VolterraâFredholm integral equation of the first ... - Semantic Scholar
Applied Mathematics and Computation 153 (2004) 141–153 www.elsevier.com/locate/amc
Volterra–Fredholm integral equation of the first kind and spectral relationships M.A. Abdou a
a,*
, F.A. Salama
b
Department of Mathematics, Faculty of Science, Om Elkora University, Mekah, Saudi Arabia b Department of Mathematics, Faculty of Education, Alexandria University, Egypt
Abstract Here, the solution in one, two and three dimensional for the Volterra–Fredholm integral equation of the first kind is obtained in the space L2 ðXÞ C½0; T , T < 1. Using a numerical method the integral equation of Volterra–Fredholm becomes a linear system of Fredholm integral equation when that the kernel of Fredholm integral takes a logarithmic form, Carleman function, generalized potential function and Macdonald function are considered as special cases. Ó 2003 Published by Elsevier Inc. Keywords: Volterra–Fredholm integral equation V-FIE; Logarithmic kernel; Potential function; Macdonald kernel
1. Introduction Many problems in mathematical physics, theory of elasticity, viscodynamics fluid and mixed problems of mechanics of continuous media reduce to the integral equations of the first or the second kind. The monographs [1,2] give many of spectral relationships in terms of orthogonal polynomials for the integral operators frequently encountered in mathematical physics, and describe a method of orthogonal polynomials based on them. Also, in [1,2], using KreinÕs method, Mkhitarian and Abdou obtained the spectral relationships for the integral operator containing logarithmic
*
Corresponding author. E-mail address: [email protected] (M.A. Abdou).
0096-3003/$ - see front matter Ó 2003 Published by Elsevier Inc. doi:10.1016/S0096-3003(03)00619-2
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kernel and Carleman kernel, respectively. Kovalenko [3] developed the FIE of the first kind for the mechanics mixed problems of continuous media, and obtained an approximate solution for the FIE of the first kind with an elliptic kernel. Abdou in [4,5] obtained the solution of FIE of the second kind with potential function kernel, and the structure resolvent for the same equation with general kernel is discussed in [5]. The potential theory method is used in [6,7] to obtain the eigenvalues and eigenfunctions for a system of FIE of the first kind with generalized potential kernel. Also, other different cases for the FIE of the first kind are solved and discussed in [8]. In [9] spectral relationships were set up, using the generalized potential theory method for an integral operator using generalized potential kernel. In [10] regular and singular asymptotic method were applied to one, two and three dimensional to obtain the solution of F-VIE of the first kind. Z Z t k0 kðx; yÞ/ðy; tÞ dy þ F ðsÞ/ðx; sÞ ds ¼ f ðx; tÞ; ð1:1Þ 0
X
where x ¼ xðx1 ; x2 ; x3 Þ, y ¼ y ðy1 ; y2 ; y3 Þ and kðx; yÞ is a kernel of Fredholm integral term, that has a singular term, while F ðtÞ is a continuous function that known as the kernel of Volterra integral term. In this work, the general solution of V-FIE of the first kind is obtained in one, two and three dimensional. Also many special cases are discussed when the kernel of position takes different forms Weber–Sonien integrals.
2. Volterra–Fredholm integral equation Consider the V-FIE if the first kind Z tZ F ðt; sÞkðx; yÞ/ðy; tÞ dy ¼ f ðx; tÞ ¼ ½cðtÞ f ðxÞ 0
X
ðx ¼ xðx1 ; x2 ; x3 Þ; y ¼ y ðy1 ; y2 ; y3 Þ; ðx; yÞ 2 X; t 2 ½0; T ; T < 1Þ ð2:1Þ under the condition Z /ðx; tÞ dx ¼ P ðtÞ:
ð2:2Þ
X
The integral equation (2.1) under the condition (2.2) can be investigated from the contact problem of a rigid surface (G, t) having an elastic material occupying the domain X where f ðxÞ describing the surface of stamp. This stamp impressed into an elastic layer surface (plane) by a variable known force with respect to time P ðtÞ whose eccentricity of application eðtÞ, t 2 ½0; T , T < 1, t that case rigid displacement cðtÞ. Here G is the displacement
M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153
143
magnitude and t is PoissonÕs coefficient. The given function cðtÞ, F ðt; sÞ are continuous in the class C½0; T , while the given function F ðx; tÞ 2 L2 ðXÞ C½0; T . The kernel of position kðx; yÞ has a singularity. The unknown function /ðx; tÞ is called the potential function of V-FIE and will be obtained in the space L2 ðXÞ C½0; T , T < 1. In order to guarantee the existence of unique solution of (2.1), we assume the following conditions: ii(i) The kernel of position kðx; yÞ 2 Cð½X ½XÞ, x ¼ xðx1 ; x2 ; x3 Þ, y ¼ y ðy R 1 ;Ry2 ; y3 Þ satisfies in L2 ðXÞ. In general, the discontinuous condition f X X k 2 ðx; yÞ dx dyg1=2 ¼ E, E is a constant, and X is the domain of integration. i(ii) The positive continuous function F ðt; sÞ 2 Cð½0; T ½0; T Þ and satisfies F jðt; sÞj < D (D is a constant) for all values of ðt; sÞ 2 ½0; T . (iii) The given continuous functions cðtÞ 2 C½0; T while f ðxÞ 2 L2 ðXÞ and f ðx; tÞ 2 L2 ðXÞ C½0; T . (iv) The unknown function /ðx; tÞ satisfies H€ older condition until respect to time and Lipschitz condition with respect to position.
3. Weber–Sonien integral forms Consider the kernel of Fredholm term in the following form: xk
t;k;e Kn;m ðx; yÞ ¼
W t ðx; yÞ ð0 6 e 6 1Þ; y eþk1 n;m Z 1 t Wn;m ðx; yÞ ¼ tt Jn ðtxÞJm ðtyÞ dt;
ð3:1Þ
0
where Jn ðzÞ is the Bessel function of the first type. Many different cases can be established from the formula (3.1) as ii(i) Logarithmic kernel, kðx; yÞ ¼ ln jx yj, t ¼ e ¼ 0, k ¼ 12, n ¼ m ¼ 12, Z pffiffiffiffiffi 1 J12 ðtxÞJ12 ðtyÞ dt ð3:2Þ kðx; yÞ ¼ xy 0
(for symmetric and skew symmetric respectively). t i(ii) Carleman kernel, kðx; yÞ ¼ jx yj , 0 6 t < 1, e ¼ 0, k ¼ 12, n ¼ m ¼ 12, Z pffiffiffiffiffi 1 t t J12 ðtxÞJ12 ðtyÞ dt ð3:3Þ kðx; yÞ ¼ xy 0
(for symmetric and skew symmetric respectively).
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pffiffiffiffiffi 2xy 1 (iii) Elliptic integral form kðx; yÞ ¼ xþy E xþy , k ¼ e ¼ t ¼ n ¼ m ¼ 0, Z pffiffiffiffiffi 1 J0 ðtxÞJ0 ðtyÞ dt ðx; yÞ 2 ½a; b: ð3:4Þ kðx; yÞ ¼ xy 0 2
2 1=2
(iv) Potential kernel, kðx n; y gÞ ¼ ½ðx nÞ þ ðy nÞ
,
1 k¼ ; 2
e ¼ t ¼ 0; n ¼ m; Z pffiffiffiffiffi 1 kðx; yÞ ¼ xy Jm ðtxÞJm ðtyÞ dt
ð3:5Þ ðm P 0Þ:
0
(v) Generalized potential kernel, kðx n; y gÞ ¼ ½ðx nÞ2 þ ðy nÞ2 t , 1 k¼ ; 2Z pffiffiffiffiffi kðx; yÞ ¼ xy
n ¼ m P 0;
e ¼ 0;
0 6 m < 1; ð3:6Þ
1
tt Jm ðtxÞJm ðtyÞ dt
ðm P 0Þ:
0
4. Numerical method The importance of this method comes from its wide applications in mathematical physics problems, where the eigenvalues and eigenfunctions of the integral equations can be studied and discussed. Also this method has wide applications in the applied sciences especially in the theory of elasticity, mixed problems of mechanics area and contact problem, where we have a linear system of FIE and the potential function can be obtained. The numerical solution of Eq. (2.1), under the condition (2.2) can be obtained, if we divide the interval ½0; T , 0 6 t 6 T 6 1 as 0 6 t0 < t1 < ti < < t‘ ¼ T , when t ¼ tk , k ¼ 0; 1; . . . ; ‘. The integral equation (2.1) takes the form Z tZ Z tk Z kðx; yÞF ðt; sÞ/ðy; sÞ dy ds ¼ kðx; yÞF ðtk ; sÞ/ðy; sÞ dy ds 0
0
X
X
¼ f ðx; tk Þ
ð4:1Þ
which can be adapted in the form Z k X ~ uj Fj;k ðtk ; tj Þ kðx; yÞ/ðy; tj Þ dy þ Oðhkpþ1 Þ ¼ f ðx; tk Þ ðhk ! 0; p~ > 0Þ; j¼0
X
ð4:2Þ where hk ¼ max0 6 j 6 k hj and hj ¼ tjþ1 tj .
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145
The values uj and the constant p~ depend on the number of derivatives of F ðt; sÞ with respect to t, for example if F ðt; sÞ 2 C 4 ð½0; T ; ½0; T Þ then, we have p~ ¼ 4, p~ ¼ k and u0 ¼ 12 h0 , u4 ¼ 12 h4 , ui ¼ hi , i ¼ 1; 2; 3; (see [11,12]). Using the following notations: /ðx; tk Þ ¼ /k ðxÞ, F ðtk ; tj Þ ¼ Fk;j , and f ðx; tk Þ ¼ fk ðxÞ we can write (4.2) in the following: k X
Z
kðx; yÞ/j ðyÞ dy ¼ fk ðxÞ:
uj Fj;k
ð4:3Þ
X
j¼0
Also, the boundary condition (2.2) becomes Z /k ðxÞ dx ¼ Nk :
ð4:4Þ
X
The formula (4.3) represents a linear system of FIE of the first kind, that can be solved using the recurrence relations. Also, the general solution of (4.3) depends on the kind of the kernel and the domain if integration X.
5. Application 5.1. One dimensional integral equation Consider a linear system of FIE. X
Z
j¼0
kðzÞ ¼
Z 0
1
uj Fj;k
K 1 1
nx /j ðnÞ dn ¼ fk ðxÞ k
LðvÞ cos vz dv ; v
ðk 2 ð0; 1ÞÞ
vþm LðvÞ ¼ ; mP1 1þt
ð5:1Þ
under the condition Z
1
/k ðxÞ dx ¼ Pk ð0Þ:
ð5:2Þ
1
The function LðV Þ is continuous and positive for t 2 ð0; 1Þ and it satisfies the following asymptotic equalities: LðV Þ ¼ m ðm 1Þv þ 0ðv3 Þ ðv ! 0Þ; m1 1 þ0 2 LðvÞ ¼ 1 ðv ! 1; m P 1Þ: u v
ð5:3Þ
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Let m ¼ 1 and k ! 1, such that the term ðn xÞ is very small, then using the relation [13] Z 1 cos vz dv ¼ ln z þ d ðd is a constantÞ ð5:4Þ v 0 using (5.4) in (5.1), we have Z a k X 1 /ðyÞ dy ¼ gk ðxÞ; uj Fj;k ln jx yj a j¼0 X gk ðxÞ ¼ fk ðxÞ uj Fj;k Pk ð0Þ:
ð5:5Þ
The formula (5.5) represents as linear system of FIE of the first kind with logarithmic kernel. To obtain the general solution of (5.5) we using the recurrence relations, where we let k ¼ 0. Then the solution can be obtained using one of the following method: KreinÕs method [14], potential theory method [15], Fourier transformation method [16] and orthogonal polynomials method [13], then let k ¼ 1 and so on. The general solution leads us to obtain the following spectral relationships: Z a k X ln jx yj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tnj ðy=aÞ dy uj Fj;k a2 y 2 a j¼0 8 Pk > < p lnð2=aÞ j¼0 uj fj;k ; ! n ¼ 0 k X ¼ ð5:6Þ uj Fj;k x > p Tnj ; nj P 1; : a nj j¼0 where Tn ðxÞ is the Chebyshev polynomial of the first type. Many important relationships can be derived from Eq. (5.6) i(i) If nj ¼ 2mj x sin n=2 ¼ ; a sin a=2
y sin g=2 ¼ a sin a=2
ð5:7Þ
and if nj ¼ 2mj þ 1 x tan n=2 ¼ ; a tan a=2
t tan g=2 ¼ a tan a=2
ða 6 n; g 6 a; a ¼ p; m ¼ 0; 1; 2; . . .Þ ð5:8Þ
we have the following integral equation: Z a k X 1 w ðnÞ dn ¼ hk ðgÞ uj Fj;k ln ng j 2 sin a j¼0 2
ð5:9Þ
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which leads us to the following spectral relations: # sin g=2 Z a" k T X 2m sin a=2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðg=2Þ dg uj Fj;k ln sin ng 2 2ðcos g cos aÞ a j¼0 2 8 P 2 > > < p ln sin a uj Fj;k ; mj ¼ 0 ¼ P uj Fj;k sin n=2 > > : p ln Tm ð2mj Þ j sin a=2
147
ð5:10Þ
and tan g=2 T cos g2 dg 2m þ1 j tan a=2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uj Fj;k ln 2 sin ng 2ðcos g cos aÞ a j¼0 2 k X uj Fj;k tan n=2 ¼p T2mj þ1 ðm P 0Þ: tan a=2 ð2mj þ 1Þ j¼1;2
k X
Z
a
!
(ii) Differentiating (5.6) with respect to x, we have Z a k X X uj Fj;k Tnj ðy=aÞ dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p Unj 1 ðx=aÞ uj Fj;k ln 2 2 yx a a y a j¼0 Z a k X dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0; uj Fj;k ðy xÞ a2 y 2 a j¼0
ð5:11Þ
ðn P 1Þ;
ð5:12Þ where Un ðx=aÞ is the Chebyshev polynomial of the second kind. Using (5.7) and (5.8) in (5.12), we have tan g=2 Z a Tnj k X gn tan a=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðg=2Þ dg uj Fj;k cot 2 2 cos g cos a a j¼0 8 0; nj ¼ 0 > > > > > P > > 2 kj¼0 uj Fj;k cosecða=2ÞU2mj 1 tan n=2 ; nj ¼ 2mj m ¼ 1; 2; . . . > > > tan a=2 < ¼ Pk tan n=2 > 2 u F cosecða=2ÞU 2mj 1 > j¼0 j j;k > tan a=2 > > > > h i > 2m 2 sin a a j > > : þ ð1Þ2 tan ; nj ¼ 2mj 1 1 þ cos a 4 ð5:13Þ
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and Z a k sec g=2: cot gn 1X tan g=2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tnj uj Fj;k dg 2 j¼0 tan a=2 cðcos g cos aÞ a 8 > < cosec ða=2Þ sec2 ðn=2Þ Pk uj Fj;k Un 1 tan n=2 ; j j¼0 tan a=2 ¼ P > k : secða=2Þ tanðn=2Þ n ¼ 0: j¼0 uj Fj;k ;
n P 1;
ð5:14Þ
5.2. Integral equation with Carleman kernel t
If we let, in Eq. (4.3), kðx; yÞ ¼ jx yj , 0 6 t < 1, X 2 ½a; a, we have a linear system of FIE of the first kind with Carleman kernel. The importance of this kernel came from the work of Arytiunian [17], who has shown that, the contact problem of the nonlinear theory of plasticity, in its first approximation reduce to a FIE of the first kind with Carleman kernel. Hence, we have k X
Z
a
t
jx yj /j ðyÞ dy ¼ fk ðxÞ jxj 6 a:
uj Fj;k
ð5:15Þ
a
j¼0
The general solution of (5.16) will take the forms k X
Z
a
uj Fj;k a
j¼0
t=2
C2nj ðs=aÞ ds jt sj
t
ða2
s2 Þ
1t 2
k X
t=2
uj Fj;k k2nj C2nj ðt=aÞ
ðn P 0Þ
ð5:16Þ
j¼0
and k X j¼0
Z
a
uj Fj;k a
t=2
C2nj 1 ðs=aÞ ds jt sj
t
ða2
s2 Þ
1t 2
k X
t=2
uj Fj;k k2nj 1 C2nj 1 ðt=aÞ ðn P 1Þ;
j¼0
ð5:17Þ where h pt i1 k2nj ¼ pCð2nj þ tÞ Cð2nj þ 1ÞCðtÞ cos 2
ðn P 0Þ:
ð5:18Þ
Here, CðÞ is defined as a gamma function, and Cnt ðxÞ is a Gegenbauer polynomial.
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149
5.3. Two and three dimensional integral equation 5.3.1. Potential kernel Assume, in Eq. (4.3) kðx n; y gÞ ¼ ½ðX nÞ2 þ ðY gÞ2 1=2 and the dopffiffiffiffiffiffiffiffiffiffiffiffiffiffi main of integration X is defined as X ¼ fðx; y; zÞ 2 X : x2 þ y 2 6 a; Z ¼ 0g. Hence, we have k X
Z Z
¼ uj Fj;k
X
j¼0
/j ðn; gÞ dn dg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ fk ðx; yÞ: 2 2 ðx nÞ ðy gÞ
ð5:19Þ
Using the polar coordinates, x ¼ r cos h, y ¼ sin h, we have X
Z
a
Kn0 ðr; qÞ/j ðqÞ dq ¼ fk ðrÞ; Z pffiffiffiffiffi 1 Jm ðtrÞJm ðtqÞ dt ðn P 0Þ kðr; qÞ ¼ rq uj fj;k
0
ð5:20Þ
0
ðJm ðxÞ is a Bessel functionÞ: The general solution of Eq. (5.21) leads to the following spectral relationships: k X
Z
1
uj Fj;k 0
j¼0
¼ rn
X
ðn;1Þ Kn ðr; qÞPmj 2 ð1 2q2 Þ dq pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2
ðn;1Þ kmj uj Fj;k Pmj 2 ð1 2r2 Þ;
ð5:21Þ
j¼0
where kmj
C2 12 þ mj ¼ ð2mj Þ!Cð1 þ nmj Þ
ð5:22Þ
and pmða;bÞ ðxÞ is a Jacobi polynomial. 5.3.2. Generalized potential kernel When the modules of elasticity of the plane is changing according to the power-law, ri ¼ K0 ehi ð0 6 h < 1Þ, where ri , ei are the stress and strain rate intensities respectively, while K0 and h are the physical constant (see [6]). For this, the kernel of Eq. (4.3) takes the form Kðx n; y gÞ ¼ ½ðx nÞ2 þ ðy gÞ2 t ;
0 6 t < 1:
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M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153
If the domain of integration takes the form X : fðx; y; zÞ 2 X : a; Z ¼ 0g, we have the integral equation Z Z k X /j ðn; gÞ dn dg uj Fj;k ¼ fk ðx; yÞ: 2 2 t ½ðx nÞ ðy gÞ X j¼0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y 2 6
ð5:23Þ
Using the polar coordinates and Fourier transformation method, we have Z 1 k X uj Fj;k Knt ðr; qÞ/mj ðqÞ dq ¼ fmk ðrÞ; ð5:24Þ 0
j¼0
where knt ðr; qÞ
pffiffiffiffiffi ¼ rq
Z
1
tt Jn ðtrÞJn ðtqÞ dt;
0 k X
Z
1
uj Fj;k
1t 2
ð1 q2 Þ
0
j¼0
¼ rn
k X
Wnt ðr; qÞ
ðnj 1tÞ Pmj 2 ð1 2q2 Þ dq
ð5:25Þ
ðnj 1tÞ uj Fj;k kmj Pmj 2 ð1 2r2 Þ;
j¼0
where kmj
¼2
2ð1t 2 Þ
C2 1þt þ mj 2 ðmj Þ!ð1 þ n þ mj Þ
ðm P 0Þ:
The spectral relationships for the integral equation of (4.3) with the elliptic integral kernel can be obtained directly from Eq. (5.22), when n ¼ 0; or from (5.26), when t ¼ n ¼ 0, in the form pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2rq Z 1 q2 dq yK p2mj k 1 rþq X pffiffiffiffiffiffiffiffiffiffiffiffiffi uj Fj;k 1 q2 0 j¼0 2 Z 1 k pffiffiffiffiffiffiffiffiffiffiffiffi p2 X ð2mj 1Þ! ¼ uj Fj;k P2mj 1 r2 ; ð5:26Þ ð2mj Þ! 4 j¼0 0 pffiffiffiffiffi 2xy where K xþy is the elliptic integral kernel and Pm ðZÞ is Legendre polynomial function. In general case, if the kernel of Eq. (4.3) takes the form Z pffiffiffiffiffi 1 t t Wn;m ðx; yÞ ¼ xy t Jn ðtxÞJm ðtyÞ dt
ð5:27Þ
0
and the domain of integration is defined as X : fðx; y; zÞ 2 X : 0 6 x; y 6 1; z ¼ 0g, then, we have the following spectral relationships:
M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153 k X
Z
1
uj Fj;k
t þÞ y 1þc Wl;c ðx; yÞPmðc:r ð1 2y 2 Þ dy j
ð1 y 2 Þ
0
j¼0
¼ xl
k X
151
rþ
ðl;r Þ k ð1 2x2 Þ; mj uj Fj;k Pmj
ð5:28Þ
j¼0
t1 Cð1 rþ þ mj ÞCðr þ mj Þ½ðmj Þ!Cð1 þ l þ mj Þ k mj ¼ 2
1
ð0 6 x 6 1; 2r ¼ 1 t ðc lÞ; 2r ¼ 1 þ t þ l þ c; Re r 0; Re t < 1Þ; where Pmða;bÞ ð1 2x2 Þ is a Jacobi polynomial. 5.3.3. Integral equation in three dimensional with Macdonald kernel 1 Assume in Eq. (4.3) kðx n; y gÞ ¼ ½ðx nÞ2 þ ðy gÞ2 2 , X ¼ fðx; y; zÞ : 1 < x < 1; jyj < a; 1 < z < 0g then, we have the following integral equation: X
Z Z uj Fj;k X
/j ðn; gÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn dg ¼ fk ðx; yÞ: 2 2 ðx nÞ þ ðy gÞ
ð5:29Þ
Using the Fourier integral transformation Z
/s ðxÞ ¼ fa ðxÞ ¼
1
/ðx; yÞeiay dy;
Z 1 1
ð5:30Þ iay
f ðx; yÞe dy;
1
and the Macdonald kernel definition Kðjajjx njÞ ¼
Z 0
1
cos ay dy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ðx nÞ þ y 2
ð5:31Þ
where a in the Fourier parameter, and K0 is the Macdonald kernel, we have the following X j¼0
Z
a
K0 ðjt vjÞwj ðvÞ dv ¼ hk ðtÞ:
uj Fj;k a
ð5:32Þ
152
M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153
The spectral relationships of (5.31) takes the form Z a C cos1 t ; q
k X e nj a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi uj Fj;k K0 ðjt vjÞ dt 2 a t2 a j¼0
1 v k u F F K ð0; qÞC X ; q j j;k e nj enj cos a ¼p Fe Kn0 j ð0; qÞ j¼0
ðq ¼ a2 =4Þ;
ð5:33Þ
where Fe Kn ð‘; qÞ, Cenj ðh; qÞ are called the Mathieu functions under the conditions, 0 6 h 6 2p, 6 ‘ < 1. If the domain of integration of (5.33)0 is considered as X ¼ fðx; y; zÞ : 1 < x; y < 1; z < 0g we will have the following spectral relations: Z 1 X K0 ðjt sjÞ s 1=2 pffiffiffi uj Fj;k e Lmj ð2sÞ ds s 0 j¼0 k p X ½ð2mj 1Þ! t 1=2 e Lmj ð2tÞ uj Fj;k ¼ pffiffiffi ð2mj Þ! 2 j¼0
ðt P 0Þ;
ð5:34Þ
where Lam ðxÞ is the Chebyshev–Laguerre polynomials. 5.3.4. Generalized Macdonald kernel As in the case 5.3.2 and according to the power law and the domain X is defined X ¼ fðx; y; zÞ : 1 < x; y < 1; 1 < z 6 0g we have the following integral equation: Z 1 k X Kh ðjsjjt sjÞwj ðsÞ ds 1 uj Fj;k ¼ g ðtÞ 0 6 h < ; ð5:35Þ k 2 jt sjh 1 j¼0 where s > 0, is the coefficient of Fourier integral transformations, and Kn ðj jÞ is the generalized Macdonald kernel, that has the following expansion: pffiffiffi X 1 p kh ðjx yjÞ ¼ h xþy Lnh1=2 ð2xÞLh1=2 ð2yÞs ¼ 1; ð5:36Þ h 2 e jx yj n¼0 where Lan ð2xÞ is the Chebyshev–Laguerre polynomial the spectral relationships of Eq. (5.34) take the form Z 1 X kh ðjt sjÞ h1=2 uj Fj;k Lm ð2sÞ ds jt sjh s1=2h j 0 j¼0
rffiffiffi p t X uj Fj;k C 12 h C 12 þ h þ mj h1=2 Lmj ð2tÞ ðt P 0; m P 0Þ; e ¼ 2 ðmj Þ! j¼0 ð5:37Þ where CðÞ is the gamma function.
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References [1] S.M. Mkhitaryan, M.A. Abdou, On various method for solution of the Carelman integral equation, Dakl. Acad. Nauk Arm. SSR 89 (3) (1990) 125–129. [2] S.M. Mkhitaryan, M.A. Abdou, On different method of solution of the integral equation for the planer contact problem of elasticity, Dakl. Acad. Nauk Arm. SSR 89 (2) (1990) 59–74. [3] E.V. Kovalenko, Some approximate methods of solving integral equations of mixed problems, J. Appl. Math. Mech. 53 (1989) 85–92. [4] M.A. Abdou, Fredholm integral equation of the second kind with potential kernel, J. Comput. Appl. Math. 107 (2000) 161–167. [5] M.A. Abdou, Fredholm integral with potential kernel and its structure resolvent, J. Appl. Math. Comput. 107 (2000) 169–180. [6] M.A. Abdou, Fredholm–Volterra integral equation and generalized potential kernel, J. Appl. Math. Comput. 131 (2002) 81–94. [7] M.A. Abdou, Spectral relationships for integral operators in contact problem of impressing stamp, J. Appl. Math. Comput. 118 (2001) 95–111. [8] M.A. Abdou, Integral equation with Macdonald kernel and its application to a system of contact problem, J. Appl. Math. Comput. 118 (2001) 83–94. [9] M.A. Abdou, Spectral relationships for integral equation with Macdonald kernel and contact problem, J. Appl. Math. Comput. 118 (2002) 93–103. [10] M.A. Abdou, Fredholm–Volterra integral equation of the first kind and contact problem, J. Appl. Math. Comp. 125 (2002) 177–193. [11] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Philadelphia, 1985. [12] K.E. Atkinson, The Numerical Solution of Integral Equation of the Second Kind, Cambridge, 1997. [13] G.Yo. Popov, Contact Problem for a Linearly Defamle Base, Kiev, Odessa, 1982. [14] M.A. Abdou, N.Y. Ezzaldin, KreirnÕs method with certain singular kernel for solving the integral equation of the first kind, Periodica, Math. Hung. 28 (2) (1994) 143–149. [15] C.D. Green, Integral Equations Method, Nelson, New York, 1969. [16] H. Hochstatadt, Integral Equations, Wiley Interscience Publication, New York, 1971. [17] N.K. Arytiunian, Plane contact problem of the theory of creef, J. Appl. Math. Mech. 23 (1959) 901–923.
Volterra–Fredholm integral equation of the first kind and spectral relationships M.A. Abdou a
a,*
, F.A. Salama
b
Department of Mathematics, Faculty of Science, Om Elkora University, Mekah, Saudi Arabia b Department of Mathematics, Faculty of Education, Alexandria University, Egypt
Abstract Here, the solution in one, two and three dimensional for the Volterra–Fredholm integral equation of the first kind is obtained in the space L2 ðXÞ C½0; T , T < 1. Using a numerical method the integral equation of Volterra–Fredholm becomes a linear system of Fredholm integral equation when that the kernel of Fredholm integral takes a logarithmic form, Carleman function, generalized potential function and Macdonald function are considered as special cases. Ó 2003 Published by Elsevier Inc. Keywords: Volterra–Fredholm integral equation V-FIE; Logarithmic kernel; Potential function; Macdonald kernel
1. Introduction Many problems in mathematical physics, theory of elasticity, viscodynamics fluid and mixed problems of mechanics of continuous media reduce to the integral equations of the first or the second kind. The monographs [1,2] give many of spectral relationships in terms of orthogonal polynomials for the integral operators frequently encountered in mathematical physics, and describe a method of orthogonal polynomials based on them. Also, in [1,2], using KreinÕs method, Mkhitarian and Abdou obtained the spectral relationships for the integral operator containing logarithmic
*
Corresponding author. E-mail address: [email protected] (M.A. Abdou).
0096-3003/$ - see front matter Ó 2003 Published by Elsevier Inc. doi:10.1016/S0096-3003(03)00619-2
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kernel and Carleman kernel, respectively. Kovalenko [3] developed the FIE of the first kind for the mechanics mixed problems of continuous media, and obtained an approximate solution for the FIE of the first kind with an elliptic kernel. Abdou in [4,5] obtained the solution of FIE of the second kind with potential function kernel, and the structure resolvent for the same equation with general kernel is discussed in [5]. The potential theory method is used in [6,7] to obtain the eigenvalues and eigenfunctions for a system of FIE of the first kind with generalized potential kernel. Also, other different cases for the FIE of the first kind are solved and discussed in [8]. In [9] spectral relationships were set up, using the generalized potential theory method for an integral operator using generalized potential kernel. In [10] regular and singular asymptotic method were applied to one, two and three dimensional to obtain the solution of F-VIE of the first kind. Z Z t k0 kðx; yÞ/ðy; tÞ dy þ F ðsÞ/ðx; sÞ ds ¼ f ðx; tÞ; ð1:1Þ 0
X
where x ¼ xðx1 ; x2 ; x3 Þ, y ¼ y ðy1 ; y2 ; y3 Þ and kðx; yÞ is a kernel of Fredholm integral term, that has a singular term, while F ðtÞ is a continuous function that known as the kernel of Volterra integral term. In this work, the general solution of V-FIE of the first kind is obtained in one, two and three dimensional. Also many special cases are discussed when the kernel of position takes different forms Weber–Sonien integrals.
2. Volterra–Fredholm integral equation Consider the V-FIE if the first kind Z tZ F ðt; sÞkðx; yÞ/ðy; tÞ dy ¼ f ðx; tÞ ¼ ½cðtÞ f ðxÞ 0
X
ðx ¼ xðx1 ; x2 ; x3 Þ; y ¼ y ðy1 ; y2 ; y3 Þ; ðx; yÞ 2 X; t 2 ½0; T ; T < 1Þ ð2:1Þ under the condition Z /ðx; tÞ dx ¼ P ðtÞ:
ð2:2Þ
X
The integral equation (2.1) under the condition (2.2) can be investigated from the contact problem of a rigid surface (G, t) having an elastic material occupying the domain X where f ðxÞ describing the surface of stamp. This stamp impressed into an elastic layer surface (plane) by a variable known force with respect to time P ðtÞ whose eccentricity of application eðtÞ, t 2 ½0; T , T < 1, t that case rigid displacement cðtÞ. Here G is the displacement
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143
magnitude and t is PoissonÕs coefficient. The given function cðtÞ, F ðt; sÞ are continuous in the class C½0; T , while the given function F ðx; tÞ 2 L2 ðXÞ C½0; T . The kernel of position kðx; yÞ has a singularity. The unknown function /ðx; tÞ is called the potential function of V-FIE and will be obtained in the space L2 ðXÞ C½0; T , T < 1. In order to guarantee the existence of unique solution of (2.1), we assume the following conditions: ii(i) The kernel of position kðx; yÞ 2 Cð½X ½XÞ, x ¼ xðx1 ; x2 ; x3 Þ, y ¼ y ðy R 1 ;Ry2 ; y3 Þ satisfies in L2 ðXÞ. In general, the discontinuous condition f X X k 2 ðx; yÞ dx dyg1=2 ¼ E, E is a constant, and X is the domain of integration. i(ii) The positive continuous function F ðt; sÞ 2 Cð½0; T ½0; T Þ and satisfies F jðt; sÞj < D (D is a constant) for all values of ðt; sÞ 2 ½0; T . (iii) The given continuous functions cðtÞ 2 C½0; T while f ðxÞ 2 L2 ðXÞ and f ðx; tÞ 2 L2 ðXÞ C½0; T . (iv) The unknown function /ðx; tÞ satisfies H€ older condition until respect to time and Lipschitz condition with respect to position.
3. Weber–Sonien integral forms Consider the kernel of Fredholm term in the following form: xk
t;k;e Kn;m ðx; yÞ ¼
W t ðx; yÞ ð0 6 e 6 1Þ; y eþk1 n;m Z 1 t Wn;m ðx; yÞ ¼ tt Jn ðtxÞJm ðtyÞ dt;
ð3:1Þ
0
where Jn ðzÞ is the Bessel function of the first type. Many different cases can be established from the formula (3.1) as ii(i) Logarithmic kernel, kðx; yÞ ¼ ln jx yj, t ¼ e ¼ 0, k ¼ 12, n ¼ m ¼ 12, Z pffiffiffiffiffi 1 J12 ðtxÞJ12 ðtyÞ dt ð3:2Þ kðx; yÞ ¼ xy 0
(for symmetric and skew symmetric respectively). t i(ii) Carleman kernel, kðx; yÞ ¼ jx yj , 0 6 t < 1, e ¼ 0, k ¼ 12, n ¼ m ¼ 12, Z pffiffiffiffiffi 1 t t J12 ðtxÞJ12 ðtyÞ dt ð3:3Þ kðx; yÞ ¼ xy 0
(for symmetric and skew symmetric respectively).
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pffiffiffiffiffi 2xy 1 (iii) Elliptic integral form kðx; yÞ ¼ xþy E xþy , k ¼ e ¼ t ¼ n ¼ m ¼ 0, Z pffiffiffiffiffi 1 J0 ðtxÞJ0 ðtyÞ dt ðx; yÞ 2 ½a; b: ð3:4Þ kðx; yÞ ¼ xy 0 2
2 1=2
(iv) Potential kernel, kðx n; y gÞ ¼ ½ðx nÞ þ ðy nÞ
,
1 k¼ ; 2
e ¼ t ¼ 0; n ¼ m; Z pffiffiffiffiffi 1 kðx; yÞ ¼ xy Jm ðtxÞJm ðtyÞ dt
ð3:5Þ ðm P 0Þ:
0
(v) Generalized potential kernel, kðx n; y gÞ ¼ ½ðx nÞ2 þ ðy nÞ2 t , 1 k¼ ; 2Z pffiffiffiffiffi kðx; yÞ ¼ xy
n ¼ m P 0;
e ¼ 0;
0 6 m < 1; ð3:6Þ
1
tt Jm ðtxÞJm ðtyÞ dt
ðm P 0Þ:
0
4. Numerical method The importance of this method comes from its wide applications in mathematical physics problems, where the eigenvalues and eigenfunctions of the integral equations can be studied and discussed. Also this method has wide applications in the applied sciences especially in the theory of elasticity, mixed problems of mechanics area and contact problem, where we have a linear system of FIE and the potential function can be obtained. The numerical solution of Eq. (2.1), under the condition (2.2) can be obtained, if we divide the interval ½0; T , 0 6 t 6 T 6 1 as 0 6 t0 < t1 < ti < < t‘ ¼ T , when t ¼ tk , k ¼ 0; 1; . . . ; ‘. The integral equation (2.1) takes the form Z tZ Z tk Z kðx; yÞF ðt; sÞ/ðy; sÞ dy ds ¼ kðx; yÞF ðtk ; sÞ/ðy; sÞ dy ds 0
0
X
X
¼ f ðx; tk Þ
ð4:1Þ
which can be adapted in the form Z k X ~ uj Fj;k ðtk ; tj Þ kðx; yÞ/ðy; tj Þ dy þ Oðhkpþ1 Þ ¼ f ðx; tk Þ ðhk ! 0; p~ > 0Þ; j¼0
X
ð4:2Þ where hk ¼ max0 6 j 6 k hj and hj ¼ tjþ1 tj .
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145
The values uj and the constant p~ depend on the number of derivatives of F ðt; sÞ with respect to t, for example if F ðt; sÞ 2 C 4 ð½0; T ; ½0; T Þ then, we have p~ ¼ 4, p~ ¼ k and u0 ¼ 12 h0 , u4 ¼ 12 h4 , ui ¼ hi , i ¼ 1; 2; 3; (see [11,12]). Using the following notations: /ðx; tk Þ ¼ /k ðxÞ, F ðtk ; tj Þ ¼ Fk;j , and f ðx; tk Þ ¼ fk ðxÞ we can write (4.2) in the following: k X
Z
kðx; yÞ/j ðyÞ dy ¼ fk ðxÞ:
uj Fj;k
ð4:3Þ
X
j¼0
Also, the boundary condition (2.2) becomes Z /k ðxÞ dx ¼ Nk :
ð4:4Þ
X
The formula (4.3) represents a linear system of FIE of the first kind, that can be solved using the recurrence relations. Also, the general solution of (4.3) depends on the kind of the kernel and the domain if integration X.
5. Application 5.1. One dimensional integral equation Consider a linear system of FIE. X
Z
j¼0
kðzÞ ¼
Z 0
1
uj Fj;k
K 1 1
nx /j ðnÞ dn ¼ fk ðxÞ k
LðvÞ cos vz dv ; v
ðk 2 ð0; 1ÞÞ
vþm LðvÞ ¼ ; mP1 1þt
ð5:1Þ
under the condition Z
1
/k ðxÞ dx ¼ Pk ð0Þ:
ð5:2Þ
1
The function LðV Þ is continuous and positive for t 2 ð0; 1Þ and it satisfies the following asymptotic equalities: LðV Þ ¼ m ðm 1Þv þ 0ðv3 Þ ðv ! 0Þ; m1 1 þ0 2 LðvÞ ¼ 1 ðv ! 1; m P 1Þ: u v
ð5:3Þ
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Let m ¼ 1 and k ! 1, such that the term ðn xÞ is very small, then using the relation [13] Z 1 cos vz dv ¼ ln z þ d ðd is a constantÞ ð5:4Þ v 0 using (5.4) in (5.1), we have Z a k X 1 /ðyÞ dy ¼ gk ðxÞ; uj Fj;k ln jx yj a j¼0 X gk ðxÞ ¼ fk ðxÞ uj Fj;k Pk ð0Þ:
ð5:5Þ
The formula (5.5) represents as linear system of FIE of the first kind with logarithmic kernel. To obtain the general solution of (5.5) we using the recurrence relations, where we let k ¼ 0. Then the solution can be obtained using one of the following method: KreinÕs method [14], potential theory method [15], Fourier transformation method [16] and orthogonal polynomials method [13], then let k ¼ 1 and so on. The general solution leads us to obtain the following spectral relationships: Z a k X ln jx yj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tnj ðy=aÞ dy uj Fj;k a2 y 2 a j¼0 8 Pk > < p lnð2=aÞ j¼0 uj fj;k ; ! n ¼ 0 k X ¼ ð5:6Þ uj Fj;k x > p Tnj ; nj P 1; : a nj j¼0 where Tn ðxÞ is the Chebyshev polynomial of the first type. Many important relationships can be derived from Eq. (5.6) i(i) If nj ¼ 2mj x sin n=2 ¼ ; a sin a=2
y sin g=2 ¼ a sin a=2
ð5:7Þ
and if nj ¼ 2mj þ 1 x tan n=2 ¼ ; a tan a=2
t tan g=2 ¼ a tan a=2
ða 6 n; g 6 a; a ¼ p; m ¼ 0; 1; 2; . . .Þ ð5:8Þ
we have the following integral equation: Z a k X 1 w ðnÞ dn ¼ hk ðgÞ uj Fj;k ln ng j 2 sin a j¼0 2
ð5:9Þ
M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153
which leads us to the following spectral relations: # sin g=2 Z a" k T X 2m sin a=2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðg=2Þ dg uj Fj;k ln sin ng 2 2ðcos g cos aÞ a j¼0 2 8 P 2 > > < p ln sin a uj Fj;k ; mj ¼ 0 ¼ P uj Fj;k sin n=2 > > : p ln Tm ð2mj Þ j sin a=2
147
ð5:10Þ
and tan g=2 T cos g2 dg 2m þ1 j tan a=2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uj Fj;k ln 2 sin ng 2ðcos g cos aÞ a j¼0 2 k X uj Fj;k tan n=2 ¼p T2mj þ1 ðm P 0Þ: tan a=2 ð2mj þ 1Þ j¼1;2
k X
Z
a
!
(ii) Differentiating (5.6) with respect to x, we have Z a k X X uj Fj;k Tnj ðy=aÞ dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p Unj 1 ðx=aÞ uj Fj;k ln 2 2 yx a a y a j¼0 Z a k X dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0; uj Fj;k ðy xÞ a2 y 2 a j¼0
ð5:11Þ
ðn P 1Þ;
ð5:12Þ where Un ðx=aÞ is the Chebyshev polynomial of the second kind. Using (5.7) and (5.8) in (5.12), we have tan g=2 Z a Tnj k X gn tan a=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðg=2Þ dg uj Fj;k cot 2 2 cos g cos a a j¼0 8 0; nj ¼ 0 > > > > > P > > 2 kj¼0 uj Fj;k cosecða=2ÞU2mj 1 tan n=2 ; nj ¼ 2mj m ¼ 1; 2; . . . > > > tan a=2 < ¼ Pk tan n=2 > 2 u F cosecða=2ÞU 2mj 1 > j¼0 j j;k > tan a=2 > > > > h i > 2m 2 sin a a j > > : þ ð1Þ2 tan ; nj ¼ 2mj 1 1 þ cos a 4 ð5:13Þ
148
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and Z a k sec g=2: cot gn 1X tan g=2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tnj uj Fj;k dg 2 j¼0 tan a=2 cðcos g cos aÞ a 8 > < cosec ða=2Þ sec2 ðn=2Þ Pk uj Fj;k Un 1 tan n=2 ; j j¼0 tan a=2 ¼ P > k : secða=2Þ tanðn=2Þ n ¼ 0: j¼0 uj Fj;k ;
n P 1;
ð5:14Þ
5.2. Integral equation with Carleman kernel t
If we let, in Eq. (4.3), kðx; yÞ ¼ jx yj , 0 6 t < 1, X 2 ½a; a, we have a linear system of FIE of the first kind with Carleman kernel. The importance of this kernel came from the work of Arytiunian [17], who has shown that, the contact problem of the nonlinear theory of plasticity, in its first approximation reduce to a FIE of the first kind with Carleman kernel. Hence, we have k X
Z
a
t
jx yj /j ðyÞ dy ¼ fk ðxÞ jxj 6 a:
uj Fj;k
ð5:15Þ
a
j¼0
The general solution of (5.16) will take the forms k X
Z
a
uj Fj;k a
j¼0
t=2
C2nj ðs=aÞ ds jt sj
t
ða2
s2 Þ
1t 2
k X
t=2
uj Fj;k k2nj C2nj ðt=aÞ
ðn P 0Þ
ð5:16Þ
j¼0
and k X j¼0
Z
a
uj Fj;k a
t=2
C2nj 1 ðs=aÞ ds jt sj
t
ða2
s2 Þ
1t 2
k X
t=2
uj Fj;k k2nj 1 C2nj 1 ðt=aÞ ðn P 1Þ;
j¼0
ð5:17Þ where h pt i1 k2nj ¼ pCð2nj þ tÞ Cð2nj þ 1ÞCðtÞ cos 2
ðn P 0Þ:
ð5:18Þ
Here, CðÞ is defined as a gamma function, and Cnt ðxÞ is a Gegenbauer polynomial.
M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153
149
5.3. Two and three dimensional integral equation 5.3.1. Potential kernel Assume, in Eq. (4.3) kðx n; y gÞ ¼ ½ðX nÞ2 þ ðY gÞ2 1=2 and the dopffiffiffiffiffiffiffiffiffiffiffiffiffiffi main of integration X is defined as X ¼ fðx; y; zÞ 2 X : x2 þ y 2 6 a; Z ¼ 0g. Hence, we have k X
Z Z
¼ uj Fj;k
X
j¼0
/j ðn; gÞ dn dg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ fk ðx; yÞ: 2 2 ðx nÞ ðy gÞ
ð5:19Þ
Using the polar coordinates, x ¼ r cos h, y ¼ sin h, we have X
Z
a
Kn0 ðr; qÞ/j ðqÞ dq ¼ fk ðrÞ; Z pffiffiffiffiffi 1 Jm ðtrÞJm ðtqÞ dt ðn P 0Þ kðr; qÞ ¼ rq uj fj;k
0
ð5:20Þ
0
ðJm ðxÞ is a Bessel functionÞ: The general solution of Eq. (5.21) leads to the following spectral relationships: k X
Z
1
uj Fj;k 0
j¼0
¼ rn
X
ðn;1Þ Kn ðr; qÞPmj 2 ð1 2q2 Þ dq pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2
ðn;1Þ kmj uj Fj;k Pmj 2 ð1 2r2 Þ;
ð5:21Þ
j¼0
where kmj
C2 12 þ mj ¼ ð2mj Þ!Cð1 þ nmj Þ
ð5:22Þ
and pmða;bÞ ðxÞ is a Jacobi polynomial. 5.3.2. Generalized potential kernel When the modules of elasticity of the plane is changing according to the power-law, ri ¼ K0 ehi ð0 6 h < 1Þ, where ri , ei are the stress and strain rate intensities respectively, while K0 and h are the physical constant (see [6]). For this, the kernel of Eq. (4.3) takes the form Kðx n; y gÞ ¼ ½ðx nÞ2 þ ðy gÞ2 t ;
0 6 t < 1:
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M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153
If the domain of integration takes the form X : fðx; y; zÞ 2 X : a; Z ¼ 0g, we have the integral equation Z Z k X /j ðn; gÞ dn dg uj Fj;k ¼ fk ðx; yÞ: 2 2 t ½ðx nÞ ðy gÞ X j¼0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y 2 6
ð5:23Þ
Using the polar coordinates and Fourier transformation method, we have Z 1 k X uj Fj;k Knt ðr; qÞ/mj ðqÞ dq ¼ fmk ðrÞ; ð5:24Þ 0
j¼0
where knt ðr; qÞ
pffiffiffiffiffi ¼ rq
Z
1
tt Jn ðtrÞJn ðtqÞ dt;
0 k X
Z
1
uj Fj;k
1t 2
ð1 q2 Þ
0
j¼0
¼ rn
k X
Wnt ðr; qÞ
ðnj 1tÞ Pmj 2 ð1 2q2 Þ dq
ð5:25Þ
ðnj 1tÞ uj Fj;k kmj Pmj 2 ð1 2r2 Þ;
j¼0
where kmj
¼2
2ð1t 2 Þ
C2 1þt þ mj 2 ðmj Þ!ð1 þ n þ mj Þ
ðm P 0Þ:
The spectral relationships for the integral equation of (4.3) with the elliptic integral kernel can be obtained directly from Eq. (5.22), when n ¼ 0; or from (5.26), when t ¼ n ¼ 0, in the form pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2rq Z 1 q2 dq yK p2mj k 1 rþq X pffiffiffiffiffiffiffiffiffiffiffiffiffi uj Fj;k 1 q2 0 j¼0 2 Z 1 k pffiffiffiffiffiffiffiffiffiffiffiffi p2 X ð2mj 1Þ! ¼ uj Fj;k P2mj 1 r2 ; ð5:26Þ ð2mj Þ! 4 j¼0 0 pffiffiffiffiffi 2xy where K xþy is the elliptic integral kernel and Pm ðZÞ is Legendre polynomial function. In general case, if the kernel of Eq. (4.3) takes the form Z pffiffiffiffiffi 1 t t Wn;m ðx; yÞ ¼ xy t Jn ðtxÞJm ðtyÞ dt
ð5:27Þ
0
and the domain of integration is defined as X : fðx; y; zÞ 2 X : 0 6 x; y 6 1; z ¼ 0g, then, we have the following spectral relationships:
M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153 k X
Z
1
uj Fj;k
t þÞ y 1þc Wl;c ðx; yÞPmðc:r ð1 2y 2 Þ dy j
ð1 y 2 Þ
0
j¼0
¼ xl
k X
151
rþ
ðl;r Þ k ð1 2x2 Þ; mj uj Fj;k Pmj
ð5:28Þ
j¼0
t1 Cð1 rþ þ mj ÞCðr þ mj Þ½ðmj Þ!Cð1 þ l þ mj Þ k mj ¼ 2
1
ð0 6 x 6 1; 2r ¼ 1 t ðc lÞ; 2r ¼ 1 þ t þ l þ c; Re r 0; Re t < 1Þ; where Pmða;bÞ ð1 2x2 Þ is a Jacobi polynomial. 5.3.3. Integral equation in three dimensional with Macdonald kernel 1 Assume in Eq. (4.3) kðx n; y gÞ ¼ ½ðx nÞ2 þ ðy gÞ2 2 , X ¼ fðx; y; zÞ : 1 < x < 1; jyj < a; 1 < z < 0g then, we have the following integral equation: X
Z Z uj Fj;k X
/j ðn; gÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn dg ¼ fk ðx; yÞ: 2 2 ðx nÞ þ ðy gÞ
ð5:29Þ
Using the Fourier integral transformation Z
/s ðxÞ ¼ fa ðxÞ ¼
1
/ðx; yÞeiay dy;
Z 1 1
ð5:30Þ iay
f ðx; yÞe dy;
1
and the Macdonald kernel definition Kðjajjx njÞ ¼
Z 0
1
cos ay dy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ðx nÞ þ y 2
ð5:31Þ
where a in the Fourier parameter, and K0 is the Macdonald kernel, we have the following X j¼0
Z
a
K0 ðjt vjÞwj ðvÞ dv ¼ hk ðtÞ:
uj Fj;k a
ð5:32Þ
152
M.A. Abdou, F.A. Salama / Appl. Math. Comput. 153 (2004) 141–153
The spectral relationships of (5.31) takes the form Z a C cos1 t ; q
k X e nj a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi uj Fj;k K0 ðjt vjÞ dt 2 a t2 a j¼0
1 v k u F F K ð0; qÞC X ; q j j;k e nj enj cos a ¼p Fe Kn0 j ð0; qÞ j¼0
ðq ¼ a2 =4Þ;
ð5:33Þ
where Fe Kn ð‘; qÞ, Cenj ðh; qÞ are called the Mathieu functions under the conditions, 0 6 h 6 2p, 6 ‘ < 1. If the domain of integration of (5.33)0 is considered as X ¼ fðx; y; zÞ : 1 < x; y < 1; z < 0g we will have the following spectral relations: Z 1 X K0 ðjt sjÞ s 1=2 pffiffiffi uj Fj;k e Lmj ð2sÞ ds s 0 j¼0 k p X ½ð2mj 1Þ! t 1=2 e Lmj ð2tÞ uj Fj;k ¼ pffiffiffi ð2mj Þ! 2 j¼0
ðt P 0Þ;
ð5:34Þ
where Lam ðxÞ is the Chebyshev–Laguerre polynomials. 5.3.4. Generalized Macdonald kernel As in the case 5.3.2 and according to the power law and the domain X is defined X ¼ fðx; y; zÞ : 1 < x; y < 1; 1 < z 6 0g we have the following integral equation: Z 1 k X Kh ðjsjjt sjÞwj ðsÞ ds 1 uj Fj;k ¼ g ðtÞ 0 6 h < ; ð5:35Þ k 2 jt sjh 1 j¼0 where s > 0, is the coefficient of Fourier integral transformations, and Kn ðj jÞ is the generalized Macdonald kernel, that has the following expansion: pffiffiffi X 1 p kh ðjx yjÞ ¼ h xþy Lnh1=2 ð2xÞLh1=2 ð2yÞs ¼ 1; ð5:36Þ h 2 e jx yj n¼0 where Lan ð2xÞ is the Chebyshev–Laguerre polynomial the spectral relationships of Eq. (5.34) take the form Z 1 X kh ðjt sjÞ h1=2 uj Fj;k Lm ð2sÞ ds jt sjh s1=2h j 0 j¼0
rffiffiffi p t X uj Fj;k C 12 h C 12 þ h þ mj h1=2 Lmj ð2tÞ ðt P 0; m P 0Þ; e ¼ 2 ðmj Þ! j¼0 ð5:37Þ where CðÞ is the gamma function.
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