On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients
aa r X i v : . [ m a t h . A P ] J u l On fundamental solutions for multidimensional Helmholtzequation with three singular coefficientsErgashev T.G.
Institute of Mathematics, Uzbek Academy of Sciences, Tashkent, Uzbekistan. [email protected]
The main result of the present paper is the construction of fundamental solutions for a class ofmultidimensional elliptic equations with three singular coefficients, which could be expressed in termsof a confluent hypergeometric function of four variables. In addition, the order of the singularityis determined and the properties of the found fundamental solutions that are necessary for solvingboundary value problems for degenerate elliptic equations of second order are found.
Key words: multidimensional elliptic equation with three singular coefficients, fundamentalsolutions, confluent hypergeometric functions of four variables.
It is known that fundamental solutions have an essential role in studying partial differential equations.Formulation and solving of many local and non-local boundary value problems are based on these solutions.Moreover, fundamental solutions appear as potentials, for instance, as simple-layer and double-layer potentialsin the theory of potentials.The explicit form of fundamental solutions gives a possibility to study the considered equation in detail. Forexample, in the works of Barros-Neto and Gelfand [1], fundamental solutions for Tricomi operator, relative toan arbitrary point in the plane were explicitly calculated. Among other results in this direction, we note a workby Itagaki [11], where 3D high-order fundamental solutions for a modified Helmholtz equation were found. Thefundamental solutions can be applied to some boundary value problems [6, 12, 13, 14, 19, 20].In the present work we find fundamental solutions for the equation p X i =1 u x i x i + 2 α x u x + 2 α x u x + 2 α x u x − λ u = 0 (1.1)in the domain R p := { ( x , ..., x p ) : x > , x > , x > } , where p is a dimension of the Euclidean space; p ≥ ; α j are real constants and < α j < ( j = 1 , , ); λ is real or pure imaginary constant.Various modifications of the equation (1.1) in the two- and three-dimensional cases were considered in manypapers [7, 8, 22, 23]. However, relatively few papers have been devoted to finding the fundamental solutions formultidimensional equations, we only note the works [16, 22]. In the paper [17], fundamental solutions of equation(1.1) with a single singular coefficient are found and investigated.In this article, at first we shall introduce one confluent hypergeometric function of four variables. Furthermore,by means of the introduced hypergeometric function we construct fundamental solutions of the equation (1.1) in anexplicit form. For studying the properties of the fundamental solutions, the introduced confluent hypergeometricfunction is expanded in products of hypergeometric functions of Gauss. With the help of the obtained expansionit is proved that the constructed fundamental solutions of equation (1.1) have a singularity of order /r p − at r → . In [3] (see also [21, p.74,(4b)]) a hypergeometric function of many variables of the form H n,p ( a, b , ..., b n , c p +1 , ..., c n ; d , ..., d p ; x , x , ..., x n )= ∞ X m ,...,m n =0 ( a ) m + ... + m p − m p +1 − ... − m n ( b ) m ... ( b n ) m n ( c p +1 ) m p +1 ... ( c n ) m n ( d ) m ... ( d p ) m p m ! ...m n ! x m ...x m n n , (2.1) ≤ p ≤ n, is considered, where ( κ ) m := Γ( κ + m ) / Γ( κ ) is the Pochhammer symbol, m is a integer number, a isa complex number, and κ = 0 , − , − , ..., if the Pochhammer symbol ( κ ) m is on the denominator.The hypergeometric function (2.1) in four variables case has a form1 , ( a, b , b , b , b , c ; d , d , d ; x, y, z, t ) = ∞ X m,n,k,l =0 ( a ) m + n + k − l ( b ) m ( b ) n ( b ) k ( b ) l ( c ) l ( d ) m ( d ) n ( d ) k m ! n ! k ! l ! x m y n z k t l , (2.2)where | x | + | y | + | z | < , | t | < / (1 + | x | + | y | + | z | ) .From the hypergeometric function (2.2) we shall define the following confluent hypergeometric functionH , ( a, b , b , b ; d , d , d ; x, y, z, t ) = lim ε → H , (cid:18) a, b , b , b , ε , ε ; d , d , d ; x, y, z, ε t (cid:19) . At the determination of the hypergeometric function H , ( a, b , b , b ; d , d , d ; x, y, z, t ) the equality lim ε → (1 /ε ) n · ε n = 1 ( n is a natural number) has been used and the found confluent hypergeometric functionrepresented asH , ( a, b , b , b ; d , d , d ; x, y, z, t ) = ∞ X m,n,k,l =0 ( a ) m + n + k − l ( b ) m ( b ) n ( b ) k ( d ) m ( d ) n ( d ) k m ! n ! k ! l ! x m y n z k t l , | x | + | y | + | z | < . (2.3)Using the formula of derivation ∂ i + j + k + l ∂x i ∂y j ∂z k ∂t l H , ( a, b , b , b ; d , d , d ; x, y, z, t ) = ( a ) i + j + k − l ( b ) i ( b ) j ( b ) k ( d ) i ( d ) j ( d ) k ×× H , ( a + i + j + k − l, b + i, b + j, b + k ; d + i, d + j, d + k ; x, y, z, t ) it is easy to show that the confluent hypergeometric function in (2.3) satisfies the system of hypergeometricequations x (1 − x ) ω xx − xyω xy − xzω xz + xtω xt + [ d − ( a + b + 1) x ] ω x − b yω y − b zω z + b tω t − ab ω = 0 y (1 − y ) ω yy − xyω xy − yzω yz + ytω yt − b xω x + [ d − ( a + b + 1) y ] ω y − b zω z + b tω t − ab ω = 0 z (1 − z ) ω zz − xzω xz − yzω yz + ztω zt − b xω x − b yω y + [ d − ( a + b + 1) z ] ω z + b tω t − ab ω = 0 tω tt − xω xt − yω yt − zω zt + (1 − a ) ω t + ω = 0 , (2.4)where ω ( x, y, z, t ) = H , ( a, b , b , b ; d , d , d ; x, y, z, t ) . Having substituted ω ( x, y, z, t ) = x τ y ν z µ t δ ψ ( x, y, z, t ) in the system of hypergeometric equations (2.4), it ispossible to find 8 linearly independent solutions of system (2.4), which is given in the table form ω ω ω ω ω ω ω ω τ − d − d − d − d ν − d − d − d − d µ − d − d − d − d δ .or in explicit form as follows ω ( x, y, z, t ) = H , ( a, b , b , b ; d , d , d ; x, y, z, t ) , (2.5) ω ( x, y, z, t ) = x − d H , ( a + 1 − d , b + 1 − d , b , b ; 2 − d , d , d ; x, y, z, t ) , (2.6) ω ( x, y, z, t ) = y − d H , ( a + 1 − d , b , b + 1 − d , b ; d , − d , d ; x, y, z, t ) , (2.7) ω ( x, y, z, t ) = z − d H , ( a + 1 − d , b , b , b + 1 − d ; d , d , − d ; x, y, z, t ) , (2.8) ω ( x, y, z, t ) = x − d y − d H , ( a + 2 − d − d , b + 1 − d , b + 1 − d , b ; 2 − d , − d , d ; x, y, z, t ) , (2.9) ω ( x, y, z, t ) = x − d z − d H , ( a + 2 − d − d , b + 1 − d , b , b + 1 − d ; 2 − d , d , − d ; x, y, z, t ) , (2.10) ω ( x, y, z, t ) = y − d z − d H , ( a + 2 − d − d , b , b + 1 − d , b + 1 − d ; d , − d , − d ; x, y, z, t ) , (2.11) ω ( x, y, z, t ) = x − d y − d z − d H , ( a +3 − d − d − d , b +1 − d , b +1 − d , b +1 − d ; 2 − d , − d , − d ; x, y, z, t ) . (2.12)2 Decomposition formulas
For a given multivariable function, it is useful to find a decomposition formula which would express themultivariable function in terms of products of several simpler hypergeometric functions involving fewer variables.For this purpose Burchnall and Chaundy [2] had given a number of expansions of double hypergeometric functionsin series of simpler hypergeometric functions. Their method is based upon the inverse pair of symbolic operators ∇ ( h ) := Γ ( h ) Γ ( δ + δ + h )Γ ( δ + h ) Γ ( δ + h ) , ∆ ( h ) := Γ ( δ + h ) Γ ( δ + h )Γ ( h ) Γ ( δ + δ + h ) , (3.1)where δ := x ∂∂x , δ := x ∂∂x Recently Hasanov and Srivastava [9, 10] generalized the operators ∇ ( h ) and ∆ ( h ) defined by (3.1) in theforms ˜ ∇ x ; x ,...,x m ( h ) := Γ ( h ) Γ ( δ + ... + δ m + h )Γ ( δ + h ) Γ ( δ + ... + δ m + h ) (3.2)and ˜∆ x ; x ,...,x m ( h ) := Γ ( δ + h ) Γ ( δ + ... + δ m + h )Γ ( h ) Γ ( δ + ... + δ m + h ) , (3.3)where δ i := x i ∂∂x i ( i = 1 , ..., m ) , (3.4)and they obtained very interesting results. For example, in the special case when m = 3 a Lauricella function inthree variables is defined by (cf.[15]; see also [21, p.33, 1.4(1)]) F (3) A ( a, b , b , b ; d , d , d ; x, y, z ) = ∞ X l,m,n =0 ( a ) l + m + n ( b ) l ( b ) m ( b ) n ( d ) l ( d ) m ( d ) n x l l ! y m m ! z n n ! and the following decomposition formula holds true [9] F (3) A ( a ; b , b , b ; d , d , d ; x, y, z ) = ∞ P l,m,n =0 ( a ) l + m + n ( b ) l + m ( b ) l + n ( b ) m + n ( d ) l + m ( d ) l + n ( d ) m + n l ! m ! n ! x l + m y l + n z m + n · F ( a + l + m, b + l + m ; d + l + m ; x ) F ( a + l + m + n, b + l + n ; d + l + n ; y ) · F ( a + l + m + n, b + m + n ; d + m + n ; z ) , (3.5)where F ( a, b ; c ; x ) = ∞ X n =0 ( a ) n ( b ) n ( c ) n n ! x n is a Gaussian hypergeometric function [3, p.56,(2)].It should be noted that the symbolic notations (3.4) in the one-dimensional case take the form δ := xd/dx and such a notation is used in solving problems of the operational calculus [18, p.26].We now introduce here the other multivariable analogues of the Burchnall-Chaundy symbolic operators ∇ ( h ) and ∆ ( h ) defined by (3.1): ˜ ∇ m,nx,y ( h ) := Γ ( h ) Γ ( h + δ + ... + δ m − σ − ... − σ n )Γ ( h + δ + ... + δ m ) Γ ( h − σ − ... − σ n ) = ∞ X s =0 ( − δ − ... − δ m ) s ( σ + ... + σ n ) s ( h ) s s ! , (3.6) ˜∆ m,nx,y ( h ) := Γ ( h + δ + ... + δ m ) Γ ( h − σ − ... − σ n )Γ ( h ) Γ ( h + δ + ... + δ m − σ − ... − σ n ) = ∞ X s =0 ( δ + ... + δ m ) s ( − σ − ... − σ n ) s (1 − h ) s s ! , (3.7)where x := ( x , ..., x m ) , y := ( y , ..., y n ) , (3.8) δ i := x i ∂∂x i , σ j := y j ∂∂y j , i = 1 , ..., m, j = 1 , ..., n ; m, n ∈ N .
3n addition, we consider operators which coincide with Hasanov- Srivastava’s symbolic operators ˜ ∇ ( h ) and ˜∆ ( h ) defined by (3.2) and (3.3) as a particular case: ˜ ∇ m, x, − ( h ) := ˜ ∇ x ; x ,...,x m ( h ) , ˜∆ m, x, − ( h ) := ˜∆ x ; x ,...,x m ( h ) , m ∈ N;˜ ∇ ,n − ,y ( h ) := ˜ ∇ − y ; − y ,..., − y n ( h ) , ˜∆ ,n − ,y ( h ) := ˜∆ − y ; − y ,..., − y n ( h ) , n ∈ N . It is obvious that ˜ ∇ , x, − ( h ) = ˜∆ , x, − ( h ) = ˜ ∇ , − ,y ( h ) = ˜∆ , − ,y ( h ) = 1 . We introduce the notation: D sz f ( z ) = X I ( k,s ) z i ...z i k k i ! ...i k ! ∂ s f∂z i ...∂z i k k , where z = ( z , ..., z k ); I ( k, s ) = { ( i , ..., i k ) : i ≥ , ..., i k ≥ , i + ... + i k = s } . Lemma.
Let be f := f ( x ) and g := g ( y ) functions with variables x and y in (3.8). Then following equalitieshold true for any m, n ∈ N : (cid:18) − x ∂∂x − ... − x m ∂∂x m (cid:19) s f ( x ) = ( − s s ! D sx f ( x ) , s ∈ N ∪ { } ; (3.9) (cid:18) y ∂∂y + ... + y n ∂∂y n (cid:19) s g ( y ) = g ( y ) , if s = 0 ,s ! s P q =1 C qs C q − s − · ( s − q )! D sy g ( y ) , if s ∈ N . (3.10)The lemma is proved by method of mathematical induction [5].In the present paper we shall use two particular cases ( m = 3 and n = 1) of the formulas (3.9) and (3.10): (cid:18) − x ∂∂x − y ∂∂y − z ∂∂z (cid:19) s f ( x, y, z ) = ( − s s ! D sx,y,z f ( x, y, z ) , s ∈ N ∪ { } ; (3.11) (cid:18) t ddt (cid:19) s g ( t ) = g ( t ) , s = 0 , s P q =1 C qs C q − s − · ( s − q )! t q g ( q ) ( t ) , s ∈ N . (3.12)Using the formulas (3.6) and (3.7), we obtain H , ( a ; b , b , b ; d , d , d ; x, y, z, t ) = ˜ ∇ , x,y,z,t ( a ) F (3) A ( a ; b , b , b ; d , d , d ; x, y, z ) F (1 − a ; − t ) , (3.13)where F ( a ; x ) = ∞ P n =0 x n ( a ) n n ! is a generalized hypergeometric function [4, Chapter IV].Now considering the equalities (3.6), (3.11) and (3.12) from the formula (3.13) we have H , ( a ; b , b , b ; d , d , d ; x , x , x , y ) = ∞ P s =0 s P q =0 P I (3 ,s ) A ( s, q ) C qs ( − s + q ( b ) i ( b ) j ( b ) k (1 − a ) q ( d ) i ( d ) j ( d ) k x i i ! y j j ! z k k ! t q q ! · F (3) A ( a + s ; b + i, b + j, b + k ; d + i, d + j, d + k ; x, y, z ) F (1 − a + q ; − t ) , (3.14)where A ( s, q ) = (cid:26) , if s = 0 and q = 0 ,q/s, if s ≥ q ≥ , I (3 , s ) = { ( i, j, k ) : i ≥ , j ≥ , j ≥ , i + j + k = s } . Applying the decomposition formula (3.5) to the expansion (3.14), we obtain H , ( a ; b , b , b ; d , d , d ; x, y, z, t )= ∞ P l,m,n,s =0 s P q =0 P I (3 ,s ) A ( s, q ) s ! C qs ( − s + q ( a ) l + m + n + s ( b ) i + l + m ( b ) j + l + n ( b ) k + m + n ( a ) s (1 − a ) q ( d ) i + l + m ( d ) j + l + n ( d ) k + m + n x i + l + m i ! n ! y j + l + n j ! m ! z k + m + n k ! l ! t q q ! · F ( a + s + l + m, b + i + l + m ; d + i + l + m ; x ) F ( a + s + l + m + n, b + j + l + n ; d + j + l + n ; y ) · F ( a + s + l + m + n, b + k + m + n ; d + k + m + n ; z ) F (1 − a + q ; − t ) . (3.15)4y virtue of the formula [4, p.64,(22)] F ( a, b ; c ; x ) = (1 − x ) − b F (cid:18) c − a, b ; c ; xx − (cid:19) , the expansion (3.15) yields H , ( a ; b , b , b ; d , d , d ; x, y, z, t ) = (1 − x ) − b (1 − y ) − b (1 − z ) − b · ∞ P l,m,n,s =0 s P q =0 P I (3 ,s ) A ( s, q ) s ! C qs ( − s + q ( a ) l + m + n + s ( b ) i + l + m ( b ) j + l + n ( b ) k + m + n ( a ) s (1 − a ) q ( d ) i + l + m ( d ) j + l + n ( d ) k + m + n i ! j ! k ! n ! m ! l ! q ! t q · (cid:16) x − x (cid:17) i + l + m F (cid:16) d − a − j − k, b + i + l + m ; d + i + l + m ; xx − (cid:17) · (cid:16) y − y (cid:17) j + l + n F (cid:16) d − a − i − k − m, b + j + l + n ; d + j + l + n ; yy − (cid:17) · (cid:16) z − z (cid:17) k + m + n F (cid:16) d − a − i − j − l, b + k + m + n ; d + k + m + n ; zz − (cid:17) F (1 − a + q ; − t ) . (3.16)Expansion (3.16) will be used for studying properties of the fundamental solutions. We consider equation (1.1) in R p . Let x := ( x , ..., x p ) be any point and x := ( x , ..., x p ) be any fixed pointof R p . We search for a solution of (1.1) as follows: u ( x, x ) = P ( r ) w ( σ ) , (4.1)where σ = ( σ , σ , σ , σ ) , r = p X i =1 ( x i − x i ) , r k = ( x k + x k ) + p X i =1 ,i = k ( x i − x i ) , P ( r ) = (cid:0) r (cid:1) − α ,α = α + α + α − p , σ k = r − r k r = − x k x k r , k = 1 , , σ = − λ r . We calculate all necessary derivatives and substitute them into equation (1.1): X m =1 A m ω σ m σ m + X m =1 3 X n = m +1 B m,n ω σ m σ n + X m =1 C m ω σ m σ + X m =1 D m ω σ m + Eω = 0 , (4.2)where A k = − P ( r ) r x k x k σ k (1 − σ k ) , C k = 4 P ( r ) r x k x k σ k σ + λ P ( r ) σ k ,B k,l = 4 P ( r ) r (cid:18) x k x k + x l x l (cid:19) σ k σ l , k = l, l = 1 , , ,D k = − P ( r ) r ( − σ k X m =1 x m x m α m + x k x k [2 α k − ασ k ] ) , A = λ P ( r ) σ ,D = 4 P ( r ) r σ X m =1 x m x m α m + λ P ( r ) α, E = 4 αP ( r ) r X m =1 x m x m − λ P ( r ) . Using the above given representations of coefficients we simplify equation (4.2) and obtain the following systemof equations: σ (1 − σ ) ω σ σ − σ σ ω σ σ − σ σ ω σ σ + σ σ ω σ σ +[2 α − ( α + α + 1) σ ] ω σ − α σ ω σ − α σ ω σ + α σ ω σ − αα ω = 0 σ (1 − σ ) ω σ σ − σ σ ω σ σ − σ σ ω σ σ + σ σ ω σ σ +[2 α − ( α + α + 1) σ ] ω σ − α σ ω σ − α σ ω σ + α σ ω σ − αα ω = 0 σ (1 − σ ) ω σ σ − σ σ ω σ σ − σ σ ω σ σ + σ σ ω σ σ +[2 α − ( α + α + 1) σ ] ω σ − α σ ω σ − α σ ω σ + α σ ω σ − αα ω = 0 σ ω σ σ − σ ω σ σ − σ ω σ σ − σ ω σ σ + (1 − α ) ω σ + ω = 0 (4.3)5onsidering the solutions (2.5)-(2.12) of the system (2.4), we define the solutions ω i ( σ ) , i = 1 , ..., of the system(4.3) and substituting those found solutions into the expression (4.1), we get some fundamental solutions of theequation (1.1) q ( x, x ) = k (cid:0) r (cid:1) − α H , ( α, α , α , α ; 2 α , α , α ; σ ) , (4.4) q ( x, x ) = k (cid:0) r (cid:1) α − α − ( x x ) − α · H , (1 + α − α , − α , α , α ; 2 − α , α , α ; σ ) , (4.5) q ( x, x ) = k (cid:0) r (cid:1) α − α − ( x x ) − α · H , (1 + α − α , α , − α , α ; 2 α , − α , α ; σ ) , (4.6) q ( x, x ) = k (cid:0) r (cid:1) α − α − ( x x ) − α · H , (1 + α − α , α , α , − α ; 2 α , α , − α ; σ ) , (4.7) q ( x, x ) = k (cid:0) r (cid:1) α +2 α − α − ( x x ) − α ( x x ) − α · H , (2 + α − α − α , − α , − α , α ; 2 − α , − α , α ; σ ) , (4.8) q ( x, x ) = k (cid:0) r (cid:1) α +2 α − α − ( x x ) − α ( x x ) − α · H , (2 + α − α − α , − α , α , − α ; 2 − α , α , − α ; σ ) , (4.9) q ( x, x ) = k (cid:0) r (cid:1) α +2 α − α − ( x x ) − α ( x x ) − α · H , (2 + α − α − α , α , − α , − α ; 2 α , − α , − α ; σ ) , (4.10) q ( x, x ) = k (cid:0) r (cid:1) α +2 α +2 α − α − ( x x ) − α ( x x ) − α ( x x ) − α · H , (3 + α − α − α − α , − α , − α , − α ; 2 − α , − α , − α ; σ ) , (4.11)where k , ..., k are constants which will be determined at solving boundary value problems for equation (1.1). Let us show that the found solutions (4.4)-(4.11) have a singularity. We choose a solution q ( x, x ) . For this aimwe use the expansion (3.16) for the confluent hypergeometric function (2.3). As a result, solution (4.4) can bewritten as follows q ( x, x ) = r − p r − α r − α r − α f (cid:0) r , r , r , r (cid:1) , where f (cid:0) r , r , r , r (cid:1) = k ∞ P l,m,n,s =0 s P q =0 P I (3 ,s ) A ( s, q ) s ! C qs ( − q ( α ) l + m + n + s ( α ) i + l + m ( α ) j + l + n ( α ) k + m + n ( α ) s (1 − α ) q (2 α ) i + l + m (2 α ) j + l + n (2 α ) k + m + n i ! j ! k ! n ! m ! l ! q ! · (cid:16) − r r (cid:17) i + l + m (cid:16) − r r (cid:17) j + l + n (cid:16) − r r (cid:17) k + m + n (cid:0) λr (cid:1) q · F (cid:16) α − α − j − k, α + i + l + m ; 2 α + i + l + m ; 1 − r r (cid:17) · F (cid:16) α − α − i − k − m, α + j + l + n ; 2 α + j + l + n ; 1 − r r (cid:17) · F (cid:16) α − α − i − j − l, α + k + m + n ; 2 α + k + m + n ; 1 − r r (cid:17) F (cid:0) − α + q ; λ r (cid:1) , (5.1) k = 4 α + α + α − Γ( α )Γ( α )Γ( α )Γ( α ) π p/ Γ(2 α )Γ(2 α )Γ(2 α ) . Following the work [8] and applying several times a well-known summation formula [4, p.61,(14)] F ( a, b ; c ; 1) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) , c = 0 , − , − , ..., c − a − b > , it is easy to show that f (cid:0) , r , r , r (cid:1) = 4 α + α + α − π p/ Γ (cid:18) p − (cid:19) , p > . (5.2)6xpressions (5.1) and (5.2) give us the possibility to conclude that the solution q ( x, x ) reduces to infinity ofthe order r − p at r → . Similarly it is possible to be convinced that solutions q i ( x, x ) , i = 2 , , ..., also reduceto infinity of the order r − p when r → . References
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