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On the Approximate Solution of Singular Integral Equations with Hilbert Kernel



M.H. Saleh and S.M. Amer
 
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ABSTRACT

The mechanical quadrature method has been applied to a certain class of nonlinear singular integral equations with Hilbert kernel in generalized Holder spaces. The rate of convergence of approximate solution has been determined. Two examples have been introduced to show that the application of mechanical quadrature method to a class of, nonlinear and linear, singular integral equations gives accurate results. These results are very acceptable compared to the exact solution. The obtained results of the mechanical quadrature method are better than the obtained results of the Toeplitz matrix method and the product Nystrom method applied to the same classes to obtain an approximate solution. The error at the interior points has been calculated.

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M.H. Saleh and S.M. Amer, 2007. On the Approximate Solution of Singular Integral Equations with Hilbert Kernel. Journal of Applied Sciences, 7: 2957-2966.

DOI: 10.3923/jas.2007.2957.2966

URL: https://scialert.net/abstract/?doi=jas.2007.2957.2966

INTRODUCTION

The theory of singular integral equations has been developed significant importance during the last years, it is arise in many problems of mathematical physics, such as the theory of elasticity, hydrodynamics, biological problems, population genetics and others. Also, nonlinear singular integral equations with Hilbert and Cauchy kernel and its related Rimann-Hilbert problems have been studied by Pogorzelski (1966), Guseinov and Mukhtarov (1980), Wolfersdorf (1985) and Wegert (1992).

Existence results and approximate solutions for certain classes of nonlinear singular integral equations are studied by Amer and Nagdy (1999 and 2000), Amer and Dardery (2005), Amer (1996, 2001 and 2005), Jinyuan (2000) and Junghanns and Weber (1993). The theory of approximation methods and its application for the solution of linear and nonlinear singular integral equations has been developed by Guseinov and Mukhtarov (1980), Kravchenko and Akilov (1982), Ladopoulos and Zisis (1996) and Mikhlin and Prossdorf (1980).

It is well known that the nonlinear singular integral equations are the much-complicated forms of the nonlinear integral equations. The mechanical quadrature method is one of the basic tools to investigate the approximate solutions of many classes of nonlinear and linear equations involving integral operator. In this research we applied the mechanical quadrature method to a certain class of Nonlinear Singular Integral Equation (NSIE) with Hilbert kernel in generalized Hölder spaces. The method has been applied to a nonlinear and a linear Singular Integral Equation (SIE) with known exact solution and the error has been calculated. The obtained results of the mechanical quadrature method of SIE are compared with the obtained results of the Toeplitz matrix method and the product Nystrom method that have been applied (Abdou et al., 2002), to obtain the approximate solution of the same problem.

FORMULATION OF THE PROBLEM

This study is devoted to investigate the approximate solution of the following nonlinear singular integral equation:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(1)

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(2)

in generalized Hölder spaces Hφ,m and Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelis a numerical parameter, under the following assumptions:

Assumption 1: Suppose that the function g(t, τ, u, (τ)) is defined on the domain

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

that has partial derivatives up to (m-1) - order and satisfy the following Hölder-Lipschitz condition for arbitrary

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(3)

where, φ, φ* are non-decreasing functions belong to the class Φ, i + j + k = β and β = 0, 1, 2, ..., m-1 and (η)β is a constant depends on β.

Assumption 2: Suppose that the function F (t, u(t), v(t)) is defined on the domain:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

that has partial derivatives up to (m-1) order and satisfy the following condition for arbitrary

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(4)

for p + q + r = v, v = 0, 1, 2, ..., m-1, where, φ1 εΦ and ξ (v) is a constant depends on v. Equation 1 with Cauchy kernel has been studied by the collocation method (Amer, 1966), the special cases of Eq. 1 have been found (Ladopoulos and Zisis, 1996; Saleh and Amer, 1987).

Some basic definitions and auxiliary results: In this section we introduce some definitions and results which will be used in the sequel.

Definition 1: (Guseinov and Mukhatarov, 1980; Mikhlin and Prossdorf, 1986).

We denote by Φ to the class of all continuous almost increasing functions φ defined on (0, π] such that Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
We denote by Φm to the class of all functions φ ε Φ such thatImage for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel implies Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel, where, c(m) is a constant depends on m.
We denote by c to the space of 2π-periodic continuous functions with the norm

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

The generalized Hölder space Hφ,m is defined as

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where, Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelis the modulus of continuity of order m of the function u and

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where, Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelis a constant depends on m.

For u ε Hφ,m we define

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

and

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

as a subspace of Hφ,m

Definition 2: (Mosaev and Salaev, 1980; Saleh and Amer, 1987).

Let the generalized Hölder space Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel,be the space of 2N-dimensional vectors z = (z0, z1, ..., z2N-1) with the norm:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

and

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

are the modulus of continuity of order m of the vector z with respect to the two sets X = {0, 2,..., 2N-2}, Y = {1, 3, ..., 2N-1} and p ε X,

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

For Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelwe define:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

as a subspace of Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

We denote the norm in the space Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelby

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Theorem 1 (Mosaev and Salaev, 1980; Saleh, 1984): Let φ ε HΦm, then the operator

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Transforms Hφ,m(M) intoImage for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel , where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where, e1(m), e2(m) and Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelare constants depend on m.

Lemma 1 (Saleh and Amer, 1987): Let the condition (3) is satisfied and then Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel.

Lemma 2 (Saleh and Amer, 1990): Let the condition (4) is satisfied andImage for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

The approximate solution in the space Hφ,m Theorem 2: Let the function F(t, u(t), v(t)) satisfy the condition (4) and the function g(t, τ, u(τ)) satisfy the condition (3), then for |λ|<λ0, (λ0 sufficiently small), the Eq. 1 has a unique solution in Hφ,m(M). The solution is uniformly convergent and can be obtained by the method of successive approximations.

Proof: Let u, v ε Hφ,m(M). Then by Lemmas 1, 2 and Theorem 1, the operator

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(5)

transforms the space Hφ,m(M) into the space Hφ,m (|λ|R). Therefore if |λ|R≤M, the operator P transforms Hφ,m(M) into itself. Using M. Riesz’s Theorem (Kravchenko and Akilov, 1982; Saleh and Amer, 1987).

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(6)

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

and from the conditions (3), (4), we obtain

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(7)

Choosing

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

and

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

then the operator P is a contraction mapping. From the completeness of Hφ,m(M) in Lp, p>1, the Eq. 1 has a unique solution in the subspace Hφ,m(M) and this solution can be found by the method of successive approximations.

The approximate solution in the space H(N)φ, m: By the mechanical quadrature formula (Saleh, 1984), the integral

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(8)

takes the following form:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(9)

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

the formula (9) at node points ti takes the form:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(10)

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Applying the quadrature formula (10) to the Eq. 1 at the node points, we obtain:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where, RN(g, tj) is the remainder term, Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel. If we put u(tj) = zj and the RN(g, tj) is negligible, we obtain the following system of nonlinear algebraic equations:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(11)

Lemma 3 (Guseinov and Mukhatarov, 1980; Amer, 2001): If the function g (t, τ, u) and its derivative gt (t, τ, u) satisfy the condition (3), then the function:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

satisfies the following condition

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(12)

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Theorem 4.1: Let the function F(t, τ, u) satisfy the condition (4) and the function g(t, τ, u) satisfy the condition (3), then the system (11) has a unique solution in the space Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelfor arbitrary N≥3 and this solution can be found by the method of successive approximations.

Proof: From definition 2, we have

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Putting

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

since the space Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelof vectors of bounded norms is a closed subspace of Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kerneland the function g(t, τ, u) satisfies the condition of Lemmas 1, 2 and Theorem 2 hence Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel.

Taking

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

let

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

and

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

(Saleh and Amer, 1987)

where, θ(m) is a constant depends on m. Thus we have:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Now, let

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Hence,

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(13)

since

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(14)

where, q(p) is a constant depends on p.

Hence from condition (3), Lemma 3 and from (Saleh and Amer, 1987) we obtain:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(15)

substituting from inequality (15) into (13), we get:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(16)

From boundedness of the operator Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kerneland by using the principle of contraction mapping at

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(17)

the system (11) has a unique solution in Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelfor arbitrary N≥3, hence the theorem is proved.

The rate of convergence of the approximate solution: From inequality (17), the Eq. 1 has a unique solution Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kerneland the system (11) has a unique solutionImage for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

The relation

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(18)

at t = tj is called the approximate solution of the Eq. 1, . The norm of the difference of the vectors z* and u* where,

u* = (u (t0), u(t1), ... , u(t2N-1)) in Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelcan be determined as follows:

Applying the quadrature formula (10) to Eq. 1 at node points tj, we obtain

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(19)

putting Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernelin (16) and using the inequality (17), we get:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

consequently, we have

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(20)

To evaluate Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel, we state the following two lemmas:

Lemma 4 (Saleh, 1984): Let Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel. Then for arbitrary natural number , we getImage for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Lemma 5 (Saleh, 1984):

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Applying formula (9) on the Eq. 1 and from Eq. 18, we obtain:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Using Lemma 5, we have:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(21)

from Lemma 4 and inequality (20), we get:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

since

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(22)

therefore,

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

taking h = Nα, p¯1 < α < 1, then

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(23)

consequently, from (21), (22) and (23), we obtain

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Hence,

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Now we present the following examples.

Example 1: Consider the integral equation:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(24)

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

It is easy to check that u* (t) = t is the exact solution of Eq. 24 at λ = 1. Applying the quadrature formula (10) to Eq. 24 at node points, we obtain the following system of nonlinear algebraic equations:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Table 1 displays the exact solution, the approximate solution and error between them for the Eq. 24 by using the mechanical quadrature method with N = 20, λ = 1 and at initial valuesImage for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel .

Now, we apply the mechanical quadrature method to a class of LSIE.

Example 2: Consider the integral equation:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(25)

under the condition

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
(26)

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

It is clear that u*(t) = sin t is the exact solution of Eq. 25 at λ = 1. Applying the quadrature formula (10) to Eq. 25 under the condition (26) at the node points tj, we obtain the following system of linear algebraic equations:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

Table 1: The results for the Eq. 24 by using the mechanical quadrature method
Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
u* = The exact solution, z* = The approximate solution, E = The error

where:

Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel

The condition u(±π) = 0 reduces the node points tj to 2N-1 points.

Table 2 displays the exact solution, the approximate solution and the error between them for the Eq. 25 under the condition (26) by using the mechanical quadrature method with N = 20, λ = 1.

CONCLUSIONS

Table 1 and 2 display that the mechanical quadrature method gives accurate results with respect to NSIE and LSIE, these results are very acceptable compared to the exact solution.

Table 2: The results of Eq. 25 by using the mechanical quadrature method
Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
u* = The exact solution, z* = The approximate solution, E = The error

Table 3: The results of Eq. 25 by using the Toeplitz matrix method and the product Nystrom method
Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kernel
u = The exact solution, = The approximate solution using product Nystrom method and R(N) = The error, = The approximate solution using Toeplitz matrix method and R(T) = The error

The Toeplitz matrix method and the product Nystrom method have been applied to the same Eq. 25 under the condition (26) by Abdou et al. (2002), Table 3 displays the values of exact solution u(t) = sin t, approximate solution Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kerneland the error R(T) at the interior points by using the Toeplitz matrix method with N = 20, λ = 1. Also it shows the approximate solution Image for - On the Approximate Solution of Singular Integral Equations with Hilbert Kerneland the error R(T) at the interior points by using the product Nystrom method with N = 20, λ = 1.
Its found that, the obtained results of the mechanical quadrature method are better than the obtained results of the Toeplitz matrix method and the product Nystrom method that have been applied in Abdou et al. (2002) to obtain the approximate solution of the same problem (25) under the same condition (26).
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1:  Abdou, M.A., I.M. Khamis and A.S. Ismal, 2002. Toeplitz matrix and product nystrom methods for solving the singular integral equation. Estratto Matematiche, LVII: 21-37.

2:  Amer, S.M., 1996. On The approximate solution of nonlinear singular integral equations with positive index. Int. J. Math. Math. Sci., 19: 389-396.
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3:  Amer, S.M. and A.S. Nagdy, 1999. On the solvability of nonlinear singular iintegral and integro-differential equations with cauchy type. Proc. Math. Phys. Soc. Egypt, 74: 115-128.

4:  Junghanns, P. and U. Weber, 1993. On the solvability of nonlinear singular integral equations. FB Math., 12: 683-698.

5:  Amer, S.M. and A.S. Nagdy, 2000. On the modified Newton's approximation method for the solution of nonlinear singular integral equations. Hokkaido Math. J., 29: 59-72.

6:  Amer, S.M., 2001. On solution of nonlinear singular integral equations with shift in generalized holder space. Chaos Solitons Fractals, 12: 1323-1334.
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7:  Amer, S.M., 2005. Existence results for a class of nonlinear singular integral equations with shift. Il Nuovo Cimento B, 120: 313-333.
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9:  Guseinov, A.L. and K.S. Mukhtarov, 1980. Introduction to the theory of nonlinear singular integral equations. Nauka Moscow (In Russian).

10:  Jinyuan, D.U., 2000. The collocation methods and singular integral equations with cauchy kernels. Acta. Math. Sci., 20: 289-302.

11:  Kravchenko, L.V. and G.P. Akilov, 1982. Functional Analysis. Pergamon Press, Oxford.

12:  Ladopoulos, E.G. and V.A. Zisis, 1996. Nonlinear singular integral approximations in banach spaces. Nonlinear Anal. Theory Methods Applic., 26: 1293-1299.

13:  Mikhlin, S.G. and S. Prossdorf, 1986. Singular Integral Operator. Academy-Verleg, Berlin.

14:  Mosaev, B.I. and V.V. Salaev, 1980. On the convergence of quadratic process for singular integrals with hilbert kernel. Coll Timely Problems in Theory of Functions. AGU, pp: 186-194.

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17:  Saleh, M.H., 1984. Basis of quadrature method for nonlinear singular integral equations with hilbert kernel in the space Hφ,k. AZ. NIINTI, No. 279, pp: 1-40.

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20:  Wolfersdorf, L.V., 1985. On the theory of nonlinear singular integral equations of cauchy type. Meth. Applied Sci., 7: 493-517.

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