LE MATEMATICHE Vol. LX (2005) – Fasc. I, pp. 41–58
MIXED TYPE OF FREDHOLM-VOLTERRA INTEGRAL EQUATION M. A. ABDOU - G. M. ABD AL-KADER In this paper, under certain conditions, the solution of mixed type of Fredholm-Volterra integral equation is discussed and obtained in the space L 2 (−1, 1) × C[0, T ], T < ∞. Here, the singular part of kernel of FredholmVolterra integral term is established in a logarithmic form, while the kernel of Fredholm-Volterra integral term is a positive continuous function in time and belongs to the class C[0, T ], T < ∞. The solution, when the mixed type integral, takes a system form of Fredholm integral equation of the first or second kind are discussed.
1. Introduction. Many problem of mathematical physics, engineering and contact problems in the theory of elasticity lead to the integral equation of the first kind, see [1], [2]. Mkhitarian and Abdou, using Krein’s method, obtained the spectral relationships for the FIE with logarithmic kernel and Carleman kernel, see [3], [4], respectively. The importance of Carleman kernel came from the work of Arytiunian [5] who has shown that, the contact problem of nonlinear theory of plasticity, in its first approximation reduce to a FIE of the first kind with Carleman kernel. Using potential theory method [6] Abdou and Hassan, in [7], Entrato in redazione il 4 Marzo 2004. Keywords: Fredholm-Volterra integral equation (F-VIE), Contact Problem, Chebyshev Polynomial, Logarithmic kernel, Potential theory method.
42
M. A. ABDOU - G. M. ABD AL-KADER
obtained the spectral relationships for the FIE of the first kind with logarithmic kernel. Also, in [8] the eigenvalue and eigenfunction are obtained for the FIE of the first kind with Carleman kernel. Abdou, in [9], [10], using potential theory method, obtained the spectral relationships for the FIE of the first kind with generalized potential kernel and Macdonold kernel, respectively. In this work, and the following work, we will consider the mixed integral equation t t F(t, τ )k(x , y)φ(y, τ ) d ydτ + G(t, τ )φ(x , τ ) dτ = f (x , t), (1.1) 0
0
x = x(x ¯ 1 , x 2 , x 3), y = y¯ (y1 , y2, y3 ), under the condition
(x , y) ∈ , t, τ ∈ [0, T ], T < ∞
(1.2)
φ(x , t) dx = P(t).
Here, the two given functions F(t, τ ) and G(t, τ ) which represent the kernels of Volterra integral term, are positive and continuous in the class C[0, T ], for all values of the time t, τ ∈ [0, T ], T < ∞. The function k(x , y), which has a term behavied badly in the domain , is called the kernel of Fredholm integral term. The given function f (x , t) is continuous with its partial dervatives with respect to position and time and belongs to the space L 2 () × C[0, T ]. The unknown function φ(x , t) is called the potential function of the mixed integral equations, and its result will be discussed in the space L 2 () × C[0, T ]. In order to guarantee the existance of unique solutions of (1.1), under the condition (1.2), we assume the following conditions: (i) The kernel of position k(x , y) ∈ C([] × []), and satisfies the following 1/2 k 2 (x , y)dx d y = A, A is a constant,
where x = x(x ¯ 1 , x 2 , x 3), y = y¯ (y1 , y2 , y3). (ii) For all values of t, τ ∈ [0, T ] the functions F(t, τ ) and G(t, τ ) belong to C([0, T ] × [0, T ]) and satisfy |F(t, τ )| < B , |G(t, τ )| < D where B and D are constants. (iii) The function f (x , t) ∈ L 2() × C[0, T ]. (iv) The unknown function φ(x , t) will satisfy H¨older condition with respect to time |φ(x , t1) − φ(x , t2)| ≤ E(x )|t1 − t2 |α ,
(0 < α < 1)
43
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
and Lipschitz condition with respect to position |φ(x 1 , t) − φ(x 2 , t)| ≤ H (t)|x 1 − x 2 |, where E(x ) and H (t) are continuous functions in x and t respectively.
2. Formulation of the problem. Consider the integral equation t (2.1)
t
F(t, τ )k(
−1
0
+
1
x−y )φ(y, τ ) d ydτ + λ
G(t, τ )φ(x , τ ) dτ = [γ (t) − f (x )] = f (x , t),
λ ∈ (0, ∞)
0
(2.2)
k(v) =
1 2
∞
−∞
tanhu iuv e du, u
i=
√ −1
under the condition (2.3)
1 −1
φ(x , t) dx = P(t)
As in [3], p. 32, the kernel of (2.2) can be written in the form ∞ (2.4) k(v) = 1/2 −∞
x−y πv tanhu iuv e du = −ln|tanh |, v = , λ ∈ (0, ∞) u 4 λ
If λ −→ ∞ and (x − y) is very small, so that the condition tanhz z, then we have (2.5)
ln| tanh
πv | = ln|v| − d, 4
d = ln
4λ , π
In this case, the integral equation (2.1) will take the form (2.6)
−
t 0
1
−1
F(t, τ )(ln|x − y| − d)φ(y, τ ) d ydτ +
44
M. A. ABDOU - G. M. ABD AL-KADER
t
+
G(t, τ )φ(x , τ ) dτ = f (x , t),
0
The integral equation (2.6) can be investigated from the contact problem of a rigid surface (G, ν) having an elastic material where G is the displacement magnitude, ν is Poisson’s cofficient. If the stamp of length 2 unit, and its surface is describing by the formula g(x ), is impressed into an elastic layer surface of a strip by a variable force P(t), 0 ≤ t ≤ T < ∞, whose eccentricity of application e(t), that cases rigid displacement γ (t). If the function F(t, τ ) represents the resistance force of material in the domain contact through the time t ∈ [0, T ], while G(t, τ ) is the external force of resistance, which is supplied through the contact domain.
3. Method of solution. To obtain the solution of (2.6) under the condition (2.3) we divide the interval [0, T ], 0 ≤ t ≤ T < ∞ as 0 = t0 ≤ t1 < ... < t N = T , where t = tl , l = 1, 2, ..., N , to get (3.1)
tl
− 0
1
−1
F(tl , τ )(ln|x − y| − d)φ(y, τ ) d ydτ +
tl
+
G(tl , τ )φ(x , τ ) dτ = f (x , tl ).
0
Hence, we have (3.2)
1 l (ln|x − y| − d)φ(y, tj ) d y+ − u j F(tl , tj ) −1
j=0
+ O(hlP+1 ) +
l vj G(tl , tj )φ(x , tj ) + O(hlP+1 ) = f (x , tl ), j=0 ˜
(hlP+1 −→ 0, hlP+1 −→ 0, P > 0, P˜ > 0) h j = tj+1 − tj . where hl¯, hl¯ = Max 0≤ j≤l,0≤ j≤l¯h j ; The values of u j , P ,vj and P˜ are depending on the number of drivatives of F(t, τ ) and G(t, τ ) with respect to t . For example, if F(t, τ ) ∈ C 4 [0, T ], then we have P = 4, l 4 in the first term of (3.2), so we get u 0 = 12 h 0 , u 4 = 12 h 4 , u i = h i ,i = 1, 2, 3. While, if G(t, τ ) ∈ C 3 [0, T ], then we have P˜ = 3, l´ and
45
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
v0 = 12 h 0 , v3 = 12 h 3 , v1 = h 1 , v2 = h 2 . More information for the characteristic points and the quadrature cofficients are found in [11], [12]. Using the following notations (3.3)
F(tl , tj ) = Fl, j , φ(y, t j ) = φj (y)
G(tl , tj = G l, j , and f (x , tl ) = fl (x ), l = 0, 1, ...N ; 0 ≤ j ≤ l,
the formula (3.2) can be adapted in the form (3.4)
l
vj G j,l φj (x ) −
j=0
l
u j Fj,l
j=0
1
−1
(ln|x − y| − d)φj (y) d y = fl (x ).
Also, the boundary condition of (2.3) becomes (3.5)
1 −1
φl (x )dx = Pl .
The formula (3.4) represents a linear system of FIE of the second kind or of the first kind according to the relations between the number of derivatives of F(t, τ ), G(t, τ ) with respect to t for all values of τ ∈ [0, T ]. So we must study the following cases: Case (1): If G(t, τ ) has i derivatives, i < l, we have (3.6)
−
l j=i+1
u j Fj,l
1
−1
(ln|x − y| − d)φj (y) d y = fl (x ) −
i
μj φj (x )
j=0
The formula (3.6) represents a linear system of FIE of the first kind, where μi , 0 ≤ i < l are constants and φj (x ), 0 ≤ j ≤ i can be obtained from the following integral equation (3.7)
i j=0
u j G j,l φj (x ) −
i j=0
u j Fj,l
1
−1
(ln|x − y| − d)φj (y) d y = fi (x )
The formula (3.7) represents a linear system of FIE of the second kind. Case (2): If the number of derivatives of F(t, τ ) and G(t, τ ) is equal, the integral equation (3.4) for all values of j , 0 ≤ j ≤ l, represents a FIE of the second kind.
46
M. A. ABDOU - G. M. ABD AL-KADER
Case (3): If F(t, τ ) has i derivatives, i < l, we have the following system i
(3.8)
vj G j,l φj (x ) == fi+1 (x ) −
j=0
i
μ∗j φj (x ) f i (x )
j=0
where μ∗j ,0 ≤ j ≤ l, are constants and ψj (x ), 0 ≤ j ≤ l is the solution of the integral equation (3.9)
i
vj G j,l ψj (x ) −
i
j=0
u j Fj,l
j=0
1
−1
(ln|x − y| − d)ψj (y) d y = fi (x )
4. Fredholm integral equation of the second kind. To obtain the solution of (3.4) for all values of j , when the two functions F(t, τ ) and G(t, τ ) have the same number of derivatives, we write it in the following (4.1)
μl φl (x ) − μ´ l
+
1
−1
(ln|x − y| − d)φl (y) d y = fl (x ) −
l−1
vj G l, j φj (x )
j=0
u j Fj,l
j=0
(μl =
l−1
hl G l,l , 2
1
−1
μ´ l =
(ln|x − y| − d)φj (y) d y
hl Fl,l , 2
G l,l = 0,
Fl,l = 0)
h j = tj+1 − tj where h¯ l , h¯ l´ = Max 0≤ j≤l,0≤ j≤l´h j ; The solution of (4.1) can be obtained by the recurrence relation. For l = 0, we have the integral equation (4.2)
μ0 φ0 (x ) − μ´ 0
1
−1
(ln|x − y| − d)φ0 (y)d y = f0 (x )
Differentiating (4.2) with respect to x , we get (4.3) 1 μ0 1 d f 0 (x ) φ0 (y) dφ0 (x ) − d y = g0 (x ), (μ = , μ´ 0 = 0) , g0 (x ) = μ dx x−y μ´ 0 μ´ 0 dx −1
47
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
The formula (4.3) represents a FIE of the second kind with Cauchy kernel. Using, in (4.3), the substitution y = 2u − 1, x = 2v − 1, we have 1 (u) 2 2 d −λ du = h(u), (λ = , h(u) = g0 (2u − 1)) (4.4) dx μ μ 0 u−v Under the boundary conditions (0) = (1) = 0, Frankel in his work [13], obtained the solution of (4.4) in the form of Chebyshev polynomials. Equation (4.4) has appeared in both combined infrared gaseous radiation and molecular conduction. A numerical method, Toeplitz matrices, is used, in [14], to obtain the solution of (4.2) and the error estimate is calculated. A series form of Legendre polynomial is used, in [15] to obtain the solution of (4.2) and the result is used to obtain the solution of the F-VIE of the second kind with logarithmic kernel with respect to Fredholm integral term and continuous function with respect to Volterra integral term. To obtain the solution of (4.1), we will use a series in the Chebyshev polynomials form. For this, set the function R(x ) which charactrizes the singular behaviour of φl (x ) in the form R(x ) = (1 + x )− 2 +α (1 − x ) 2 +β 1
1
where α, β = −1, 0, 1 such that −1 < − 12 + α < 1, −1 < 12 + β < 1. Let 1 α = 0, β = −1, then R(x ) = (1 − x 2 )− 2 , which is called the weight function of the Chebyshev polynomials Tn (x ), n = 0, 1, 2, .... Introduce new unknown functions G l (x ), 0 ≤ l ≤ N , where φl (x ) will behave like (4.6)
R(x ) = (1 − x 2)− 2 1
φl (x ) = R(x ) G l (x ),
Now, for the numerical solution of (4.1), we express G l (x ) as G l (x ) =
(4.7)
∞
anl Tnl (x )
nl =0
Hence, we have φl (x ) =
(4.8)
∞
a nl
nl =0
Tnl (x ) (1 − x 2 )
which can be truncated to (4.9)
φlM (x )
=
M nl =0
a nl
Tnl (x ) (1 − x 2)
48
M. A. ABDOU - G. M. ABD AL-KADER
So, (4.1) tends to μl
(4.10)
M
Tn (x ) − μ´ l a nl l 2) (1 − x nl =0 M
π a0i (ln2 − d)
T (x) π nMl =1 anl nnl l
l−1
M
Tn (x ) − f nl vj G l, j a nl l (1 − x 2 ) j=0 (1 − x 2) nl =0 nl =0 l−1 π a0j (ln2 − d) nj = 0
M Tnj (x) + u j Fj,l nj ≥ 1 π nj =1 anj nj
=
Tnl (x )
nl = 0 nl ≥ 1
j=0
where f nl =
(4.11a)
2 π
1
−1
fl (x ) Tnl (x ) dx , (1 − x 2)
and 1 f nl = π
(4.11b)
1
−1
nl ≥ 1
fl (x ) dx (1 − x 2)
The previous results of (4.10) is obtained after using the following spectral relation [16] 1 Tn (y) π (ln2 − d) n=0 dy = π (ln|x − y| − d) (4.12) T (x ) n ≥1 (1 − y 2) n n −1 The formula (4.10) leads us to discuss the following cases: Case (i): For nl = 0, l = 1, 2, ..., we have
(4.13)
μl a0l (1 − x 2 )
+ π (ln2 − d)
l−1
− μ´ l π a0l (ln2 − d) =
a0j u j Fj,l −
l−1
j=0
vj G l, j
j=0
f 0l (1 − x 2 )
a 0l (1 − x 2 )
,
0 ≤ l ≤ N.
Integrating (4.13) with respect to x , we get f0l + 2(ln2 − d) (4.14)
a 0l =
l−1
j=0
a0j u j Fj,l −
μl − 2(ln2 − d)μ´ l
l−1
j=0
vj G l, j a0j , 0 ≤ l ≤ N.
49
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
Eq. (4.14) represents the zero level of the eigenvalue of the potential function φl (x ), 0 ≤ l ≤ N , and must satisfy the relations for all values of l μl = 2(ln2 − d) μ´ l
(4.15)
a) when G(t, τ ) and F(t, τ ) have the same order of derivatives, the formula (4.14) becomes (4.16)
a 0l =
f 0l +
l−1 j=0
bj Hj,l a0j
hl G l,l (1 − 2(ln2 − d))
,
d =
1 [1 − 2ln2] 2
where bj Hj,l = 2(ln2 − d)u j Fj,l − vj G l, j . b) If G(t, τ ) = 0 for all values of t, τ ∈ [0, T ], we get (4.17)
a 0l =
f0l + 2(ln2 − d)
l−1 j=0
a0j u j Fj,l
−hl (ln2 − d)Fl,l
,
hl = tl+1 − tl
c) If F(t, τ ) = 1, G(t, τ ) = 0, we have a 0l =
(4.18)
f0l + 2(ln2 − d)
l−1 j=0
Hj
−hl (ln2 − d)
,
d =
1 [1 − 2ln2] 2
Case (ii): For nl = 0, l = 0, the formula (4.13) becomes
(4.19)
μl a00 (1 − x 2 )
− μ´ l π a00 (ln2 − d) =
f 0l (1 − x 2 )
Integrating (4.19) with respect to x , finally, we have a 00 =
(4.20)
f 00 , μ0 − 2(ln2 − d)μ´ 0
Case (iii): For nl ≥ 1, l = 0, 1, ..., N , the formula (4.10) becomes μl
(4.21)
M nl =0
=
M nl =0
a nl
Tnl (x ) (1 −
Tnl (x )
x 2)
− f nl (1 − x 2)
− μ´ l π
M l−1
M nl =1
a nl
Tnl (x ) nl
Tn (x ) vj G l, j anl l (1 − x 2) j=0 nl =0
50
M. A. ABDOU - G. M. ABD AL-KADER
+π
M l−1
u j Fj,l anj
j=0 n j =1
Tnj (x ) nj
Multiplying both sides of (4.21) by the term Tnj (x )dx , and integrating with respect to x from -1 to 1, then using the following [17] Tm (x ) Tn (x ) =
(4.22)
1
−1
Tn (x ) dx =
1 [Tm+n (x ) + Tm−n (x )], 2
2 , 1−n2
n = 0, 2, 4, ... n = 1, 3, 5, ...
0
we get μl am l − 2μ´ l
(4.23)
M Anl ,m l a nl = H m l , nl n =0 l
where (4.24)
Hm l = f m l −
l−1
vj G l, j am l + 2
M l−1
u j Fj,l
j=0 n j =1
j=0
ml , nl = 1, 2, ..., M;
Anl ,m l an j , nl
0≤l ≤N
and (4.25)
Anl ,m l =
1 1−(nl +m l )2
0
+ 1−(nl1−m l )2 nl + ml nl + ml odd
even
Abdou and Bassen, in [18] proved that, for the system (4.23), for l = 0, is bounded and has a unique solution. By following the same way of Abdou and Bassen, in [18], we can prove that the system of (4.23), for all values of 0 ≤ l ≤ N , is bounded, when M −→ ∞. To prove that the system (4.26) has a unique solution for M −→ ∞, we write it in the form am l = L m l + λl
(4.26)
∞
Rnl ,m l anl
nl =1
where L ml =
1 Hm l , μl
λl =
2μ ´l , μl
Rnl ,m l =
1 An ,m nl l l
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
Assume S´ m = λ´ l
∞
|Rnl ,m l | = λl
51
∞ 1 |Anl ,m l | nl
nl =1
n=1
Applying Caudy- Minkowski inequality, we get (4.27)
∞ ∞ 1 1 2 1 2 Sm ≤ λl [ ] [Anl ,m l ] 2 n2 nl =1
n=1
Finally, we get μ´ l ≤ 0.3gμl ,
(4.28)
which represents the condition to have a unique solution, and the values of |am l | must satisfy the inequality |am l | ≤
(4.29)
μl |L m l | μl − 2μ´ l
As special case, when nl = ml , 0 ≤ l ≤ M , using uniqueness condition μ´ l 1 < k to determine k , we get μl |
(4.30) Therefore (4.31)
k = max
nl (4nl2 − 1) μ´ l |< μl 4(nl2 − 1) 4(2n2l −1) , nl (4n2l −1) 1 , ln2−d
n ≥ 1, 0 ≤ l ≤ M n=0
Lemma 4.1. This method is said to be convergent of order r in [-1, 1], if and only if for N sufficiently large, there exist a constant D > 0 independent of N such that (4.32)
(x ) − N (x ) ≤ D N −r
Also, the error term E N can be given by the relation E N = | − N | where N −→ as N −→ ∞.
52
M. A. ABDOU - G. M. ABD AL-KADER
5. Fredholm Integral Equation of the first kind.. According to case (1), in Section 3, when the derivatives of G(t, τ ), will respect to t , for all τ ∈ [0, T ] is less than the derivatives of G(t, τ ) with respect to the same argument, we have the integral equation (5.1)
l
u j Fj,l
1
−1
j=i+1
[ln
1 + d]φj (y) d y = Hl (x ) |x − y|
where Hl = fl (x ) −
i
l ≥ i +1
μi i (x )
j=0
and i (x ) represent the potential function of the integral equation (3.7). The integral equation (5.1) represents a FIE of the first kind with logarithmic kernel, and its solution will be discussed, using potential theory method, under the condition 1 φl (x ) dx = Pl , 0≤l ≤N (5.2) −1
Introduce the logarithmic potential function (5.3)
Ul (x , z) =
l
u j Fj,l
j=i+1
1
−1
[ln
1 (x − y)2 + z2
+ d]φj (y) d y
Eq. (5.3), under the condition (5.2) reduces to the Dirichlet boundary value problem ∂2 ∂2 + , (x , z) ∈ (−1, 1) ∂ x 2 ∂ z2 x ∈ (−1, 1) Ul (x , z)|z=0 = Hl (x ) 1 r = x 2 + z2 Ul (x , z) Pl (ln + d), r 1 Pl (ln + d) −→ finite term, as r −→ ∞ r Ul (x , z) = 0,
(5.4)
=
The solution of the integral equation (5.1) is equivalent to the solution of the Dirichlet problem (5.4). After constructing the functions Ul (x , z), the potential functions φj (x ), i + 1 ≤ j ≤ l, will be determined from the formula (5.5)
φl (x ) = −
1 ∂Ul (x , z) lim sgn z , z−→0 π ∂z
x ∈ (−1, 1)
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
53
Assume the density source functions 1 Wl (x , z) = Ul (x , z) − Pl (ln + d) r So, the boundary value problem (5.4) becomes
(5.6)
(5.7)
(x , z) ∈ (−1, 1) Wl (x , z) = 0, Wl (x , z)|z=0 = Hl (x ) − Pl (ln r − d) Wl (x , z) −→ 0, as r −→ ∞
Consequently, Eq. (5.5) takes the form 1 ∂ Wl (x , z) − Pl δ(x )]}, x ∈ (−1, 1) φl (x ) = − sgn z−→0 {z[ π ∂z where δ(x ) is the Dirac-delta function. The boundary value problem (5.7) can be constructed by the method of conformal mapping, see [9], that transforms a given complicated region into a simpler one. For this aim, we use the mapping function, (5.8)
(5.9)
v=
1 1 ω(ζ ) = (ζ + ζ −1 ), v = x + iy, 2 2 √ i = −1, v = ρeiθ , 0 ≤ θ ≤ 2π,
which maps the region in (x , y) plane into the region outside the unit circle γ , ) does not vanish or becomes infinite outside γ . The mapping such that dω(ζ dζ function (5.9) maps the upper and the lower half-plane (x , z) ∈ (−1, 1) into the lower and the upper of the semi-circle ρ = 1, respectively. Moreover the point v −→ ∞ will be mapped into the point ζ = 0. Using the mapping (5.9), the function of Eq. (5.6) takes the following form (5.10)
Nl (ρ, θ ) = Ml (ρ, θ ) − Pl (2ρ + d)
where Nl (ρ, θ ) = Wl (x , y) = Wl ( 12 (ρ + ρ1 ) cos θ, 12 (ρ − ρ1 ) sin θ ) and Ml (ρ, θ ) = Ul ( 12 (ρ + ρ1 ) cos θ, 12 (ρ − ρ1 ) sin θ ). In view of Eq. (5.10), the boundary value problems of (5.7) will be transformed to
(5.11)
1 ∂ 2 Nl 1 ∂ Nl ∂ 2 Nl + + , (ρ ≤ 1, ∂ρ 2 ρ ∂ρ ρ 2 ∂θ 2 Nl (0, θ ) = 0, Nl (1, θ )| = H˜ l (θ ) − Pl (ln2 + d) H˜ l (θ ) = Hl (x ) = Hl (ρ cos θ ),
ρ = 1.
−π ≤ θ ≤ π )
54
M. A. ABDOU - G. M. ABD AL-KADER
Consequently, after using the chain rule, Eq. (5.8) is transformed to φl (cos θ ) = (π | sin θ |)−1 [Pl +
(5.12)
∂ Nl ]ρ=1 ∂ρ
To solve the Dirichlet problem of (5.11), we use the Fourier series method, see [7] ∞ anl ρ nl cos nl θ , (0 ≤ l ≤ N ) Nl (ρ, θ ) = nl =0
where (5.13)
1 a 0l = 2π
π
−π
1 Nl (θ ) dθ, nl = 0 a0l = 2π
π
−π
Nl (θ ) cos nl θ dθ, nl ≥ 0.
substitute (5.13) in (5.11) after setting the given function Hl (x ) = νnl Tnl (x ), where νnl are known constants, Tnl (x ) are the Chebyshev polynomials of the first type of order n; n = 0, 1, 2, ..., then use the result in (5.12), we get nl νnl cos nl θ(π sin θ )−1 , nl = 1, 2, ... (0 ≤ l ≤ N ) (5.14) φl (cos θ ) = nl = 0 νnl Pl (π sin θ )−1 and −1
Pl = [π (ln2 + d)]
π
−π
Hl (cos θ ) dθ,
nl = 0
. Using (5.14) in (5.1), where, θ = cos−1 x , we have the following spectral relationships (5.15)
l j=i+1
1
u j Fj,l −1
1 1 − y2
[ln
1 + d]Tnl (y) d y = |x − y| =
π (ln2 + d), nl = 0 π T (x ) nl nl
which represents the spectral relationships for the Fredholm integral equations of the first kind with logarithmic kernl. Numerical Results. To consider the behaviour of the solution function φl (x ) which is represented numerically by φlM (x ) in Eqn (4.9) with Eqns (4.14), (4.21) and (4.24). The general behaviour can be described in figure 1. Little difference may be occurred for changing the parameter values.
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
55
FIGURE 1. Conclusion. From the above results and discussions, the following may be concluded:
56
M. A. ABDOU - G. M. ABD AL-KADER
(1) The contact problem of a rigid surface having an elastic material, when a stamp of length 2 unit is impressed into an elastic layer surface of a strip by a variable force P(t), 0 ≤ t ≤ T < ∞, whose eccentricity of application e(t), represents a Fredholm-Volterra integral equation of mixed type. (2) The numerical method used gives us a system of Fredholm integral equation, which it’s solution can be obtained using the recurrence relations. (3) The kind of the system of Fredholm integral equation depends on the relation between the number of derivatives of F(t, τ ), which represents the resistance force of material in the domain contact through the time t ∈ [0, T ] and G(t, τ ), which represents the external force of resistance supplied through the material of contact domain. (4) When the value of G(t, τ ) −→ 0, t ∈ [0, T ], T < ∞, i.e., their is no external force of resistance, the integral equation of mixed type takes a form of a system of Fredholm integral equation of the first kind. (5) The displacement problems of antiplane deformation of an infinite rigid strip with width 2 unit, putting on an elastic layer of thickness h is considered as a special case of this work, when G(t, τ ) = 0,
F(t, τ ) = 1,
t = 1,
F(x , t) = H.
where H represents the displacement magnitude and φ(x , 1) = ψ(x ) is the unknown displacement stress. (6) The problems of infinite rigid strip, with width 2 unit immpresed in a viscous liquid layer of thickness h, when the √ strip has a velocity resulting from the impulsive force v − v0 e−iωt , i = −1, where v0 is the constant velocity, ω is the angular velocity resulting from rotating the strip about z-axis, are considered as special case of our work, when G(t, τ ) = 0, F(t, τ ) =contant. In the discussion (5) and (6), we note that, when h −→ ∞, this means the depth of the liquid or the thickness of the elastic material becomes an infinite . (7) The three kinds of the displacement problems, in the theory of elasticity and mixed contact problem which discussed in [1] are considered special case of this work. (8) The potential function method has a large application in mathematical physics problems, where the integral equation becomes equivalent to a boundary value problem, which can be soved easily. Also the potential function method enables us to discuss the spectral relation of the integral equation.
MIXED TYPE OF FREDHOLM-VOLTERRA. . .
57
(9) The conformal mapping, in the theory of elasticity, or in the applied sciences transforms a given complicated region into a simpler one. (10) The mixed integral equation with Carleman kernel can be established from this work, using the following relation ln|x − y| = h(x , y)|x − y|−ν ,
0 ≤ ν < 1,
where h(x , y) = |x − y|ν ln|x − y| is a smooth function. (11) The Fredholm integral equation of the first or second kind with logarithmic and Carleman kernels, are considered, now, as special cases of this work. (12) Many spectral relations, that has a large applications in mathematical physics, were established from this work.
REFERENCES [1]
V.M. Aleksandrov - E.V. Kovalenko, Problems in Mechanics Media with Mixed Boundary Conditions, Nauka Moscow 1986.
[2]
G.Yo. Popov, Contact Problems for a Linearly Deformable Base, Odessa Kiev, (1982).
[3]
S.M. Mkhitarian - M.A. Abdou, On different method of solution of the integral equation for the planer contact problem of elasticity, Dokl. Acad. Nauk. Arm. SSR, 89 (2) (1990), pp. 59–74.
[4]
S.M. Mkhitarian - M.A. Abdou, On various method for the solution of the Carleman integral equation, Dakl. Acad. Nauk. Arm. SSR, 89 (3) (1990), pp. 125–129.
[5]
N.k. Arytiunian, Plane contact problem of the theory of creef, J. Appl. Math. Mech., 23 (1959), pp. 901–923.
[6]
C.D. Green, Integral Equation Methods, Nelsson New York (1969).
[7]
M.A. Abdou - S.A. Hassan, Boundary value of a contact problem, PU.M.A., 5 (3) (1994), pp. 311–316.
[8]
M.A. Abdou, Spectral relationships for integral operators in contact problem of impressing stamps, Appl. Math. Comput., 118 (2001), pp. 95–111.
[9]
M.A. Abdou, Integral equation and contact problem for a system of impressing stamps, Appl. Math. Comput., 106 (1999), pp. 141–148.
[10]
M.A. Abdou, Integral equation with Macdonald kernel and its application to a system of contact problem, Appl. Math. Comput., 118 (2001), pp. 83–94.
[11]
L.M. Delves - J.L. Mohamed, Computational Methods for Integral Equations, Cambridge (1985).
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[12]
M.A. Golberg (Ed.), Numerical Solution of Integral Equations, New York (1990).
[13]
J.I. Frankel, A Galerkin solution to a regularized Cauchy singular integrodifferential equation, Quart. Appl. Math., LIII (2) (1995), pp. 245–258. M.A. Abdou - S.A. Mahmoud - M.A. Darwish, A numerical method for solving the Fredholm integral equation of the second kind, Korean J. Comput. Appl. Math., 5 N. 2 (1998), pp. 251–258.
[14]
[15] [16]
[17] [18]
[19]
M.A. Abdou, Fredholm-Volterra integral equation with singular kernel, J. Appl. Math. Comput., 137 (2003), pp. 231–243. M.A. Abdou - N.Y. Ezzeldin, Kreins method with certain singular kernel for solving the integral equation of the first kind, Periodica Math. Hung., 28 (2) (1994), pp. 143–149. I.S. Gradshteyn - I.M. Ryzhik, Table of Integrals Series and Products, fourth edition England Academic press (1980). M.A. Abdou - M. Bassem, A numerical method for solving Fredholm integral equation of the second kind with Cauchy kernel, Accepted in condian J. of Physics, (2002). A.H. England, Complex Variable Method in Elasticity, London, NewYork (1971).
M. A. Abdou, Department of Mathematics Faculty of Education Alexandria University, (EGYPT) e-mail: abdella 77 @yahoo.com G. M. Abd Al-Kader, Department of Mathematics, Faculty of Science, Al-Azhar University Nasr City 11884, Cairo (EGYPT) e-mail:
[email protected]
