Method of Riesz potentials applied to solution to nonhomogeneous singular wave equations Текст научной статьи по специальности «Математика»

Математические заметки СВФУ Июль—сентябрь, 2018. Том 25, № 3

УДК 517.9

METHOD OF RIESZ POTENTIALS APPLIED TO SOLUTION TO NONHOMOGENEOUS SINGULAR WAVE EQUATIONS E. L. Shishkina and S. Abbas

Abstract. Riesz potentials are convolution operators with fractional powers of some distance (Euclidean, Lorentz or other) to a point. From application point of view, such potentials are tools for solving differential equations of mathematical physics and inverse problems. For example, Marsel Riesz used these operators for writing the solution to the Cauchy problem for the wave equation and theory of the Radon transform is based on Riesz potentials. In this article, we use the Riesz potentials constructed with the help of generalized convolution for solution to the wave equations with Bessel operators. First, we describe general method of Riesz potentials, give basic definitions, introduce solvable equations and write suitable potentials (Riesz hyperbolic B-potentials). Then, we show that these potentials are absolutely convergent integrals for some functions and for some values of the parameter representing fractional powers of the Lorentz distance. Next we show the connection of the Riesz hyperbolic B-potentials with d'Alembert operators in which the Bessel operators are used in place of the second derivatives. Next we continue analytically considered potentials to the required parameter values that includes zero and show that when value of the parameter is zero these operators are identity operators. Finally, we solve singular initial value hyperbolic problems and give examples.

DOI: 10.25587/SVFU.2018.99.16952 Keywords: Riesz potential, Bessel operator, Euler—Poisson—Darboux equation, singular Cauchy problem.

1. Introduction

M. Riesz [1,2] has created a new method for solution to nonhomogeneous hyperbolic equations by generalization of the fractional Riemann- Liouville integral. We generalize and apply this method to solution to hyperbolic equations with Bessel operators acting by all variables. This method involves overcomes difficulties within the theory of hyperbolic differential equations which are due to the occurrence of divergent integrals. Namely, following to [2] we give solutions to the hyperbolic nonhomogeneous equations with Bessel operator acting by all variables using if nesessary the analytical continuation of an expression which depends analytically on a parameter. In the present paper we apply method of Riesz potentials only to equation in two dimensional case. The construction of analytical continuation presented in this paper can be applied to the multidimensional case but formulas will be cumbersome and we hope in future, we will be able to construct more conviniant technique of analytic continuation in the multidimensional case.

© 2018 E. L. Shishkina and S. Abbas

We consider two differential equations

□y u(x) = f (x) (1)

and

(□Y - c2)u(x) = f (x), c e R, (2)

where x = (xi, X2), xi > 0, X2 > 0, 7 = (71, 72), 71 > 0,72 > 0,

□Y = (BY1 )x1 — (BY2 )X2 , d2 v d 1 d d

is the singular Bessel differential operator Bv. These equations appear in various areas of mathematics and physics, such as the electromagnetic field and the meson field [3], the propogation of sound [4], random motions [5] etc.

Algorithm of constructing of Riesz potential generalized by operator L and application to solution of differential equations with this operator L is next.

(1) An integral transform J?"l convenient for working with operator L is chosen (for example, ^l is Fourier transform when L = □, ^l is Hankel transform when L = □). For sutable functions f we have J?"LLf = PJ^f, where P is a symbol of operator L.

(2) Fractional negative power of L or Riesz potential is constructed by formula

f = r<PLf. Here can be generalised functions, for

example when P is indefinite quadratic form.

(3) Integral representation of Riesz potential for operator L is realized in the form of convolution Iaf = (J^P^t * f )^. The convolution (• * -)l must correspond to the chosen integral transform J?"l .

(4) The obtained integral Iaf is studyied for absolute convergence for some class of functions f. It is found out at what values of the a this integral converges absolutely. Other properties, such as boundedness, semigroup property, etc. can be studied.

(5) Additional conditions on the function f for which equality /a+kLf = Iaf for some natural k (for example, k = 2 when P is a quadratic form) is true are clarified.

(6) By constructing an analytic continuation (or without it if possible) should be shown that for a = 0 the potential Iaf is the identity operator I0f = f for some class of functions.

(7) Using obtaining results one can easily write a solution to the equation Lu = f for some class of functions f. It is just nesessary

to apply Ia+k to

the both side of Lu = f: Ia+kLu = Iau = Ia+kf. Then putting a = 0 we get u = Ikf. Here an analytic continuation Iaf is used of if needed.

It is easy to see that using this scheme, we can also obtain a solution to equation Lmu = f with iterated operator L.

Algorithm of constructing of Riesz potential is closed to the compositional method developed by S. M. Sitnik (see [6-9]).

2. Basic definitions

We deal with the subset of the Euclidean space

R+ = {x = (X1,X2) € R2, xi > 0, X2 > 0}.

Let x = (xi, X2), |a;| = \]x\ + x\ and il be finite or infinite open set in R2 symmetric with respect to each of coordinate axes x1 =0 and x2 = 0, = £ n R+ and n+ = tt nR2_, where

= {x = (x1,x2) G R2, xi>0,a;2>0}.

Consider the class Cm (£+) consisting of m times differentiable on functions and denote by Cm(il+) the subset of functions from Cm(il+) such that all derivatives of these functions with respect to x1 and x2 are continuous up to x1 = 0 and x2 = 0. Class C™(i2+) consists of functions from Crn{VL+) such that

Q2k+1 f

Oxf+1

ß2k+1 f 0, J

x=0 dx2

dx2k+1

= 0

x=0

for all nonnegative integer k (see [10, p. 21]). In the following, we will denote C™( 1+) by C™ We set

with intersection taken for all finite m. Let C^(R^) = C

The symbol jv is used for the normalized Bessel function:

. , 2Vr (v + 1) . ,

j At) = ——LMt),

where Jv (t) is the Bessel function of the first kind of order v (see [11]). The function jv (t) is even by t and

^(o) = i, im

= 0. (3)

t=0

Using formulas 9.1.27 from [12] we obtain

(Bl,)t3^i(rt) = -T23^1(Tt). (4)

We deal with multi-index 7 = (71, 72), 71 > 0, y2 > 0, |y| = 71+72. The operator kTtT for k > 0 is generalized translation acts by a variable t defined by the next formula (see [13, p. 122, formula (5.19)])

n

Y(h±l) r

kT?ip(t) = J / tp{ Vi2 + r2 - 2tr cos tp) sin*-1 V dtp

v^U) J

(5)

and

YT9 = Yi TVi Y2 TV2

I x xi X2

X Xi X2

where each of the one-dimensional generalized translations YiTXj,1, i = 1, 2, acts by a variables x1 and x2 according to the formula (5).

It is known that

fcTtT7fc-i (at) = jk-i (at)jk-i (a,T), (6)

where a G R.

We introduce hyperbolic Riesz B-potentials in two dimentional case (see [14])

1 f c-|-r| -,

^!){x) = ^TJ0) J 2 CnfX^dy, y^yfyf, (7)

and

2+ | Y I — a

n- 2- ( a-l-rl-2 j-

= J (vl-vl) 4 Cnf)(x)K._±M^(c^y22-y21)y~< dy

«2,7 (a)

(8)

where c G R, 7 = (71, 72), 7i > 0, 72 > 0, |y| = 71 + 72, a > |y|,

= {0 < yi, 0 < y2 < yi}, = {0 < y2, 0 < yi < y2}. In (8) Kv (x) is the modified Bessel functions of the second kind

Kv{x) = ^-I-Ax)-I;{x\ V<£Z, (9)

2 sm(vn)

and (x) is the modified Bessel functions of the first kind

1 / x \ 2m+v

W=E iri + Ml j ' (10)

^^ mil (m + v +1) \2/

m=0 v '

When v is an integer the limit is used.

For the modified Bessel function Kv asymptotic for large argument x is

Ku(x) ~

and asymptotic for small argument x is if v > 0.

3. Absolute convergence

Theorem 1. Let f €Sev. Then integrals Jg f and Jg cf converge absolutely on R+ when |y| < a.

Proof. We have

TO yi

1 C C c-l-rl -,

= J^J J yT ^ J (y* 2 " №)(*)*? dy2 = 0 0

{y2 = y1t}

TO 1

1 f ,,a-1 1, [ f-i ,2\

■ J vr1dy1 J(1 -i2)^-1^^^^/)^ dt.

Using formula

(a)

00

v TT g(h) = v T^T)

and property

vTTg(h)|<sup |g(h)|

R+

(see [13, p. 124]) we obtain

TO 1

C f iia~ f o-|t|_,

\(mj)(x)\<—— dyihi-t2) ^ -w<oo.

Similarly for Jg cf we get

2+|Y|-a TO y2

- /" /" c-l-,1-2

№,,c/) (*) = ^r^y J y? dy2 J (y2 - y\) 4 CTlf) (*) ' 0 0

x K2+H-c - yf^yj1 dyi =

il

{y1 = y2t}

c 2 / S+M / o. c-l-,1-2

' 2 J.. / /1 _ -

J yf^ dy2 J (I- t2)^1 CTy? t*nf!){x)

(a)

00

x K 2+1-,1-e »(cyavl -i2)i71 dt.

Taking into account (11) we obtain

2+H-a TO „ 1

i j, _!_!_U_ n Q;—X /.

So integrals Jg f and Jg cf converge absolutely on R+ when |y | < a and f € Se

4. Connection of the Riesz hyperbolic B-potential with QY and QY — c2

In this section we show that Ig+2QYf = Ig^f and Ig+2c(QY - c2)f = Ig^f. Theorem 2. For a > |y| +2, f e Sev, such that

lim x!?j{x) = 0 and lim = 0 (12)

X2 >0 2 X2^0 2 dx2

the next formula is valid

Ig+2Qy f = IS f, (13)

where QY = BY1 — BY2.

Proof. Using property 1.8.3 Yi(By.)Xi = (By.)£(see [10]) for i = 1,2 we obtain

1 f c-l-r

(I^n,f)(x)= ^{a + 2) J ijjl-vl) 2 Cn^)xf(x))ytdy

JT+

1 /" c-l-r

= ^ (a + 2) J 2 ((B^)yi - (B72)y2)CTvj)W dy

1+

1 ^ ^^-y2)^^),^7^/)^^

(a + 2)

+

a— | ~y |

I-Vl) 2 (B72)y2CTyj)(x)y^

1+

tt CO

1 li i c-l-r

y272 dy2 J (yi - yl) 2 (B71 )V1 Cnf) {x)vT dy!

(a + 2)

0 V2

yi

- J yV dy 1 J{yl-yD^iB^CTlf^yf dy2 I • 00

Let

h = ¡(yl- yl) 2 (Bn )yi (7T*/) (x)yi1 dj/!

and

Vi

a— | -y |

h= J (yi - yi) 2 (S72)y2 0

Now we consider Ji:

a— | Y |

vl-vl

y2

dy1

a— | ~y |

2 , dv = —tf — (>T«f)(x)dy1

a —|Y | i d

= {v\-vi)~^~v? ¡Jj-CTVfXx)

d pryy

dy1

f TXf (x)

a— | Y |

yT^ivl-yD^dy,.

Taking into account that f € Sev and |y| + 2 < a we get

J1 =

¿Ct;/)(i>

a— | y|

vT^ivi-vD^d.y,

y2

/^J a — | y |

9 ',2 „2

dy1

+ / ( 7TX f )(x) yTXf )(x)

a— | y|

(.B7l)Vl(yi-y22) 2

y2 TO

(BYi )yi (y2 - y|)

yi=y2

y 71 dy1 y71 dy 1

(|7|-a) CTyj)(x)(yf-yi) 2 ((1 + 72 — ot)y\ + (1 + 7i)j/|)yi1 <^yi

y2

The same transformations to the integral J2 with considering (12) gives

J2 = y12 - y22

a—| Y | 2 2

dy2 =

a— | y|

«=(y?-y22) 2 , dv = —y? — CTyj)(x)dy2

a —| y I r)

(vi-vD^vf^Tim

y2 =0

d

vT-^ivl-yl) 2 dy2

a— I Yl 2

TO

1

2

¿C t;?/)(,)

Q PL— |-y|

V22Q^{V1-Vl) 2 dy2 =

/^J a — | y |

■=yr 4{yl -yl) ^' dv=wSTVj) {x) dy2

+

0 Vi

=y

TV f (x)

TXf)(x)

a— |~y|

(B-r^yM-vi) 2

a— |~y|

V2=0

yl>2 dy2 y^T dy2

0

Vi

/* a— I -y _q

Then we have (Ig+2Qy f )(x)

ItI - Q

¿C,7(a + 2)

x \ yfdy2 ^Tyj)(x)(yl-yl) 2 z((l+72-a)y2 + (l+7i)y22)y71^i

0

CO

V2 Vi

- J yf dVl J (^Tlf)(x)(yl -yf)^-2((l +72)y2 + (1 +71 - a)y2) J/71 dj/1 00

Vi

ItI ~ Q

¿C,7(a + 2)

tt

yYi dyi / (YTXf) (x)(y2 — y!)

2

2

X [(1 + Y2 — a)y2 + (1 + 7i)y2 — (1 + Y2)y2 — (1 + Yi — a)y|)]yY2 dy2

tt Vi

a(a — |Y|)

Jf2,Y (a + 2)

00

Here we took into account that

Jf2,7(a + 2) _ 2a

-m .

y?1 dj/! / ( - y2) 2 ^ dy2 = (Pn/IW.

a(a — |y|) na(a — |y |)

sin

nw'^W^+iW^M+i

i=1

ia-2

sin

n v 2

-sin

n

7l + n2rfa+i

2 a V 2

2 tT(^M + 1

' — IYI

n

2

x

n

Theorem 3. For a > 171 +2, f G Sev, such that

„„J n:__s^+72 df

lim 1, 2 fix) = 0 and lim x92 = 0, (14)

X2 —>0 2 w X2—0 2 dx2 v 7

1S+;C(°7 + c2)f = ISy f (15)

the next formula is valid

where Q = BYi — BY2.

Proof. Using property 1.8.3

Yi TV. (By. )xi = (BY. )Yi TV. (see [10]) for i = 1, 2 we obtain

H-Q .. . .

(^:2c(a7 + ^2)/)W = ^7(Q + 2) J (vl-vi) 4 №d7+ *%/(*)]

2+

C 2 I I . I , ^ a~ I "y I

^ Hvl ~ vl)v^ dy = J V? dy2 J (yi - yf) <

00 (lrry fV^l K-, (n. A,2_„2^

x [(B7l )yi ÇTvj) (a;)] K^ {cy/y£^)y? dy 1 - JyT dy 1 J(y\ - yï)^ [(B72)y2CTyj)(x)]Khl^ (c^yl - yj)y? dy2

u X J , _

2

0 yi

+ c2 (y!-yî) 4 [^J{x)]K^{cJyl-yl)y'dy

Let

y2

2+

h = I (vl - y2i) [(B7l)yi (7TIf^K^ (cJyl-yl)yT dVl

and

xJ , . . _

2

0

q— l~y|

h = I (yl - yi) 4 [(B,2 )y2 (7T If) (a;)] K ^ (c^ y\ - yl)yf dy2

yi

Taking into account that f G Sev and conditions (12) we get

f a — | Y I

h = J(yl-yï)^ 0

K 1-,1-c (c\J yl ~ y\) dy1 = s^M , d <9

/ oc— I -y I f)

K^c^yl-ylM-y^yT^TlDix)

dy1

C TXf (x)

d '„2 „.2^^

dy1 d

dy1

(y22-y?) 4 K^tcJyi-yf)

yi=0

y?1 dy1

g— l~y|

(y22-y2) 4 K^icJyl-yl)

Yi

u = y1

A

dy1

^M^ {cJVl ~ vi){vl ~ vi)

dy1

+ C TX f) (x)

dv =

It

y?1 dy1 = dy1

-J/71(C\M - 2/?) (y22 - 2/?)] №)(*)

0

y2

= i (?

d Y d

It

y1=0

dy1

J (TT If) (x) [(B71 )yi K^ (c^yi-yj) (yi - yi) ] yT dy!

0

g-|-y|-2 /-

c I +7i)(y22 - J/?) 4 K^^yi - yf)

o-H-4

hl^+2(cyy2 ~ vi)]vT dyi

and

J2

7Tvxf){x) [(Bl2)V2K\_j\-o_ (cJyi - yf)(y22 - y2) - " dy2

It

cJ ^Tlf)(x)[yl(yi-y\) 4 K^^yi-yl) y1

a— |-y| — 2

Then we have

2+ItI-C

(Jg+2(nY+cf (x)

ac 2

2+

x ^2+1-,1-c (W- y\)y7 dy.

Here we took into account that

(a + 2) 2a . /71 + 1

—-- = — sin -7r

a na V 2

nr(^)r(f + i

d

4

TO

v ' i=1 v ' ">a— 1 / . \ 2

5. Identity operators

Theorem 4. Let f € Sev, 2+y2 — Y1>0, Y1 is not odd, then hyperbolic Riesz B-potentials Jg f and Jg cf can be analytically continued to all values a > — 1 and (JgYf)(x) = f(x), (JgY,cf)(x) = f(x).

Proof. Let S > 0. We denote K+ the triangle bounded by the lines y1+y2 = S, y1 = y2 and y1 = 0. Then + = K+ U (jf + \ K+). We consider first (Jg f) (0):

1 /" c-|-r|

+

=^2,7 (a)

where

1 r c-l-rl -,

K+

1 /" c-l-rl -I

JT+ \K+

We will show that Jf and J2* are holomorphic for a > —1 and J0 = f (0), J° = 0 which gives that (Jg f)(0) = f (0).

Consider Jf. Changing variables by formulas

yi = ^(l+x), № = (16)

noticing that ^ = ■5°' and V = + xc), where b, c £ R^ we obtain y\ — y\ =

and

5 1

0 0

We develop f + xc)) by the Teylor formula in x:

N-1 p

/(y) = /(*(& + xc)) = ^FP(<T, 9) + RN(x), p=0 p'

where

dp

x=0

and

X

1 r dN

RM = (]v3T)T J + Xc))(x - X)"-1 dx-0

Then

7° = 91+H1 E ^ [ F^e^-U*

1 21+I7l^2i7(a) ^ p! ^ 1

0

5 1

+ J cr^dcr J X^-Hl + x)71 (1 - x)l2RN{x)dx ) • (17)

Integral

J X^~1+P(1+X)71(1 -xV*dX 0

is integral representation of the Gauss hypergeometric function

1

2F1(a,b;c;z) = T{b^-b)J ^^ ~ ^"'(l ~ dt> c > b > °>

0

for b = + p > 0, c-6 = 72 + 1>0, c = "-71+72 + p+l, a = -71. It is

known that 2F1(a, b; c; z) is defined for z = —1 when c — a — b > 0. Since in our case c — a — 6 = I7I + 1 > 0 it can be analytically continued to " J7 + p < 0 as the power series. So we have

1

x^-1+p(i+x)71(i - x)72 dx =

r(°+P + 1)

~ t? 1 a ~ M , Q - 71 + 72 . , ,

x 2F1 ( -7i, —o---2--P ' ~

It means that we have analytic continuation of If to all a > 0. Integrating by parts the integral by <r in (17) we obtain analytic continuation of If to all a > —1. Let

Kp(a) 11 1

Oa+ I7I — 1 2

i=1

r(^+p)r(72 + l) / «- ItI .a-71+72

r(a±7i+p+l) "2-1 ^ /1. 2 ' i-, 2

pi -7i, —tt11 +P>-n-—h p + 1; -1 .

The most important term in (17) is the term with p = 0 has the form

f^ J fi^a-Ha. (18)

Using formula 15.1.21 from [12] of the form

2F1(a,b;a-b+ 1;-1) =-^^ ~ b. + l) . , a - b + 1 ^ 0, -1, -2,...

n . . . ) 2ar(l + f -6)r(*±i)'

and taking into account the Euler's reflection formula

n

r(l - z)T(z) =-—-sin (nz)

and

T(z)T (z + ^J = 21-2^T(2z).

we get for 2i±i £ Z

n r (72 + 1)

Ko(0) =

2|y |-1 2

i=1

_7T__r(72 + 1)_+

2'7hl sin (2i±i^) r (2i±i) r (M + i) r (f + !) r №)

i=1

V5F r(72 + l) _2 (ig)

272-1 r (12±1) r (^ + 1)

Now we carry out the analytic continuation of the expression (18). The factor K0(a) has no singularity at a = 0 and for ^ Z and the formula (19) is valid for Ko(0).

5

Integrating J f (<T&)<ra-1 da by parts we get 0

^ J f{vb)aa-1da = |Lo " J

and

5

H ff^j I = fw -1 da = fw - fw + =

a—>0 ^ ,

o o

à

Now we show that for p = 1, 2,... all summands in (17) are equal to zero. lying f we obtain

Applying formula + to + 1) = z(z + 1) • • • (z + m)T(z), to (e N to r(" J7 + p)

Kp(a)

Oa+lfl-l 2

sin(^)nr№)

i=1

„ r(72 + 1) „ A a - ITI , a - 71 + 72 . , ,

x —;-\-r 2-ri —7i,--h p;--h p + 1; —1

r(a-7i+72 +P+1) 1 ^ 2 2 1 '

That means that Kp(0) has no singularity at a = 0. For any positive integer p we have p

+ = f, (20) where = y = a(b + tc). Hence all intergals

diIk '

1

0

converge for a > — 1 when p = 1,2,..., Kp(0) is finite and lim , a > = 0 we get

that all summands in (17) for p = 1, 2,... are equal to zero.

Now we consider I^. Making the change of variables (16) in the last expression we obtain

1

<2 -01 + i-vi*. /-A / X^-'H+XrHl-X^dx J fiaib + xc^-Ua.

21+Iy^2y (a)

0

Since f G Sev and S > 0 then the function

tt

G(x,6,a) = J (a(b + Xc)) da

5

is in Sev by x as well as by 6 and holomorphic in a. Assuming n 1

^TH^T-2-(! + (! " xV2G(x, e, a) = W(X)

n r (*±i)

i=1

we get

1

1

p- 1

l2

r(f)r(2d2l)

The expression

1

1 / l~r

r№)0

JX^-'WU) dX

n

can be continued analytically as a holomorphic function of a to any a > a0 by

integrating by parts, where ao is arbitrary. Since I2 contains a factor } it

rv?)

vanishes when a ^ 0. This completes the proof of the fact that )Jg f)(0) = f (0). Taking g(x) = (7TXf (x)) instead of f (x) we can write (Jg f) (x) = f (x) that means that Jg0 is the identity operator. For Jg cf the proof is analogous.

6. Analitical continuation

In Theorem 1 we proved that (7) and (8) converge absolutely for function f €Sev when a > |y|. But for solution to (1) and (2) we should to construct analytical continuations of (7) and (8), defined for a < |y| and show that Jg f = f and

Jg y ,cf = f .

Lemma 1. For |y| < a next formulas are valid

= MTJo) J dy, (21)

'7 R+

1 i , , c-|-r

(m„cf)(x) = / Iy\72-1KI±hl^(cy2)(^Tyi T*TMf)(x)y?y2 2 dy.

2

(22)

Proof. We have

1 r C-2-H

(Igj)(x) =J (yl-yl) 2 {y)FTZf){xWdy 1+

CO yi

1 f f a-2-l-r

VT dyi J (y\ ~ yl) 2 (71 Tl\ 72 T^2/) W2 dV2-

^2,7 (a)

00

Let

yi

f C-2-H

h = J (yi - yl) 2 (y) (7lT£ ^TIU) [x)yf dy2 0

yi n

Я/--o-2-H

f{x 1, y/x* + vl + 2V2 cos <p) (yl - yl) 2

00

x y?2 sin72-1 ^d^dy2

yi n

/--C-2-H

= C(72) J J 7lT^f(Xl, y2 cos v + x2f + yf sin2 <p) (yl - yl) 2 00

x y?2 sin72-1 ^d^dy2.

Passing to the polar coordinates

z1 = y2 cos z2 = y2 sin 0 < ^ < n, (23)

+

in I and putting

{|z| <yi}+ = {zi G R, Z2 G R+ : |z| < yi}

we obtain

ct-2-H

h=C( 72) J ^1f(x1,y/(z1+x2^+zi)(y21-\z\2) 2 Z72dzidz2

{|z|<yi} +

{(zi + x2) ^Zi}

ct-2-H

= C(72) J ^/K^z2 + z2)(y2- |z~|2) 2 z72 dzidz2,

{|S|<yi} +

where |z| = -y/~~ X2)2 + z|.

Consider a part of a sphere in a space of dimension 3 of points (zi, z2,£) with center in (x2, 0, 0) and with radius yi:

(zi - x2)2 + z2 + £2 = y2 or |z|2 + £2 = y2, zi G R, z2 G R+, £ > 0.

The projection of this part of the sphere onto the plane £ = 0 and the surface element are |z| < yi and dS = ^j-dz respectivly. Considering that £ = (y2 — |z|2)2 we can rewrite the integral by |z| < y1 in the form

h = C(72) J y/z* + z2) (y2 - |z~|2) dzidz2

{|S|<yi} +

= C(72) f

yi J

{|^|2+c2=y2}+

YiTfif (xi, Jz? + z2)£a-i-|Y|z2Y2 dS.

Using the obtained representation for I we get

(ISy f )(x)

C(l2) J yf-1 dVl J yjzl + zD^-^zf dS :

0 {|s|2+c2=y2}+

{zi - x2 ^ zi}

C(72) J yf-1 dy ! J ^Tgf(Xl, yj{z1+x2)2 + zD^-^zf dS

0 {|z|2+c2=y 2}+

= C(72) /(M2 + C2)21^ + X2)2 + Z2)

x £a-i-|Y|zY2 dzidz2d£ = {zi = y2 cos z2 = y2 sin

C»

—

= C(72) / r~1H7' ^ / (y22 + C2) 2 y272

+

x J f{x!, yj(2/2 cos f + x2)2 + yl sin ^) sinl^ipdtp

o

oo oo

= J r-1"171^ j 11T^W2^Tyif(xl,x2)(yl+e)llTly'l2dy2

oo

Renaming the integration variable £ by y1 we obtain (21). For /g f we have the same situation:

C 2 I , ^ ^ "-ItI-2

2

!,Y (.....

JT2+

f^-')"'= few J M-rfJ^-r-p/Jw

"-2

x jfa+i-ri-c (cy yf - y?)y7 dy

OO y2

[(yl-yD^^CTVfXx)

(a)

oo

x K^2+ ¡-yi—o; (c Jyf - y2)y7'

Let

^2

a-|-y|-2 y-

/2 = J (yl - vl) 4 (7lT£ ^Tllf) (x)K2_±1^ (c\jyl - yi)yT dy ! o

1/2 7T _

= C(7i) J J "2Tgf(y/a* + y2 + 2xlVl cos x2) (y2 - yf) ^^ oo

x (cy y% - y^yj1 sin71 1(fid(fidy1

V2 IT

= C(71 ) J J 72T«V( v(yi cos ^ + + y2 sin2 ^ (y2 _ y2)^^ oo

x JCa+i-Ti-c (cyyl - yf)yT sin71 1ipd,ipd,yi.

Passing to the polar coordinates z1 = y1 cos z2 = y1 sin 0 < ^ < n, in J and putting {|z| < y2}+ = {z1 € R, z2 € R+ : |z| < y2} we obtain

h = C{7i) j ^Tyif(^(z1+x1f + zlx2)(y22 - Izl2)^

{|z|<y2} +

x K2+hl-a (cyy I — |z|2)z21 dzidz2 =

{(Z1 + X1) ^ Z1}

C(71) j l2Tyif(^zl+zlx2)(yl - |z~|2)"

{|^|<y2} +

i-2-H

1

x K2+ h-c {cJy2 — \z\*)z? dz\dz2,

where \z\ = \/(zi — xi)2 +

Consider a part of a sphere in a space of dimension 3 of points (zi, z2,£) with center in (xi, 0, 0) and with radius y2:

(zi - xi)2 + z2 + £2 = y2 or |z|2 + £2 = y2, zi G R, z2 G R+, £ > 0.

The projection of this part of the sphere onto the plane £ = 0 and the surface element are \z\ < y2 and dS = ^dz respectivly. Considering that £ = (yf — |z|2)2 we can rewrite the integral by |z| < y2 in the form

ct-2-H

h = C(7i) j "»TVZfiyJz* + zlx2)(yl - |z~|2)

{|2|<V2} +

x K2+ 1-,1-c {c\jy\ - \z\2)z^ y dzidz2

= ^ J ^Ttljj^z2 + zlx2)K2+ira (cQt^z? dS.

{|®|2+€2=y|}+ Using the obtained representation for I we get

œ

xjyf-1dy2 f ^f^/z2 + zlx2)K2+lll-a dS :

0 {|s|2+C2=y|}+

{zi - xi ^ zi}

œ

J vTXdy2 j ^f(y/(z1+x1)^+zlx2)

CM

«2.7(a)

0 {|z|2+i2=y|} +

x JCg+iTi-c (c^^-z71 2

- i

/ (M2 + n22-1 + X,)2 + 4 *a)

q— l~y|

x ^2+i-yi-c 2 ¿J1 dzidz2<i£ =

2

{zi = yi cos z2 = yi sin

œ œ

c(7l) Ui+^icm^dï, [{yi+e^yTdy,

oc> t \ I " ±XLU -01

»2,7(a) J 2

00

x J 727^1+« /(y/(y1 cos</7 + X1)2 + yfsin2</?, x2) sin71 1 ipdip

+

n

OO CO

— 1 /..2 I /-2

_I_ /V-., , i.iw^ ^f f l^^&Y2^ tltw 72^^?+«'

,7(a)

(yl+e) 2 f(xi,x2)y? dVl

X i X2

00

2,7(a)

7

R+

/a— | -y |

K2±M^(cy2)y2 2 Ij/r2-1!71^!172^1/)^?1^.

So we obtain (22). This completes the proof.

Integrating by parts (21) and (22) we obtain analitical continuation of Jg f and

Jg1,cf.

First let a > |y|. For Jg f we have

O CO

= ^TM / / Ifl71-1^1 ^/K^r1-'7' dVl.

2,7(a)

00

Let in the inner integral

u = |y|7i-1(7i TXy1 72 TXy22 f )(x) dv = yr1-171 dy1

then

CO

J |y|7i-1(7iTXy1 72TXy22f)(x)ya-1-171 dy1

0

= M71 ~1 (71 172 Ty22 /) (x) y i

yi=0

/(¿lyl71-^71^72^2/)^^171^! 0

CO

We have

where

n

F = C (71)|y1|7i-1/

yi|y| -Xiyicosip \yW\y\2 +x\ — 2xi|y| cos </?

1

Then

x /Kl/M2 + x? - 2x 11y| cos,J,x2) sin7i-1 ^d^ + (Y1 - 1)|s|7'-37iT^/(*1,*2).

2,7 0 0

The resulting integral converges for a > |y | — 2 and can be considered as a continuation of Jg f. Further integration by parts allows us to extend the integral to other values a.

For Ja f we have

gY

(JgY,Cf )(x)

CO CO

^r^y j VT dyi j 2 dy2.

00 Let in the inner integral

= |y^-'i71^ 72TlyJf)(x) dv = K,+ M-a(cy2)yr^ dy2

then

CO

j \yV2-1K1±]^{cy2)^Tyxl f){x)yT^ dy2

a— I ~y I

—K^{cy2)y2 2 |yV^^Ty^T^f^x)

C 2

O

y 2=0

/a-|7|

a— t C 2

+ 1

r ^ -»!%/)(*)

0

So we can use the formula

CO

+ -J K^{cy2) (A^-I^i "T^fi^y^ dy2.

O

!,7V

0

where

O

1 /* /A , , \ Q— I ~y I

F2 = -J K^(cy2) 72TlvJf)(x)) y2 2 dy2

02

as an analitical continuation of /g cf for |y| — 1 < a < |y|.

u

O

7. Solution to nonhomogeneous singular wave equations

Let we have equations

□7 u(x) = f (x),

(□7 - c2)u(x) = f (x), c G R,

where x = (xi,x2) G R2, f (x) G Sev (or some f from a wider class of functions such that the corresponding potentials exist and have the properties desired for the solution u). We will use considered Riesz type potentials to invert operators □ and □7 — c2 by application of the corresponding Riesz potentials to both sides of the considered equations. Function f also should satisfy to conditions (12) and (14), correspondingly (if we take function f not from Sev we should verify the validity of (13) or (15), accordingly). Then we apply to the both side of the equation □yu(x) = f (x) an operator 2, we get

Y

(/^□7u) (x) = (/§+2f) (x).

Now puttinq a = 0, taking into account theorem 4 we obtain a solution of □yu(x) = f (x) in the form

u(x) = (/§ Y f )(x).

j2

V

T2

V

Here if |y| < 2 we use formula (7) for Ig f if |y| > 2 we construct analytical continuations (see section 6).

Similarly we obtain that the solution to (d7 — c2)u(x) = f (x) is defined by

u(x)= (/2 Y C f )(x),

where f should satisfy to (14). If |y| < 2 we use formula (8) for Ig f if |y| > 2 we construct analytical continuations (see section 6).

Also the obtained solutions satisfies the conditions

u(0,x2) = f (x)|x1=0 and u(0,x2) = f)(x)|Xl=o,

correspondingly.

Example 1. Consider a problem

((B2)xi — (By)x2)u(x)= x2e-Xlj7(x2), Y> 0, (24)

u(0,x2) = 3jY(x1; b), uxi (0,x2) = 0. (25)

A solution to (24), (25) obtained by described method is

u(x) = 2e~Xl (xi + ^xi + 3)j7(x2).

The plot of u(x) for y = ^ is presented on Fig. 1. Checking we have

№k (xl + 3xx + 3)j7(x2) = (x2 - 3xx - 3)j7(x2),

and

Fig. 1. u(x) = (x\ + 3xi + 3)j7(x2)

(B7)X2{x\ + 3xi + 3) j7(x2) = ~\e~Xl {x\ + 3a?i + 3)j1(x2)

((B2)X1 - (B7)X2)-e-^ {x\ + 3xi + 3) j7(x2) = xie'^j^x2),

u(0,x2) = 3j7(x2), Uxi (0,x2) = 0. Example 2. Consider a problem

((Bi2)Xl - (B2)X2 - l)u(x) = j_i(x1)xl 7 > 0,

^(0, x2) = ^ (3 - , uXl (0, x2) = 0. A solution to (26), (27) obtained by described method is

u(x) = 7^(3 — x^)j_x(xi).

The plot of u(x) is presented on Fig. 2. Checking we have

(Bi)Xl Q(3 - a*)) =\(A- 3)j-i(®i),

(26) (27)

Fig. 2. u(x) = k3-x1)j_1(x1)

and

(b2)x2 Q(3 - ^i)j-i(^i)^) =-3j_i(x1)

((Bh)Xl-(B2)X2-l) Q(3 =j-i(xl)xl

^(0, x2) = ^ (3 - x\), uXl (0, x2) = 0.

REFERENCES

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2. Riesz M., "L'integrale de Riemann-Liouville et le probleme de Cauchy," Acta Math., 81, No. 1-2, 1-223 (1949).

3. Fremberg N. E., "Some applications of the Riesz potential to the theory of the electromagnetic field and the meson field," Proc. R. Soc. London, Ser. A, Math. Phys. Sci., 188, No. 1012, 18-31 (1946).

4. Darboux G., Lecons sur la Theorie Generale des Surfaces et les Applications Geometriques du Calcul Infinitesimal, 2, Gauthier-Villars, Paris (1915).

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6. Katrakhov V. V. and Sitnik S. M., "Composition method for constructing B-elliptic, B-hyperbolic, and B-parabolic transformation operators," Russ. Acad. Sci., Dokl. Math., 50, No. 1, 70-77 (1995).

7. Sitnik S. M., "Transmutations and applications: a survey," arXiv: 1012.3741 [math.CA] (2010).

8. Sitnik S. M., "A short survey of recent results on Buschman-Erdelyi transmutations," J. Ine-qual. Spec. Funct., Spec. Issue to honor Prof. Ivan Dimovski's contributions, 8, No. 1, 140—157 (2017).

9. Fitouhi A., Jebabli I., Shishkina E. L., and Sitnik S. M., "Applications of integral transforms composition method to wave-type singular differential equations and index shift transmutations," Electron. J. Differ. Equ., 130 (2018).

10. Kipriyanov I. A., Singular Elliptic Boundary Value Problems [in Russian], Nauka, Moscow (1997).

11. Watson G. N., A Treatise on the Theory of Bessel Functions, Camb. Univ. Press, Cambridge (1922).

12. Abramowitz M. and Stegun I. A., eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bureau of Standards, New York (1964). (Appl. Math. Ser.; V. 55).

13. Levitan B. M., "Expansion in Fourier series and integrals with Bessel functions [in Russian]," Uspekhi Mat. Nauk, 6, No. 2, 102-143 (1951).

14. Shishkina E. L., "On the boundedness of hyperbolic Riesz B-potential," Lith. Math. J., 56, No. 4, 540-551 (2016).

Submitted May 22, 2018

Elina L. Shishkina Voronezh State University,

Faculty of Applied Mathematics, Informatics and Mechanics, Universitetskaya square, 1, Voronezh 394006, Russia ilina_dico@mail. com

Syed Abbas

School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, H.P., 175005, India sabbas.iitkSgmail.com