pmm(cog8~), qm=const
Hence, condition (5) implies the closure of a functional system of the form G,'"(M)= @(M,4=
k~'(kRas.,)P,m(cose~,)e~~.
(6)
Here, M=(p, cp, z),Rf =p2+(z--a,)Z. Sin 8z,=p/&.w For fixed m, system (6) constitutes a system of "lower" multipoles,nwhere each function is a combination of elementary functions. Remark 2. Complete functional systems of vector functions can be constructed in a completely analogous manner. In order to do this, it is sufficient to carry out the following constructions: a) for the boundary condition [II, E]ls=O, using Green's vector formula, we obtain the fundamental relationship of type (1) b) for the Cartesian components of an analytical vector function of type (2) which satisfy the Helmholtz equation, we carry out all the reasoning outlined above. As a result, we obtain the following complete systems of vector functions: Vxvx(M,m(P)ei),vx(MP(P)e>), I,m=O,I,..., m
vxvx(Gnm(P)eJ,
vx(G?Im(P)eJ, n=f, 2,. . ,
m=O,i,...,
where e, is a basis of unit vectors of a Cartesian coordinate system, i=i,?.3.We note that the completeness of the system of functions VxVx(G,"e,) was established in /2/ on the basis of other considerations. However, use of the scheme in /2/ to establish completeness does not make it possible to establish the completeness of a system of functions in the method of fundamental solutions. As an example, let us construct complete systems of functions of the method of fundamental solutions /?/ in the case of structure of D, which indicate that there is an ideally conducting plane or dielectric laminar medium in D ,,assuming that the boundaries of separqtion of the media with different parameters donotintersect the surface of the local body S. Let the ozaxis be perpendicular to the ideally conducting plane lI=D, and the point of accumulationof _ the set {M,),,,be Mo=DI. Problem 3. To construct a complete system for the method of fundamental solutions when the set (M,)cD, is located in an interval of the oz-axis of a cylindrical system of coordinates. Let us assume that the bounary condition on 11 has the form u(,,=O.In this case the fundamental relationship takes the form of (1) with rl, replaced by g, that is, by Green's function for an ideally conducting half-space. By applying the technique used in the solution of problem 2, we obtain a complete system of the form x""(M)=Gm(M,z.)-C~(M,z;), n=l,Z, ...I m=O, I,.... (7) where G" are the functions from (6) and zn*is the conjugate point on 0: with respect to Il. Let (M") be the same as in problem 3 and DZ be perpendicular to the boundary of
203 separation of the layers. Problem 4. TO construct a complete system of the method of fundamental solutions when there is a laminar medium in D, In this case Green's function for a laminar medium S (see /7/j is used in the fundamental relationship (1). Its Fourier coefficients can be represented in the form
gmW,P)= s
J,(hp)J,(hpp)u(z,zp,a)adh,
0
where u(z, ZP,h) is a function which ensures that the joining conditions on the boundaries of separation of media with different parameters are satisfied. By using the condition of the theorem, we have, apart from a constant factor,
lb k”‘~~,~)/p~“‘~PP-+O
Hence,
~J,(hp)h’“v(z, 0
zp, h)h dh.
in this case, the complete system of functions takes the form
U.~(M)=(G~(M,Z.)-G~(I,Z,‘)+
jJ,,,(hp)h~kdlie~” 0
The occurrence of am in theintegral (8) results from the magnitude of the singularity of the functions Gm when the arguments are identical. We note that the completeness of the functional systems (7) and (8) has not been previously established, although a numerical algorithm for investigating diffraction problems based on the method of fundamental solutions has been previouslv used successfullv in the solution of problems of diffraction bv dielectric solids of rotation located in a homogeneous half-space /8/. The author thanks A.G. Sveshnikovforhis help and interest. REFERENCES 1. EREMIN YU.A. and SVESHNIKOV A.G., Foundation of the method of non-orthogonal series and the investigation of certain inverse diffraction problems, Zh. vychisl. Mat. mat. Fiz., 23, 3, 738-742, 1983. 2. EREMIN YU.A., LEBEDEV O.A. and SVESHNIICOV A.G., The use of multipole sources in the method of non-orthogonal series in diffraction problems, Zh. vychisl. Mat. mat. Fir., 25, 3, 474-418, 1985. 3. KUPRADZE V.D., On the approximate solution of problems of mathematical physics, Uspekhi Matem. Nauk, 22, 2, 58-107, 1967. 4. KRAVTSOV V.V., Approximation of functions of many variablts by an antenna potential, Zh. vychisl. Mat. mat. Fiz., 18, 3, 672-680, 1978. 5. SVESHNIKOV A.G., EREMIN YU.A. and CORLOV N-V., The foundation of realization of the method of non-orthogonal series in problems of diffraction by dielectric bodies, Vestnik Moskovsk. Gos. Univ (MGU), Ser. 15, Vychisl. Matem. i Kibernetika, 3, 13-20, 1983. 6. VASIL'YEV E.N., GORELIKOV A.I. and FALULIN A.A., Green's tensor function in rotating coordinates, in: Collection of Scientific-MethodologicalArticles on Applied Electrodynamics, Vyssh. Shkola, Moscow, Issue 3, 3-25, 1980. 7. EREMIN YU.A., ZAKHAROV E.V. and NESMEYANOVA N.I., The method of fundamental solutions in problems of the diffraction of electromagneticwaves by bodies of rotation, in: Computational Methods and Programming, Izd. Moskovsk. Gos. Univ. (MGU), XxX11, 28-43, 1980. 8. SVESHNIKOV A.G., EREMIN YU.A. and ORLOV N.V., Investigationof some mathematical models in the theory of diffraction by the method of non-orthogonal series, Radiotekhn. i elektronika, 30, 4, 697-704, 1985.
Translated by E.L.S.