On a Certain Class of Integral Equations Associated ... - Springer Link

Acta Appl Math (2007) 98: 135–152 DOI 10.1007/s10440-007-9151-9

On a Certain Class of Integral Equations Associated with Hankel Transforms P. Malits

Received: 14 July 2005 / Accepted: 8 March 2007 / Published online: 15 August 2007 © Springer Science + Business Media B.V. 2007

Abstract This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber–Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber–Orr integral transforms of functions belonging to L1 and L2 . These operators together with the proven Paley–Wiener’s theorem for the Weber–Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke–Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations. Keywords Fredholm integral equation · Dual integral equations · Hankel transform · Weber–Orr transform · Paley–Wiener theorem Mathematics Subject Classification (2000) 45B05 · 45F10 · 44A15

1 Introduction This paper deals with a class of Fredholm integral equations of the first kind arising from applying the Hankel integral transform to mixed boundary value problems in various branches of applied mathematics, physics and engineering 0

a

xyn (x)

∞

S(p)H (p)[1 + L(p)]Jn (pr)Jn (px)dp dx = fn [r],

0 ≤ r ≤ a. (1.1)

0

P. Malits () Department of Communication Engineering and Center for Appl. Indust. Mathematics at Department of Sciences, HIT-Holon Institute of Technology, 52 Golomb Street, Holon, Israel e-mail: [email protected]

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P. Malits

Here: yn (x) is an unknown function to be determined, xyn (x) ∈ L1 [0, a]; n is a non-negative integer; the function L(p) is bounded at p = 0, L(p) ∈ L1 (0, ε) and L(p) = c/p +O(1/p σ ) as p → ∞, σ > 1; ⎧ N 2 2 ⎪ ⎨ p + am 2 , N ≥ 1, 2 2 H (p) = ; (1.2) S(p) = p + bm 2 m=1 ⎪ πpλ[Jν (pλ) + Yν2 (pλ)] ⎩ 1, N = 0, am and bm are positive distinct numbers; λ > 0; ν > 0; Jν (p) and Yν (p) are the Bessel functions of the first and second kind, respectively. The right-hand side fn (r) is a continuous piecewise differentiable function whose derivative is the sum of a piecewise constant function and an absolutely continuous function, fn (r) = O(r n ) as r → 0. Upon interchanging the order of integration this equation is rewritten in the form of the integral equation ∞ An (p)S(p)H (p)[1 + L(p)]Jn (pr)dp = fn (r), 0 ≤ r ≤ a, (1.3) 0

in which the integral converges uniformly and a xyn (x)Jn (px)dx. An (p) =

(1.4)

0

Since the Bessel function Jn (px) is an entire function of exponential type x [3], An (p) is an entire function of exponential type ≤ a. On the real axis, p −n An (p) is bounded and An (p) → 0 as p → ∞. Once An (p) is found, the solution may be obtained with the inverse Hankel transform. In the simplest case N = 0, ν = 1/2, we have H (p)S(p) = 1. Then (1.3–1.4) become equivalent to the well-known dual integral equations which were studied by many authors. The most important results have been obtained in the original papers by Cooke [2] and Lebedev [5], which gave rise to a large stream of applications in physics and engineering. The formal results established by Cooke and Lebedev can be also derived with the multiplying factor method [4, 9, 10, 12, 14] based on the Erdelyi–Kober operators. Explicit solutions for ν = 1/2, L(p) = 0 were discussed in [1]. Some formal procedures regularizing the dual integral equations which are equivalent to (1.1) as ν = 1, 2 and n = 0, 1 were first given by the author [7, 8]. In this paper we give a rigorous method to regularize (1.1). It will be demonstrated in Sect. 6 that the structure of the kernel of the integral equation (1.1) provides us with more flexibility and enables highly efficient regularizations for equations arising in applications. These regularizations are efficient in many situations where the Cooke–Lebedev method does not work efficiently. Our approach is based on new results for Hankel and Weber–Orr integral transforms and on integral operators that transform their kernels ones into other (Sect. 2). In particular, we prove new inversion theorems for Hankel and generalized Weber–Orr transforms and establish Paley–Wiener’s theorem for the classic Weber–Orr transform. Also, four inversion formulas are indicated for the above-mentioned operators. These formulas are non-trivial analogues of the inversion formulas for a special case of the Erdelyi–Kober operators (the Abel transforms). The above results are employed to find certain general representations of the Hankel transforms of integer order for functions from L1 possessing compact supports (Sects. 2 and 5). In Sects. 3, 4 and 5, the integral equations of the first kind (1.1) are reduced

On a certain class of integral equations

137

to equivalent Fredholm integral equations of the second kind whose solutions can be found exactly when L(p) = 0. Explicit formulas for the solutions are written down for n = 0 and n = 1 as L(p) = 0, H (p) = 1.

2 Integral Transforms Involving Bessel Functions and Generalized Abel Equations We commence with the L1 theory of the Hankel transform

q (p) =

∞

(2.1)

xq(x)Jα (px)dx. 0

Theorem 2.1 Suppose

√

rq(r) ∈ L1 (0, ∞). Then the inversion formulas

q(r) = r −α−1

d α+1 r dr

q(r) = −r α−1

d 1−α r dr

∞

q (p)Jα+1 (pr)dp,

α≥−

0

∞

q (p)Jα−1 (pr)dp,

0

1 2

and

1 α≥ , 2

(2.2) (2.3)

are valid at every point r > 0 where q(r) is the derivative of its integral. Proof Using Fubini’s theorem, we write

∞

q (p)Jα+1 (pr)dp = lim

∞√

R→∞ 0

0

xq(x)

R

√

xJα (px)Jα+1 (pr)dp dx.

(2.4)

0

A direct evaluation on the basis of the asymptotic expansions for the Bessel functions [3] shows that the inner integral has finite bounds for R ≥ R0 > 0, x ≥ 0 as r > 0 is fixed. This permits us to look for the limit R → ∞ inside the integration sign. Then utilizing the integral [13]

∞

Jα+1 (pr)Jα (px)dp =

0

x α r −α−1 ,

0 < x < r,

0,

x > r > 0,

α > −1,

(2.5)

and differentiating (2.4) yield the inversion formula (2.2). The formula (2.3) can be proved in an analogous manner. Remark 1 It is readily seen that formula (2.2) holds also for −1 < α < −1/2 if, in addition, r α+1 q(r) ∈ L1 (0, ε). Formula (2.3) holds for 0 < α < 1/2 if, in addition, r 1−α q(r) ∈ L1 (ε, ∞) for some ε > 0. We proceed with the generalized Weber–Orr integral transform [6, 11] G(p) =

∞

λ xg(x)χν,μ (p, x)dx,

λ > 0,

(2.6)

λ λ (p, x) = Y μ(px)Jν (pλ) − Yν (pλ)Jμ (px), ν = μ + m and m is a non-negative where χν,μ integer.

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P. Malits

Theorem 2.2 Suppose

√

tg(t) ∈ L1 (λ, ∞). Then the inversion formulas

λ χν,μ−1 (p, t) d 1−μ ∞ t dp, G(p) 2 dt J (pλ) + Yν2 (pλ) 0 ν ∞ λ χν,μ+1 (p, t) d dp, G(p) 2 g(t) = t −μ−1 t 1+μ dt J (pλ) + Yν2 (pλ) 0 ν g(t) = −t μ−1

μ≥

1 2

1 μ≥ , 2

(2.7) (2.8)

or μ > −1, ν = μ,

are valid at every point where g(t) is the derivative of its integral. Moreover, if additionally t 1−μ g(t) ∈ L1 (λ, ∞t), then the inversion formulas (2.7) holds for μ > 0 and the inversion formulas (2.8) holds for μ > −1, ν ≥ 0. If g(t) is smooth at the point t = r, then g(r) =

∞

pG(p) 0

λ χν,μ (p, r)

Jν2 (pλ) + Yν2 (pλ)

μ > −1, ν ≥ 0.

dp,

(2.9)

Proof The proof of the inversion formulas (2.7) and (2.8) is completely analogous to the proof of the preceding theorem. We employ the integral J=

∞

λ λ χν,μ+1 (p, t)χν,μ (p, x)

dp =

Jν2 (pλ) + Yν2 (pλ) 0

1, μ = ν, μ > −1, ν ≥ 0. hμ = 0, μ < ν,

μ −μ−1 λ2ν hμ x t , + x μ t 1+μ 0,

λ ≤ x < t, x > t ≥ λ,

(2.10)

This integral is evaluated by means of the identity [16]

∞ 0

λ λ χν,μ−1 (p, t)χν,μ (p, x)


=

∞

dp

Jμ+1 (pt)Jμ (px)dp − Re

0

0

∞

Jν (pλ) Hν(1) (pλ)

(1) (pt)dp, Hμ(1) (px)Hμ+1

(2.11)

where Hν(k) (p) is the Hankel function. Examining the loop integral L

Jν (zλ) Hν(1) (zλ)

(1) (zt)dz = 0, Hμ(1) (zx)Hμ+1

(2.12)

where L consists of segments [ε, R], [R, R + iR], [iR, R + iR], [iε, iR] and the arc |z| = ε, 0 ≤ arg z ≤ π2 , we obtain as R → ∞, ε → 0 that the second item is λ2ν hμ x −μ t −1−μ . Then the integral (2.5) yields (2.10). Invoking asymptotic expansions for Bessel functions and well-known results for Fourier transforms, one might ascertain that the integral in (2.9) converges uniformly as g(t) is smooth. Thus, we are permitted to differentiate the integral in (2.8) inside the integration sign to establish (2.9). Theorem 2.3 Suppose μ = ν − 1/2 ± 1/2, ν ≥ 0, √ tg(t) ∈ L2 (λ, ∞), λ > 0. Then the formulas

√

pG(p)/|Hν (pλ)| ∈ L2 (0, ∞) or


139

N

G(p) = lim

λ xg(x)χν,μ (p, x)dx,

N→∞ λ

g(r) = lim

N→∞ 0

N

pG(p)

λ χν,μ (p, r)


(2.13) (2.14)

dp

define the reciprocal isometric operators with the isometric relation ∞ ∞ G1 (p)G(p) dp = p 2 xg1 (x)g(x)dx. Jν (pλ) + Yν2 (pλ) 0 λ

(2.15)

The inversion formula g(t) = t −μ−1

d dt

∞

G(p) 0

λ λ (p, t) − ρ 1+μ χν,μ+1 (p, ρ) t 1+μ χν,μ+1


dp,

(2.16)

is valid at every point t > ρ ≥ λ where g(t) is the derivative of its integral. Proof This theorem is covered by the general Sturm–Liouville theory and, with the exception of the inversion formula (2.16), is well known [17]. The inversion formula (2.16) is derived from (2.15) by taking g1 (x) = x μ H (t − x)H (x − ρ), where H (t) is the Heavλ (p, t) − iside unit function. The Weber–Orr transform of g1 (x) is pG1 (p) = t 1+μ χν,μ+1 λ ρ 1+μ χν,μ+1 (p, ρ). Our proof of the generalized Paley–Wiener’s theorem for the Weber–Orr transform is based on the theorem stated above. A real-value version of such a theorem was discussed in [18]. The classical Paley–Wiener’s theorem corresponds to ν = 1/2. Theorem 2.4 The class of even entire functions of exponential type a belonging with the weight z1−m , m = 0, 1, to L2 [0, ∞) coincides with the class of functions of the form a+λ λ sg(s)χν,ν−m (z, s)ds,

(z) = zm λ √ sg(s) ∈ L2 [λ, a + λ], ν > 0.

(2.17)

λ (z, s) are even Proof The relations for Bessel functions [3] indicate that the kernels zm χν,ν−m entire functions of exponential type s − λ. Then Schwartz’s inequality immediately shows (2.17) to be an even entire function of exponential type a. Conversely, in order to prove that an arbitrary even entire function (z) of exponential type a, z1−m (z) ∈ L2 [0, ∞), can be represented in the form (2.17), we use its contraction

(p) onto the real axis. It is sufficient to prove that the Weber inversion (2.16) of G(p) = p −m (p) yields a certain function g(s) with the compact support [λ, a + λ]. This objective can be achieved by means of the contour integral

z−m (z) L

(1) (zs) Hν−m+1

Hν(1) (zλ)

dz = 0,

s > a + λ.

(2.18)

Since z1−m (z) ∈ L2 [0, ∞), there is the estimate | (z)| ≤ A exp(a| Im(z)|) [15, Sect. III.4]. Thus, when |z| → ∞, the integrand is O(exp[(a + λ − s) Im(z)]) if Im(z) > 0 and tends

140

P. Malits

to zero if Im(z) = 0. In a neighborhood of the origin, the integrand is λν s m−ν−1 (0)z−1 + O(1). The limits R → ∞, ε → 0 and portioning the imaginary part lead to the integral

∞

G(p) 0

λ χν,ν−m+1 (p, s) πλν dp =

(0), Jν2 (pλ) + Yν2 (pλ) 2s ν−m+1

s > a + λ.

(2.19)

√ Hence, according to Theorem 2.3, g(s) = 0 for s > a + λ and sg(s) belongs to L2 [λ, a + λ]. Consequently, for real z = p, the function (p) is of the form (2.17) that gives its unique analytical extension as (z) in the complex plane. Further we shall exploit the discontinuous integrals ⎧ ⎨ 2 cosh(νθ ) , 0 < r < t − λ, λ pχν,ν (p, t)J0 (pr)dp = πλt sinh(θ ) R0 (t, r) = ⎩ 0 0, r > t − λ, 2 λ + t2 − r2 θ = arcosh , t ≥ λ, 2λt

∞

Re ν > −1.

(2.20) and R1 (t, r) =

∞

λ pχν,ν−1 (p, t)J1 (pr)dp −

0

2t ν−1 πrλν

⎧ ⎨ λ cosh[(ν − 1)θ ] − t cosh(νθ ) , 2 = πλrt sinh(θ ) ⎩ 0,

0 < r < t − λ,

Re ν > 0,

(2.21)

r > t − λ,

as well as the integrals

∞

Ql (t, r) = 0

λ χν,ν−1+l (p, t)Jl (pr) dp, Jν2 (pλ) + Yν2 (pλ)

ν > 0, l = 0, 1.

(2.22)

The first integral is evaluated by utilizing the known integral involving the product of three Bessel functions [13]. The second integral can be derived by integration by parts on λ λ (p, t)] = [λpχν−1,ν−1 (p, t) − the basis of the relations rJ1 (pr)dp = −d[J0 (pr)], d[pχν,ν−1 λ (p, t)]dp and the integral (2.20). tpχν,ν The functions Ql (t, r) are the Weber–Orr inverses of the functions of exponential type r. Then Ql (t, r) = 0 for r < t − λ. It is established by the method of contour integration employed in the proof of Theorem 2.4. Introduce the integral operators Rl =

t−λ

(·)ρRl (t, ρ)dρ 0

and

r+λ

Ql =

(·)sQl (s, r)ds,

ν > 0.

(2.23)

λ

It is readily established by means of Fubini’s theorem that the operators Rl , l = 0, 1, map the space of functions belonging with the weight ρ l−1 to L1 [0, c] onto L1 [λ, d] and the operators Ql map L1 [λ, d] onto L1 [0, c]. As ν = 1/2, these operators become the Abel transforms which are related to the Riemann–Liouville fractional integral of order 1/2. By inversion of the integrals (2.20–2.22), the operators Rl and Ql are shown to transform the


141

kernels of the Weber–Orr and Hankel integral transforms ones into other λ R0 J0 (pr) = χν,ν (p, t),

λ R1 J1 (pr) = χν,ν−1 (p, t) −

λ Q0 pχν,ν−1 (p, t) = J0 (pr),

λ Q1 pχν,ν (p, t) = J1 (pr).

2t ν−1 , πpλν

(2.24) (2.25)

The above formulas remain valid in the complex plane in virtue of the analytical continuation. The rigorous proof of these obvious relations is omitted. Equations involving Rl or Ql generalize the well-known Abel equation. The corresponding inverse operators are established in the explicit form. Theorem 2.5 Suppose ψ(t) ∈ L1 [λ, d] and ϕ(r) ∈ L1 [0, c]. Then the relations d ν t R0 Q0 ψ(s), dt d ψ(t) = −t ν−1 t 1−ν R1 Q1 ψ(s), dt 1 d ϕ(r) = rQ1 R0 ϕ(ρ), r dr d ϕ(r) = − Q0 R1 ϕ(ρ), dr

ψ(t) = t −ν

t ∈ [λ, d],

(2.26)

t ∈ [λ, d],

(2.27)

r ∈ [0, c],

(2.28)

r ∈ [0, c],

(2.29)

are valid for ν > 0 at every point where ψ(t) and ϕ(r) are the derivatives of their integrals. Proof We begin with the case ψ(t) ∈ C 1 [λ, d]. Let g(t) ∈ C 1 [λ, ∞) be a function coinciding with ψ(t) on [λ, d] and having a compact support. Theorem 2.2, where μ = ν − 1, holds for this function. Its Weber–Orr integral transform G(p) is a continuous function on (0, ∞) and G(p) = O(1/p) as p → 0. Integration by parts and asymptotic expansions for Bessel functions show that G(p) = O(1/p 2 ) as p → ∞. Substituting the representation λ (p, t) = R0 J0 (pρ) into formula (2.8), we are permitted to interchange the order of inteχν,ν gration because the arising integrals are uniformly convergent. We have

∞

∞

λ χν,ν−1 (p, s) ds 2 Jν (pλ) + Yν2 (pλ) 0 λ √

∞ ∞ λ sλχν,ν−1 (p, s) s −ν d ν =t t R0 g(s) 2 − cos p(s − λ) ds J0 (pρ)dp dt λ Jν (pλ) + Yν2 (pλ) 0 λ 1 ∞ s g(s) sin p(s − λ)ds . − (2.30) p λ λ

g(t) = t −ν

d ν t R0 dt

J0 (pρ)dp

sg(s)

On interchanging the order of integration and evaluating the arising integrals this becomes 1 −ν d ν g(t) = t t R0 Q0 g(s) − √ Qg , dt λ ρ+λ √ ρ+λ (s − λ) √ sg(s)ds = Qg d sg(s) − arcsin ρ ρ 2 + (s − λ)2 λ λ π + ρ + λg(ρ + λ). (2.31) 2

142

P. Malits

≡ 0, we arrive at relation (2.26). Upon interchanging the order of Finally, noting that Qg integration in (2.26) we obtain g(t) = t −ν

d ν t dt

t

g(s)It (s)ds, λ

(2.32)

t

It (s) = s

(x − λ)R0 (t, x − λ)Q0 (s, x − λ)dx. s

Here the kernels Rl (t, r) and Ql (t, r) are of the form Fl (t, r)|(t − λ)2 − r 2 |−1/2 , where Fl (t, r) are bounded functions. One could see that It (s) ∈ C[λ, t]. Integrating from λ to t, we rewrite the above equation at the form of the functional

t

g(s)(s ν − t ν It (s))ds = 0. λ

Since g(s) is an arbitrary function from C 1 , this implies It (s) = s

t

(x − λ)R0 (t, x − λ)Q0 (s, x − λ)dx = s ν /t ν .

(2.33)

s

Now the general case ψ(t) ∈ L1 [λ, d] can be proved by interchanging the order of integration in (2.26) that is justified by Fubini’s theorem. Relation (2.27) may be derived in the same way from (2.7), where μ = ν. Relations (2.28) and (2.29) are found in an analogous manner. We define a function q(r) ∈ C 1 [0, ∞) having a compact support and make use of Theorem 2.1 (α = 0, 1) together with the representations (2.25) and integrals (2.20, 2.21). For example, in the case of (2.3) and α = 1, this leads to q(r) = −

∞ d 2(r + λ)ν C(r) q(s)ds , Q0 R1 q(r) − dr πλν 0

(2.34)

where the integral

∞

C(r) = 0

λ χν,ν (p, r + λ)J0 (pr) dp = Im p[Jν2 (pλ) + Yν2 (pλ)]

0

∞

Hν(1) (p(r + λ)) pHν(1) (pλ)

J0 (pr)dp =

πλν 2(r + λ)ν (2.35)

is evaluated in the same way as the integral (2.19). Then we obtain (2.29) and

t

(x + λ)R0 (x + λ, s)Q1 (x + λ, t)dx = −1.

s

(2.36)

s

Finally, the formula (2.29) for ϕ(r) ∈ L1 [0, c] is proved by interchanging the order of integration. We note that the inversion formula (2.28) remains valid if rϕ(r) ∈ L1 [0, c]. Corollary 2.6 Bounded functions do not belong to the null spaces of the operators t −ν dtd t ν R0 and t ν−1 dtd t 1−ν R1 .


143

Proof The estimates √ |Rl ϕ(r)| ≤ Bl (t − λ), Bl are positive constants, are valid for any bounded function rϕ(r). They show that ϕ(r) belongs to the null space of the operator t −ν dtd t ν R0 , (t ν−1 dtd t 1−ν R1 ), if and only if R0 ϕ(r) = 0, (R1 ϕ(r) = 0). But this is impossible because of (2.28) and (2.29). Corollary 2.7 Let f (ρ) be an absolutely continuous function. Then ν−1 t d ν t R0 f (ρ) = −R1 f (ρ) + f (0) , dt λ f (ρ) ν−1 d 1−ν t t R1 f (ρ) = R0 f (ρ) + . dt ρ t −ν

(2.37) (2.38)

Proof We insert ϕ(ρ) = f (ρ) and ϕ(ρ) = (ρf (ρ)) /ρ into (2.28) and (2.29), respectively, and integrate from 0 to ρ. Now one may ascertain that every absolutely continuous function f (ρ) is represented in the form f (ρ) = Qn ψn (t) + Cn , n = 0, 1, where ψ0 (t) = −R1 f (ρ) and ψ1 (t) = R0 [f (ρ) + f (ρ)/ρ] belong to L1 [λ, d], Cn = limρ→0 ρ n f (ρ). Then the desirable result follows from the inversion formulas (2.26, 2.27) and the relation t

−ν

d ν d λ t R0 1 = lim t −ν t ν R0 J0 (pr) = lim πλpχν,ν−1 (p, t)/2 = p→0 p→0 dt dt

ν−1 t . λ

(2.39)

N 2 2 2 2 Lemma 2.8 Let xy(x) ∈ L1 [0, a] and H (p) = N k=1 (p + ak )/ k=1 (p + bk ), ak = bk . 1 Then there exist a function n (s) ∈ L [λ, a + λ] and an even meromorphic function (p) with the simple poles ±iak solely, p n (p) ∈ L2 (0, ∞), lim|p|→∞ |(p)| exp(−a|p|) = constant, such that the representation

a

xy(x)Jn (px)dx = 0

p H (p)

a+λ λ sn (s)χν,ν+n−1 (p, s)ds + p n (p)

(2.40)

λ

is valid in the complex plane for n = 0, 1. Here n (s) and (p) obey the conditions

a+λ λ

λ sn (s)χν,ν+n−1 (iam , s)ds + p n−1 H (p)(p) p=iam = 0,

m = 1, 2, . . . , N, (2.41)

and the function n (s) is unique if (p) is given. Proof By substituting the representations (2.25) into the left-hand side of the relation (2.40) and interchanging the order of integration, we arrive at

a

a+λ

xy(x)Jn (px)dx = p 0

φn (s) =

λ sφn (s)χν,ν+n−1 (p, s)ds,

(2.42)

λ a+λ

(x − λ)Qn (s, x − λ)y(x − λ)dx.

(2.43)

s

The kernels Ql (s, x −λ) ∈ L1 [λ, a +λ] as functions of the variable s for every x ∈ [λ, a +λ]. Then applying Fubini’s theorem, we ascertain that φn (s) ∈ L1 [λ, a + λ]. This fact justifies the preceding operation as well.

144

P. Malits

Then, taking φn (s) = n (s) + n (s), n (s) ∈ L2 [λ, a + λ], (2.42) can be written, according to our generalization of Paley–Wiener’s theorem, as

a

a+λ

xy(x)Jn (px)dx = p 0

λ (p), sn (s)χν,ν+n−1 (p, s)ds + p n

(2.44)

λ

(p) ∈ L2 (0, ∞). (p) is an even entire function of exponential type a, p n where Now, one might see that the representation (2.40) follows from (2.44). Conditions (2.41) are evident if we set p = iam and take into account that the integral on the left-hand side of (2.44) is an entire function. If (p) is given, then the uniqueness of n (s) follows from Theorems 2.2 and 2.3. Remark 2 There is no converse of this lemma. In other words, it is not correct that for any function n (s) ∈ L1 [λ, a + λ] may be found xy(x) ∈ L1 [0, a] and (p), p n (p) ∈ L2 (0, ∞), satisfying (2.40). For example, if one sets n (s) = Qn (s, a) into (2.40), then

a

xy(x)Jn (px)dx = 0

Jn (pa) + p n (p), H (p)

n = 0, 1.

(2.45)

This equation has no solutions xy(x) ∈ L1 [0, a]. Indeed, on the real axis, the Hankel be o(p −1/2 ) as p → ∞. But the function on transform of a function from L1 [0, √ a] must −1/2 cos(pa + πn/2 − π/4) + o(p −1/2 ) for any the right-hand side of (2.45) is 2/πp n 2 p (p) ∈ L (0, ∞).

3 Case n = 0 In this case the operator R = equation t −ν

d ν t dt

∞

A0 (p) 0

π λ −ν d ν t dt t R0 2

transforms the integral equation (1.3) into the

H (pλ)[1 + L(p)] λ χ (p, t)dp = Rf0 (r), p[Jν2 (pλ) + Yν2 (pλ)] ν,ν

with

λ ≤ t ≤ a + λ, (3.1)

a

A0 (p) =

(3.2)

xy0 (x)J0 (px)dx. 0

It is important to note that, according to Corollary 2.6, the operator R is a left equivalent transformation for (1.3). Further we are based on the representation following from Lemma 2.8

A0 (p) =

p H (p)

α λ sω(s)χν,ν−1 (p, s)ds − λ

N k=1

xk

Jl (pα0 ) , l p (p 2 + ak2 )

where α = a + λ, l ≥ 0, 0 < α0 ≤ a, (t) ∈ L1 [λ, α] and xk are some numbers.

(3.3)


145

Conditions (2.41) give the complementary relations connecting (s) with the undetermined coefficients xk xk =

akl+1 k Il (ak α0 )

α

1 ≤ k ≤ N,

sω(s)Wk,ν,ν−1 (s)ds,

(3.4)

λ

in which Wk,μ,ν (s) = Iμ (ak s)Kν (ak λ) − (−1)μ−ν Iν (ak λ)Kμ (ak s), Iν (t) and Kμ (t) are the modified Bessel functions; k =

N N 2 −1 2 2 bm − ak2 am − ak2 . π m=1 m=1 m=k

Substituting (3.3) into (3.1) coupled with the inversion formula (2.8) yields the Fredholm integral equation of the second kind (I + K0 )ω = Rf0 (r) +

N

xk Fk (t),

K0 ω =

α

(s)sK0 (t, s)ds,

(3.5)

λ

k=0

Here:

∞

K0 (t, s) =

pL(p)

0 ∞

Fk (t) = 0

λ λ χν,ν−1 (p, s)χν,ν−1 (p, t) dp, 2 Jν (pλ) + Yν2 (pλ)

λ H (p)[1 + L(p)]Jl (pα0 )χν,ν−1 (p, t)

p l (p 2 + ak2 )[Jν2 (pλ) + Yν2 (pλ)]

(3.6) dp, k ≥ 1.

Invoking asymptotic expansions for Bessel functions, one might ascertain that K0 (t, s) possesses a logarithmic singularity as c = 0 or is a continuous function as c = 0. The functions Fk (t) are continuous. By inserting (3.4), we can write (3.5) as the Fredholm integral equation of the second kind containing the unknown ω(t) only

α

ω(t) = Rf0 (r) −

s)ds, λ ≤ t ≤ a + λ, ω(s)s K(t,

λ

s) = K0 (t, s) − K(t,

N akl+1 k Fk (t)Wk,ν,ν−1 (s). I (a α ) k=0 l k 0

(3.7)

Lemma 3.1 If the Fredholm integral equation (3.7) is solvable, then its solution is an absolutely continuous function. Proof According to Corollary 2.7, we get for the right-hand side of the integral equation 2 2t ν−1 f0 (0) 2t ν−1 f0 (0) Rf0 (r) = − R1 f0 (r) = − R1 f10 (r) − R1 f20 (r), ν πλ πλ πλν

(3.8)

where f10 (r) is a linear combination of the Heaviside unit functions H (rk − r), rk ≤ a, and f20 (r), f20 (0) = 0, is an absolutely continuous function.

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A straightforward computation manifests that the second item is an absolutely continuous function ⎧ ⎪ ⎨ λt2 λP|ν−1|− 1 (cosh θ0 ) − tPν− 1 (cosh θ0 ) , t ≤ rk + λ, 2 2 (3.9) R1 H (rk − r) = ⎪ ⎩ 1 2 λPν−1 (θa , θ0 ) − tPν (θa , θ0 ) , t ≥ rk + λ. π λt 2

2

2

2 +λ2 t +λ −r Here Pν (x) is the Legendre function, cosh θ0 = t 2tλ , cosh θa = 2tλ k , Pν (θa , θ0 ) is the integral η sinh (θa /2) cosh[2ν arsinh(sinh (θ0 /2) cos s)] . (3.10) ds, η = arccos 2 sinh (θ0 /2) 0 1 + sinh (θ0 /2) cos2 s

The third item is an absolutely continuous function in virtue of Corollary 2.7. It is well known that if the right-hand side of the integral equation (3.7) is continuous, then its solution is continuous as well. If one showed that the operator K0 transforms any continuous function g(t) into an absolutely continuous function, the lemma would be proven. We may establish this property of K0 by invoking Theorem 4 and noting that the function α λ sg(s)χν,ν−1 (p, s)ds (3.11) pG(p) = p λ

is bounded and belongs to L (0, ∞). Now pG(p)L(p) ∈ L1 (0, ∞) and p 2 G(p)L(p) = L2 (δ, ∞) for some δ > 0. Then p 3/2 G(p)L(p)/|Hν(1) (pλ)| may be represented in the form p 3/2 G(p)L(p)/Hν(1) (pλ) = L1 (p) + L2 (p), (3.12) 2

where L1 (p) ∈ L1 (0, ∞) is a function with a compact support and L2 (p) ∈ L2 (0, ∞). Hence simple analysis on the basis of Theorem 2.3 brings in the conclusion that

∞

K0 g = 0

λ pG(p)L(p)χν,ν−1 (p, t) dp 2 Jν (pλ) + Yν2 (pλ)

(3.13)

is a continuous almost everywhere differentiable function whose derivative belongs to L2 (λ, a + λ). This means that the right-hand side of (3.7) is an absolutely continuous function. Theorem 3.2 The Fredholm integral equation of the first kind (1.1) for n = 0 is equivalent to the Fredholm integral equation of the second kind (3.7). Proof Since (3.1, 3.2) are equivalent to (1.1), it is sufficiently to prove that the substitution (3.3) coupled with conditions (3.4) is a right equivalent regularization. If one established with the inverse Hankel transform that xy0 (x) ∈ L1 [0, a], this objective would be achieved (see Remark 2). Lemma 3.1 permits to write α λ sω(s)χν,ν−1 (p, s)ds p λ

α

λ = αω(α)χν,ν (p, α) − λ

λ s ν (s 1−ν ω(s)) χν,ν (p, s)ds.

(3.14)


147

This relation and the integral (2.20) yield y0 (x), y0 (x) = αω(α)R0 (a, x) − T s ν (s 1−ν ω(s)) + α T[g] = g(s)R0 (s, x)ds, y0 (x) ∈ C.

(3.15) (3.16)

r

It may be readily proved that T : L1 → L1 and the desirable result is obtained.

Another approach is to represent ω(t) = ω(t) +

N

(3.17)

xk ωk (t),

k=0

where ω(t) and ωk (t) are solutions of the Fredholm integral equations (I + K0 ) ω(t) = Rf0 (r) and (I + K0 )ωk (t) = Fk (t), respectively. Substituting the expression (3.17) into (3.4) yields the system of N linear algebraic equations for N unknowns x1 , . . . , xN N m=1

xm

Il (ak α0 ) akl+1 k

α

δk,m −

sωm (s)Wk,ν,ν−1 (s)ds = gk ,

k = 1, 2, . . . , N,

(3.18)

λ

where δk,m is the Kronecker delta and the right-hand side is defined by α s ω(s)Wk,ν,ν−1 (s)ds. gk =

(3.19)

λ

If 1+ L(p) is a non-negative bounded function, then the integral operator in (1.1) is positive definite and the solution is unique. In this case, the operator I + K0 is also positive definite: 2 α ∞ α p[1 + L(p)] λ tω(t)[(I + K0 )ω]dt = tω(t)χν,ν−1 (p, t)dt dp > 0, Jν2 (pλ) + Yν2 (pλ) λ λ 0 (3.20) and, according to the Fredholm alternative, bijective. Hence, the solution of the system of the linear algebraic equations is unique as well. In the special case L(p) = 0, the integral operator K0 does not occurs and the solution may be found exactly. In particular, as H (p) = 1, we can write down the explicit formula s−λ a+λ πλ d x s 1−ν R1 (s, x)d s ν tf0 (t)R0 (s, t)dt . (3.21) y0 (x) = 2x dx x+λ 0 4 Case n = 1 π ν−1 d 1−ν t t R1 transforming the Bessel function Here we employ the operator M = − 2λ dt λ (p, t). Then (1.3) becomes J1 (pr) into the kernel of the Weber transform pχν,ν

−t ν−1

d 1−ν t dt

∞

A1 (p) 0

λ ≤ t ≤ a + λ.

H (pλ)[1 + L(p)] λ χ (p, t)dp = Mf1 (r), p[Jν2 (pλ) + Yν2 (pλ)] ν,ν−1 (4.1)

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P. Malits

According to Lemma 2.8, the solution may be taken in the form A1 (p) =

1 p H (p)

α λ sϕ(s)χν,ν (p, s)ds −

N

λ

xk

k=1

Jl+1 (pα0 ) , p l (p 2 + ak2 )

where l is a non-negative integer, ϕ(s) ∈ L1 [λ, α] and subjected to the conditions α a l+1 k xk = k sϕ(s)Wk,ν,ν (s)ds, 1 ≤ k ≤ N. Il+1 (ak α0 ) λ

(4.2)

(4.3)

Substituting (4.2) into (4.1) leads to the Fredholm equation of the second kind (I + K1 )ϕ(t) = Mf1 (r) +

∞

pL(p) 0

xk Um (t),

∞

ϕ(s)sK1 (t, s)ds, λ

λ λ χν,ν (p, t)χν,ν (p, s) dp, 2 Jν (pλ) + Yν2 (pλ)

λ H (p)[1 + L(p)]Jl+1 (pα0 )χν,ν (p, t)

0

α

K1 ϕ(t) =

k=1

K1 (t, s) = Uk (t) =

N

p l (p 2 + ak2 )[Jν2 (pλ) + Yν2 (pλ)]

(4.4) dp,

or, equivalently, to the equation

α

ϕ(t) +

ϕ(s)sK2 (t, s)ds = Mf1 (r),

(4.5)

λ

where the kernel K2 (t, s) is given by the expression K2 (t, s) = K1 (t, s) −

N akl+1 k Wk,ν,ν (s)Uk (t). I (a α ) k=1 l+1 k 0

(4.6)

The following theorem can be proved in a fashion of Theorem 3.2. Theorem 4.1 The Fredholm integral equation of the first kind (1.1) for n = 1 is equivalent to the Fredholm integral equation of the second kind (4.5) whose solution, if it exists, is an absolutely continuous function. If N = 0, we take ϕ(t) = ϕ (t) +

N

(4.7)

xk ϕk (t),

k=0

and derive, in an analogous to Sect. 3 manner, the system of the linear algebraic equations N m=1

xm

Il+1 (ak α0 ) akl+1 k

α

δk,m −

sϕm (s)Wk,ν,ν (s)ds = gk ,

k = 1, 2, . . . , N.

(4.8)

λ

Here: ϕ (s) and ϕm (s) are solutions of the Fredholm equations of the second kind (I + K1 ) ϕ (s) = Mf1 (r), (I + K1 )ϕm (s) = Um (t), respectively, and α gk = s ϕ (s)Wk,ν,ν (s)ds. λ


149

If L(p) = 0, then K1 (t, s) ≡ 0 and we can obtain an exact solution. In particular, as H (p) = 1, the explicit solution is y1 (x) = −

πλ d 2 dx

a+λ x+λ

s ν R0 (s, x)d s 1−ν

s−λ

tf1 (t)R1 (s, t)dt .

(4.9)

0

5 Case n ≥ 2 The integral equation (1.3) for any integer n ≥ 2 may be equivalently transformed into a form involving either J0 (pr) if n = 2k or J1 (pr) if n = 2k + 1, k = 1, 2, . . . . Introduce the operators a 1 (·)t (r 2 − ρ 2 )n−1 r 1−n dr, (5.1) Fn = n−1 2 (n − 1)! ρ M d sn 2 1 d d ρ Gn = − cm I− , (5.2) dρ ρ dρ dρ m=1 where M = [n/2]; sn = (1 − (−1)n )/2; cm = ak are positive distinct numbers, cm = bm as m ≤ N and cm are arbitrary as m > N . Utilizing the relation [r −l Jl (pr)] = −pr −l Jl+1 (pr) and integrating by parts, we find Fn [Jn (pr)] =

n−1 1 Jk (pa) J (pρ) − (a 2 − ρ 2 )k , 0 n k k p n−k p 2 k!a k=0

n ≥ 1.

(5.3)

By making use of the Bessel differential equation and (5.3), the integral equation (1.3) can be transformed with the operator Gn Fn into the form

∞

An (p) 0

n−1 (p)[1 + L(p)]Jsn (pρ) 2H dp − Bnk Dnk (ρ) = fn (ρ), πpλ[Jν2 (pλ) + Yν2 (pλ)] k=0

0 ≤ ρ ≤ a, (5.4)

where fn (ρ) = Gn Fn [fn (r)] satisfies the conditions indicated above for fn (r), Dnk (ρ) = Gn [(a 2 − ρ 2 )k ] are polynomials, ∞ 1 2H (p)[1 + L(p)]Jk (pa) dp, (5.5) Bnk = k k An (p) n−k+1 2 k!a 0 πp λ[Jν2 (pλ) + Yν2 (pλ)] M N1 2 2 2 2 (p 2 + c2 ) N m=1 (p + am ) m=1 (p + dm ) (p) = m=N+1 m = , (5.6) H N N 1 1 2) 2) p 2M m=M+1 (p 2 + bm p 2M m=M+1 (p 2 + bm N1 = max(M, N ), dm = am as m ≤ N and dm = cm as m > N. Lemma 5.1 The function fn (r) does not belong to the null space of the operator Gn Fn . / ker Fn and the function g(ρ) = Proof It is readily seen that under our assumptions fn (r) ∈ Fn [fn (r)] possesses n derivatives, g (k) (a) = 0 for k = 0, 1, . . . , n − 1. Since the point ρ = a is regular for the differential operator Gn , Fn [fn (r)] can not be an eigenfunction of the operator Gn .

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Thus the operator Gn Fn is a left equivalent transformation. Equation (5.4) may be reduced to an equivalent Fredholm equation of the second kind in the same way as in Sects. 3 and 4. The only difference is that we should replace the substitutions (3.3) and (4.2) by An (p) =

1 p (p) H

α λ

λ sω(s)χν,ν+s (p, s)ds − n −1

N1

xk

k=1

Jl+n (pα0 ) p l (p 2 + dk2 )

(5.7)

and express Bnk via ωn (t). When L(p) ≡ 0, the arising Fredholm equations of the second kind possess degenerate kernels. The substitution (5.7) is based on the next generalization of Lemma 2.8. Lemma 5.2 Let xy(x) ∈ L1 [0, a]. Then there exist a function (s) ∈ L1 [λ, a + λ] and an even meromorphic function (p) with the simple poles ±idk solely, p sn (p) ∈ L2 (0, ∞), lim|p|→∞ |(p)| exp(−a|p|) = constant, such that the representation

a

xy(x)Jn (px)dx = 0

p H (p)

a+λ λ

λ s (s)χν,ν+s (p, s)ds + p sn (p), n −1

(5.8)

is valid in the complex plane for any non-negative integer n. Here (s) and (p) obey the conditions

a+λ

λ

λ s (s)χν,ν+s (idm , s)ds + [p sn −1 H (p)(p)]p=idm = 0, n −1

m = 1, 2, . . . , N1 , (5.9)

lim p sn −n (p) = constant,

(5.10)

p→0

and the function (s) is unique if (p) is given. Proof We note that if PM (p 2 ) = p 2M + κ2M−2 p 2M−2 + · · · + κ0 is an even polynomial n of order 2M, then the function x (p) = p −n PM (p 2 )Jn (px) − (−1)[ 2 ] p −sn Jsn (px) is an even entire function of exponential type x possessing the following asymptotic behavior as p → ∞: 3 3

x (p) = exp(ipx)O p − 2 −sn + exp(−ipx)O p − 2 −sn . a The integral mean theorem gives that 0 (p) = 0 xy(x) x (p)dx is an even entire function of exponential type ≤ a and p sn 0 (p) ∈ L2 [0, ∞). According to our generalization of Paley–Wiener’s theorem, such functions can be represented in the form of the Weber– Orr transforms of certain functions having the compact support [λ, a + λ] and belonging to L2 [λ, a + λ]. Thus, we may write PM (p 2 ) pn

a 0

n

xy(x)Jn (px)dx = (−1)[ 2 ] p −sn

a

xy(x)Jsn (px)dx + 0 (p)

(5.11)

0

and prove the lemma in the same arguments as Lemma 2.8.


151

6 Example Consider the well-known integral equation arising in the potential theory and the theory of elasticity ∞ a π pλ Jn (pr)Jn (px)dp dx = fn (r), 0 ≤ r ≤ a, λ > 0, n = 0, 1. xyn (x) tanh 2 0 0 (6.1) Upon denoting 2H (p)[1 + L(p)] π pλ = , tanh 2 πpλ[Jν2 (pλ) + Yν2 (pλ)]

H (p) =

N p 2 + am2 , 2 p 2 + bm m=1

(6.2)

this integral equation is rewritten in the form of the integral equation (1.1). Choosing different values of ν and different rational functions H (p), one can obtain Fredholm integral equations of the second kind possessing different kernels. α Introduce the functional space with the inner product (ϕ, ) = λ tϕ(t) (t)dt. Employing Parseval’s equality (2.15) for the Weber–Orr integral transform of functions with the compact support (λ, α), we find the following simple estimates for the norms of the operators Km in the Fredholm equations of the second kind obtained in Sects. 3 and 4 Km = sup (ω, Km ) ≤ max |L(p)|,

(6.3)

min[1 + L(p)] ≤ I + Km ≤ max[1 + L(p)].

(6.4)

ω=1

If we take ν = 1/2, N = 0, then L(p) = tanh(πpλ/2) − 1, the correspondent Weber– Orr integral transforms become the cosine and sine Fourier transforms and the kernels of our Fredholm equations of the second kind are the same that arise in the Lebedev– Cooke method. The operators I + Km are positive definite. Their norms have the estimates Km ≤ 1, I + Km ≤ 1. As λ → 0, the sequence of the operators I + Km weakly converges to zero lim (g, (I + Km )g) = lim

λ→0

∞

λ→0 0

πpλ pG2 (p) tanh dp = 0 2 2 2 J1/2 (pλ) + Y1/2 (pλ)

(6.5)

and the algorithm becomes inefficient. For ν = 1, N = 0, we obtain the much better estimates Km ≤ 0.16765, 1 ≤ I + Km ≤ 1.16765, which manifest that the Fredholm equations of the second kind are efficiently solvable with both projective and iterative methods for all positive values of the parameter λ. It is important for numerical methods that, in contrast to the Cooke–Lebedev method, the operator I + Km is strictly positive definite for any λ ≥ 0. One might attain a more rapid convergence of algorithms by an appropriate choice of the function H (p). For example, as ν = 1 and H (p) =

(p 2 π 2 λ2 + 16)(p 2 π 2 λ2 + (p 2 π 2 λ2

+ 4)(p 2 π 2 λ2

+

π2 8

π2 2

+ 3.648)

+ 15.648)

,

(6.6)

we have the very good estimates Km ≤ 0.0517, 0.9487 ≤ I + Km ≤ 1.0517 for any λ ≥ 0.

152

P. Malits

References 1. Anderssen, R.S., De Hoog, F.R., Rose, L.R.F.: Explicit solution of a class of dual integral equations. Proc. Roy. Soc. Edinb. A 91, 1031–1041 (1982) 2. Cooke, J.: A solution of Tranter’s dual integral equations problem. Q. J. Mech. Appl. Math. 9(1), 103– 110 (1956) 3. Erdelyi, A. (ed): Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1954) 4. Erdelyi, A., Sneddon, I.: Fractional integration and dual integral equations. Can. J. Math. 14(5), 685–698 (1962) 5. Lebedev, N.N.: The distribution of electricity on a thin parabolic segment. Dokl. Akad. Nauk SSSR 114(3), 513–516 (1957) 6. Malits, P.: Transformation an arbitrary function into an integral in cylindrical functions and its application in theory of elasticity. In: Stability and Strength of Constructions. Dnepropetrovsk University, Dnepropetrovsk (1973) (in Russian) 7. Malits, P.: Effective approach to the contact problem for a stratum. Int. J. Solids Struct. 42, 1271–1285 (2005) 8. Malits, P.: Indentation of an incompressable inhomogeneous layer by a rigid circular indenter. Q. J. Mech. Appl. Math. 59(3), 343–358 (2006) 9. Mandal, B.N.: A note on Bessel function dual integral equations with weight function. Int. J. Math. Math. Sci. 9, 543–550 (1988) 10. Mandal, B.N., Mandal, N.: Advances in Dual Integral Equations. Chapman & Hall/CRS, London/Boca Raton (1999) 11. Nassim, C.: Associated Weber integral transforms of arbitrary orders. Indian J. Pure Appl. Math. 20, 1126–1138 (1989) 12. Noble, B.: The solution of Bessel function dual integral equations by a multiplying-factor method. Proc. Camb. Phil. Soc. 59(2), 351–362 (1963) 13. Prudnikov, A.P., Brychkov, Y. A., Marichev, O.I.: Integrals and Series, vol. 2. Gordon and Breech, London (1986) 14. Rahman, M.: A note of the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity. Z. Angew. Math. Phys. 46, 107–121 (1995) 15. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971) 16. Srivastav, R.P.: A pair of dual integral equations involving Bessel functions of the first and second kind. Proc. Edinb. Math. Soc. 14, 149–158 (1964) 17. Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second Order Differential Equations. Clarendon, Oxford (1924) 18. Tuan, V.K., Zaed, A.: Paley–Wiener-type theorems for a class of integral transforms. J. Math. Anal. Appl. 266(1), 200–226 (2002)