Applied Mathematics
Fundamental solution of the Cauchy problem
for an anisotropic electrodynamic system
V. Yakhno
1, A. Sevimlican
21 Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia
e-mail:[email protected]
2 Dokuz Eylul University, Faculty of Arts and Sciences, Buca, 35160 Izmir, Turkey
e-mail:[email protected]
Received: March 01, 2001
Summary.
This paper deals with Maxwell's system. An Explic-it formula for a fundamental solution of the Cauchy problem for Maxwell's system with the constant dielectric and magnetic perme-abilities, and the matrix conductivity is constructed.Key words:
Cauchy problem,fundamental solution, Maxwell'ssys-tem, telegraph equationMathematics Subject Classication (1991): 35A08, 35L15, 78A25
1. Introduction
This paper is concerned with a fundamental solution of the Cauchy problem for Maxwell's system. Dielectric and magnetic permeabilities of this system are constants, conductivity is a matrix with constants elements. Maxwell's system is reduced to the telegraph vector equa-tion by means of the vector and scalar electrodynamics potentials. This telegraph vector equation is a hyperbolic system. Constructing fundamental solutions of the Cauchy problem for hyperbolic equa-tions and systems is a very important problem because by means of these fundamental solutions we nd solutions of the Cauchy prob-lems for equations and systems with arbitrary nonhomogeneous terms and initial data. Basic features of the structure of the fundamental solutions for hyperbolic equations are well known (see 1{7]). This
structure can be easily traced in the works of J. Hadamard 3] and S. L. Sobolev 7].
The fundamental solution of the Cauchy problem for hyperbolic equations consists of a singular part with the support on a charac-teristic conoid and a regular part with the support in the closure of the internal part of this conoid. The regular part of the fundamental solution is a smooth function inside the conoid if the coecients of the equation are suciently smooth.
Consider, for instance, a scalar hyperbolic equations of the form u tt = 3 X i=1 3 X j=1 a ij( x) @ 2 u @x i @x j + 3 X j=1 b j( x) @u @x j +c(x)u+(x;x 0 t) where x = (x 1 x 2 x
3) is the three dimensional space variable from R
3,
tis the one dimensional variable fromRx 0 = ( x 0 1 x 0 2 x 0 3) is the three dimensional parameter from R
3 and
(x;x 0
t) is the Dirac delta function concentrated atx =x
0,
t= 0. Assume that all coe-cients of this equation are smooth functions. A fundamental solution of the Cauchy problem for this equation may be written in the form
u(xtx 0) = (t) N X k =;1 a k( xx 0) k( t 2 ; 2( xx 0)) + u N( xtx 0) where(xx
0) is the distance in the geodesic metric
d = 3 X i=1 3 X j=1 b ij( x)dx i dx j 1=2 the matrixB = (b ij( x)) is inverse to A = (a ij( x))a ij( x) are coe-cients of the hyperbolic equation
0(
t) is the Heaviside step-function ( 0( t) = 1 for t 0 and 0( t) = 0 for t < 0), k( t) = t k k! 0( t) for k = 12::: ;1(
t) = (t), (t) is the Dirac delta function concen-trated at t = 0, and u
N(
xt) is the regular part of a fundamental solution.
The procedure of the successive determination of the coecients was described by V. G. Romanov 4, 5, 6]. In our article this procedure is applied to construct an explicit formula for the telegraph vector equation. The main result of the present paper is an explicit formula for a fundamental solution of the Cauchy problem for Maxwell's sys-tem which is constructed by the fundamental solution of the Cauchy problem for the telegraph vector equation.
2. Equations of the electromagnetic eld
The complete set of Maxwell's equations has the form
(1) curlx H= 1 c @D @t + 4 c J (2) curlx E=; 1 c @B @t (3) divx B= 0 (4) divx D= 4
where vectors D, B, and J are electric displacement, the magnetic induction, and current density is the density of electric charges. The valuesand J satisfy the relation
(5) @
@t
+ divx J = 0
and hence equations (1) and (2) are related to each other. This re-lation expresses the law of the conservation of the electric charge. In addition there are constitutive relations that expresses B, D, andJ in terms ofE and H. These equations are
(6) D="E B= H J = E+j
where"is the dielectric permeability, is the magnetic permeability, is the conductivity, andjis the density of currents arising from the action of the external electromagnetic forces. We assume that (7) E = 0 H = 0 = 0 j= 0 for t<0:
This means there is no electromagnetic eld, currents, or electric charges at the timet <0. We note that equation (3) follows imme-diately from (2) and (6), and equation (4) can be obtained from (1), (5), and (6). Really, applying the operator divx to (2) we have
@ @t
divx B = 0 and from (6) we can nd divx
Bj
t<0= 0. The last two relations imply (3). Applying the operator divx to (1) we nd
divx @D
+ 4 div x
and using (5) we have @ @t
divx
D;4 ] = 0:
Equation (4) follows from the last relation, (6), and (7). Let us con-sider now the problem of determining the vector functionsE,H, and the function satisfying (1){(7) if", are constants, is a matrix, and j is the known vector function. It is easy to show that a solu-tion of this problem may be found successively. On the rst step we determineE and H appearing in the equations
(8) curlx H= 1 c @ @t ("E) + 4 c E+ 4 c j (9) curlx E=; 1 c @ @t ( H) subject to conditions (10) Ej t<0= 0 Hj t<0= 0 and then recover the functionfrom relations
(11) @ @t =;div x E+ 4 c j] (12) j t<0= 0 : Equations (8)-(10) may be written as follows
(13) curlx H= " @E @t + E+ j curl x E=; @H @t (14) Ej t<0= 0 Hj t<0= 0 where "= " c , = c , = 4 c , j= 4 j c
. Further we shall omit bars over letters" , for the simplicity of writing.
3. Scalar and vector electrodynamics potentials
Applying divx to the second equation of (13), we get the following Cauchy problem @ @t (divx( H)) = 0 divx( H)j t<0= 0 : Therefore divx(
H) = 0 fort2R. Further we seekHin the following form
(15) H = 1
curlx
A:
Here A is the vector function which is called the vector electrody-namic potential. In this case the property divx(
H) = 0 holds imme-diately. Substituting (15) into the second relation of (13) we get
curlx E+ curl x @A @t = curlx( E+ @A @t ) = 0: We look forE such that
(16) E =; @A @t +r x ' where 'is a scalar function and curl
x r
x
'= 0. This scalar function is called the scalar electrodynamic potential.
Substituting (15) and (16) into the rst equation of (13) we have (17) curlx(curlx(1 A)) =;" @ 2 A @t 2 + " @ @t (r x '); @A @t + r x '+J: Using the property curlx(curlx
A) = r xdivx A; x A equation (17) becomes (18) 1 r xdivx A;" @ @t ; +" @ 2 A @t 2 ; 1 x A+ @A @t =J where = r x
'. The equality (18) will hold if A and are chosen from the relations
(19) @ @t + 1 " =a 2 r xdivx A @ 2 A 2 ;a 2 x A+ 1 @A =a 2 J
where a= 1 p
"
. For holding (14) the conditionsj
t<0= 0, Aj
t<0 = 0 are sucient. If the vector potential A is found then the scalar potential can be de ned by
(20) (xt) =a 2 1 Z ;1 (t;)exp(; 1 " t)r xdivx A(x)d:
H andE are found by (15) and (16).
4. The fundamental solution of the Cauchy problem
for the telegraph equations system
This section deals with fundamental solution of the Cauchy problem for the telegraph vector operator
L= @ 2 @t 2 ;a 2 x+ 2 M @ @t whereM = 12 " .
Denition 1.
A matrix E is called the fundamental solution of theCauchy problem for the operator L if each column E j,
j = 123, of
this matrix satises the following equations
(21) LE j = e j (xt) (22) E j j t<0= 0 : Here e 1= (1 00),e 2 = (0 10),e 3 = (0 01).
Theorem 1.
Let E(xt) = (t)(;)exp(;Mt) 2 a 3 + (t;jxj=a)Mexp(;Mt)I 1( M p ;=2) 4 a 3 p ;where exp(Mt) and I 1(
Mt) are matrices dened by exp(Mt) = 1 X k =0 (Mt) k k! I 1( Mt) = 1 X k =0 (Mt) 2k +1 k!(k+ 1)! and ; =t 2 ;jxj 2 =a
2. Then the matrix
E(xt)is a fundamental
Proof. We seek the jth column of a fundamental solution of the Cauchy problem in the following expansion
(23) E j( xt) =(t) 1 X k =;1 j k( xt) k( ;) where k(
xt) are unknown vector functions. We have to determine these functions. For this aim we use the following properties of gen-eralized functions: ; k ;2( ;) = (k;1) k ;1( ;) 0 k( ;) = k ;1( ;) (t) k( ;) = 0 for k;1 ; 0( t) ;1( ;) = 2 a 3 (xt) and the following expressions forE
j t, E j tt, r x E j, x E j: (24) @E j @t =(t) 1 X k =;1 j k k( ;) +(t) 1 P k =;1 @ j k @t k( ;) +(t) 1 P k =;1 j k @ @t k( ;) =(t) 1 P k =;1 @ j k ;1 @t +@; @t j k k ;1( ;) (25) 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : @ 2 E j @t 2 = ; 0( t) 1 P k =;1 j k k( ;) + 2 @ @t (t) 1 X k =;1 j k k( ;) +(t) @ 2 @t 2 1 X k =;1 j k k( ;) = 2 a 3 j ;1(0 0)(xt) +(t) 1 P k =;1 k ;1( ;) @ 2 j k ;1 @t 2 + 2 @ j k @t @; @t + j k @ 2 ; @t 2 + @; @t 2 j k (k;1) ; (26) r x E j = (t) 1 X k =;1 r x j k ;1+ j k r x ; k ;1( ;) (27) x E j = (t) 1 P k ;1 x j k ;1+ 2 r x j k r x ; + j k x ; + j k( x ;) 2( k;1) k ;1( ;)
where j ;2 0, @; @t = 2t, r x ; = ; 2x a 2, x ; = ; 6 2. Substituting (23) into (21) we get 2 a 3 j ;1(0 0);e j (xt) +(t) 1 X k =;1 L x j k ;1+ 2 @ j k @t @; @t ;2a 2 r x j k r x ; +L x ; + 4(k;1) j k k ;1( ;) = 0: Equating to zero the expressions by (xt) and
k ;1( ;) for k;1, we have relations (28) j ;1(0 0) = e j 2 a 3 (29) x @ @x j k+ t @ @t j k+ j k(( k+ 1) +Mt) =; 1 4L j k ;1 : Considering (29) along the curve de ned by
dx dt
= x()
t=p
where pis a constant, is a parameter, and multiplying (29) by k, k=;101, the relation (29) may be transformed to the form (30) d d k +1 j k( x()p) +Mp k +1 j k =; k 4 L j k ;1 j x=x()t=p : Integrating (30) from 0 to (x) and using (28) we nd
(31) j ;1( xt) = exp(;Mt) j ;1(0 0) k +1( x) j k( x()p) =; 1 4 exp(;Mp) (x) Z 0 kexp( Mp) L j k ;1( z)j =x()z =p d k= 01:::
Making the change of variable =(x)s, the last equation may be written as follows j k( xt) =; 1 4 1 Z s kexp( Mt(s;1))L j k ;1( z)j z =st =sx ds:
We can show that L j ;1( z)j =sxz =st = ;M 2exp( ;Mst) j ;1(0 0) (32) j 0( xt) = M 2 4 1 Z 0 exp(Mst;Mst;Mt)ds j ;1(0 0) = M 2 2 j ;1( xt): Then, using the previous formulas, we nd
L j 0( z)j =sxz =st = ; M 4 4 exp(;Mst) j ;1(0 0) (33) j 1( xt) =; 1 4 1 Z 0 sexp(Mt(s;1))L z j 0 ds = 12! M 2 4 j ;1( xt)
and so on continuing the reasoning, for arbitrary naturalk we have: L j k ;1( z)j =sxz =st = ; M 2 2k +2 exp(;Mst) 1 (k+ 1)! j ;1(0 0) (34) j k( xt) =; 1 4 1 Z 0 s kexp( Mt(s;1))L j k ;1 j =sx z =st ds = 1( k+ 1)! M 2 2k +2 j ;1( xt): Therefore we found j k( xt),k=;101:::, by means of (31){(34). Substituting formulas (31){(34) in (23) we get
(35) E j( xt) =(t) ;1( ;) j ;1( xt) +(t) 1 P k =0 (M=2) 2k +2 (k+ 1)! j ;1( xt) ; k k! (;):
Using (t)(;) = (t;jxj=a), ;1(
;) = (;), the expression (35) may be written in the form
E(xt) = (t)(;)exp(;Mt) 2 a 3 + (t;jxj=a)Mexp(;Mt)I 1( M p ;=2) 4 a 3 p ; : Here I 1(
Mt) is the matrix de ned in the formulation of the theorem
I 1( Mt) = 1 X k =0 (Mt) 2k +1 k!(k+ 1)! : u t Remark 1.I 1(
t) is the modi ed Bessel function.
5. The fundamental solution of the Cauchy problem
for Maxwell's system
Denition 2.
A matrix G(xt) = 0 B B B B B B @ G 1j G 2j G 3j G 4j G 5j G 6j 1 C C C C C C A j=123 = H j E j j=123the j-th column of which satises the following conditions
curlx H j ;" @E j @t ; E j = e j (xt) (36) curlx E j+ @H j @t = 0 (37) H j j t<0= 0 E j j t<0= 0 where e 1 = (1 00), e 2 = (0 10), and e 3 = (0 01)), is called a
Thus, by the reasoning which we used for obtaining (15), (16), and (18), if we chooseJ =e
j
(xt) we can get the following equalities
(38) H j = 1 curlx A j (39) E j= ; @A j @t + j (40) @ j @t + 1 " j = a 2 r xdivx A j j j t<0= 0 (41) @ 2 A j @t 2 ;a 2 A j+ 2 M @A j @t = e j (xt) A j j t<0= 0 :
Theorem 2.
LetM be a matrix. Then a fundamental solution of theCauchy problem for Maxwell's system is given by formulas (38) and (39), where A j( xt) =a 2 (E(xt)e j), j( xt) =a 2 1 Z ;1 (t;)exp(;2M(t;))r xdivx A j( x)d
and E(xt) is a fundamental solution of the Cauchy problem for the
telegraph vector equation (see Theorem 1).
Proof. Using a fundamental solution of the telegraph vector equation, the solution of (41) can be represented by the following formula
A j( xt) =a 2 E(xt)e j j= 123: Calculate j(
xt),j = 123, by means of the formula
j( xt) =a 2 1 Z ;1 (t;)exp(;2M(t;))r xdivx A j( x)d: FinallyH j and E
j can be found by (38) and (39). u t
6. Conclusion
Using V. G. Romanov's procedure 4{6] the explicit formulas for fun-damental solutions of the Cauchy problem for the telegraph vector equations and Maxwell's system were constructed.
Formulas for generalized solutions of the Cauchy problems for the above mentioned systems with arbitrary nonhomogeneous terms and initial data may be obtained. For example, the following theorem holds.
Theorem 3.
LetJ(xt)be a vector function with componentsJ k( xt) 2 D 0( R 4), J k( xt)j t<0= 0G(xt) be a fundamental solution of the
Cauchy problem for Maxwell's system then the vector function
(42) H E = Z R 4 G(x;t;)()J()d
is a solution of the Cauchy problem (13) and (14).
The proof of this theorem is based on checking that vector function (42) satis es equations (13) and (14).
V. G. Romanov's procedure may be used for the construction of a fundamental solution of the Cauchy problem for Maxwell's system in which", are scalar smooth functions, is a matrix whose elements are smooth functions.
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