Journal of Applied Mathematics

Volume 2012 (2012), Article ID 365904, 14 pages

doi:10.1155/2012/365904

Research Article

Numerical Solution of Weakly Singular Integrodifferential Equations on Closed Smooth Contour in Lebesgue Spaces

Feras M. Al Faqih1,2

¹ Department of Mathematics and Statistics, King Faisal University, Saudi Arabia

²Department of Mathematics, Al-Hussein Bin Talal University, P. O. Box 20 Ma'an, Jordan

Received 18 June 2012; Accepted 3 September 2012

Academic Editor: Yongkun Li

Copyright © 2012 Feras M. Al Faqih. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces.

1. Introduction

Singular integral equations (SIE) and singular integro-differential equations with Cauchy kernels (SIDE) and systems of such equations model many problems in elasticity theory, aerodynamics, mechanics, thermoelasticity and queuing analysis (see [1–6] and the literature cited therein). The general theory of SIE and SIDE has been widely investigated over the last decades [7–11]. It is known that the exact solution for SIDE can be found only in some particular cases. That is why there is a necessity to elaborate approximation methods for solving SIDE.

In the past, there was a lot of research in literature devoted to an approximate solution of SIE and SIDE by collocation and mechanical quadrature methods. The equations are defined on the unit circle centered at the origin or on the real axis, see for example [12–15]. However, the case when the contour of integration is an arbitrary smooth closed curve has not been studied enough.

It should be noted that conformal mapping from the arbitrary smooth closed contour to the unit circle does not solve the problem. Moreover, it makes it more difficult. In the present paper we consider the collocation and mechanical quadrature methods for the approximate solution of weakly SIDE. We use the Fejér points as collocation knots. In Section 2 we introduce the main definitions and notations. We present the numerical schemes of collocation and mechanical quadrature methods in Section 3. In Section 4 we formulate the auxiliary results. We use these results to prove the convergence theorems in Section 5.

We note that the convergence of the collocation method, reduction method and mechanical quadrature method for SIDE and systems of such equations in generalized Hölder spaces has been obtained in [16–18]. The equations are given on an arbitrary smooth closed contour (not weakly SIDE).

2. The Main Definitions and Notations

Let be an arbitrary smooth closed contour bounding a simply connected region of the complex plane and let , , where is the complex plane. Let be a function, mapping conformably the outside of unit circle on the domain so that

We assume that the function has the second derivative, satisfying on the Hölder condition with some parameter ; the class of such contours is denoted by [19, 20].

Let be the space of complex functions with norm

where is the length of .

Let be the Lagrange interpolating polynomial

3. Numerical Schemes of the Collocation Method and Mechanical Quadrature Method

In the complex space we consider the weakly singular integro-differential equation (SIDE):

where , , , and are known functions; is an unknown function; ( is a positive integer). Using the Riesz operators , , (where is the identity operator, and is the singular operator (with Cauchy kernel)), we rewrite (3.1) in the following form convenient for consideration:

where , , .

We search for a solution of (3.1) in the class of functions, satisfying the condition

In order to reduce the numerical schemes of collocation method we introduce a new integro-differential equation from the initial one. The weakly singular kernels are substituted by continuous ones. We obtain the new approximate equation

where

is an arbitrary positive number, is characteristic part of weakly SIDE. Equation (3.1) with the conditions (3.3) we denote as problem “(3.1)–(3.3)”. We search for the approximate solution of problem (3.1)–(3.3) in polynomial form

where are unknown complex numbers. We note that the function , constructed by formula, obviously satisfies the condition (3.3). Let be residual of SIDE. The collocation method consists in setting it equal to zero at some chosen points ,   on and thus obtaining a linear algebraic system for unknowns which is determined by solving it:

Using the (3.7) we obtain a system of linear algebraic equations (SLAE) for collocation method:

where , are distinct points on and , . We approximate the integrals in SLAE (3.8) by quadrature formula:

where , at and , for , and is the Lagrange interpolation operator defined by formula (2.3).

Thus, we obtain the following SLAE from (3.8):

4. Auxiliary Results

We formulate one result from [21], establishing the equivalence (in sense of solvability) of problem (3.1)–(3.3) and SIE. We use this result for proving Theorems 5.3 and 5.4. The functions and can be represented by integrals of Cauchy type with the same density :

Using the integral representation (4.1) we reduce the problem (3.1)–(3.3) to the equivalent (in sense of solvability) of SIE

for unknown where

where , are Hölder functions. An obvious form for these functions are given in [21]. By virtue of the properties of the functions , , , , , the function is a continuous function in both variables.

Lemma 4.1. The SIE (4.2) and problem (3.1)–(3.3) are equivalent in the sense of solvability. That is, for each solution of SIE (4.2) there is a solution of problem (3.1)–(3.3), determined by formulae

where (, , , and are the binomial coefficients). On the other hand, for each solution of the problem (3.1)–(3.3) there is a solution

to the SIE (4.2). Furthermore, for linearly independent solutions of (4.2), there are corresponding linearly-independent solutions of the problem (3.1)–(3.3) from (4.6) and vice versa.

In formulas (4.6) by we understand the branch which vanishes as and by the branch which vanishes as .

4.1. Estimates for Weakly Singular Integral Operators

Lemma 4.2. Let , and , . Then the function , satisfies the inequality

By , ,…, we denote the constants.

The proof can be found in [22].

Lemma 4.3. Let the assumptions of Lemma 4.2 be satisfied; then , where ,  .

The proof of this lemma can be found in [22].

5. Convergence Theorems

Define as

The norm in is determined by the equality

We denote by the image of the space with respect to the map equipped with the norm of . We formulate Lemmas 5.1 and 5.2 from [23]. We use these lemmas to prove the convergence theorems.

Lemma 5.1. The differential operator , is continuously invertible and its inverse operator is determined by the equality

From Lemma 5.1   Lemma 5.2 follows.

Lemma 5.2. The operator is invertible and

The proofs of Lemmas 5.1 and 5.2 can be found in [23].

The convergence of collocation method and mechanical quadrature method are given in the following theorems.

Theorem 5.3. Let the following conditions be satisfied:

(1), ;(2)the functions and belong to the space , ;(3), ;(4)the index of the function is equal to zero; (5) , , function ;(6)the operator is linear and invertible; (7)the points form a system of Fejér knots on [24, 25]:

Then, the SLAE (3.8) of collocation method has the unique solution , for numbers that are large enough and for numbers small enough. The satisfies the following inequality:

The approximate solutions , constructed by formula (3.6), converge when in the norm of space to the exact solution of the problem (3.1)–(3.3) in sense of

and the following estimation for convergence holds:

The and are modules of continuity, where

Proof. Using the conditions of Theorem 5.3 we have that the operator is invertible. We estimate the perturbation of depending on . Using Lemma 4.3 and the relation we obtain

Let us show that the operator is invertible for sufficiently small values such that the inequality (5.6) is valid. Using the representation and (5.10), we obtain from Banach theorem that the inverse operator exists. The following inequalities hold:

The SLAE (3.8) of the collocation method for SIDE (3.1) for is equivalent to the operator equation

where , is defined by formula (3.5). Using the integral presentation (4.1), (5.12) is equivalent to the operator equation

where operator is defined in (4.2), substituting by and by (where is calculated by formula (3.5)). Equation (5.13) represents the collocation method for SIE

We should show that if is large enough and satisfies the relation (5.6) the operator is invertible. The operator acts from the subspace (the norm as in ) to the subspace

(the norm as in .)

Using formulas (4.1) the and can be represented by Cauchy-type integrals with the same density :

Using the formulas

and relations (4.1) we obtain from (5.16)

We obtain from previous relation that .

The collocation method for SIE was considered in [19, 20, 26], where sufficient conditions for solvability and convergence of this method were obtained. From (5.16), Lemma 4.1, and we conclude that if function is the solution of (5.13) then the function is the discrete solution for the system and vice versa. We can determine the function from relations (4.6):

From the conditions , , and of Theorem 5.3 and Lemmas 5.1 and 5.2, the invertibility of operator follows. From Banach theorem and Lemma 4.3 for small numbers ( satisfies the relation (5.6)) we have that the operator is invertible. We should show that for (5.13) all conditions of the Theorem 1 are satisfied from [19, 20]. Theorem 1 [20] gives the convergence of the collocation method for SIE in spaces . From condition 3 of Theorem 1 [20] and from (4.3) we obtain the condition 3 of Theorem 5.3. From the equality

we conclude that the index of the function is equal to zero, which coincides with condition of Theorem 5.3. Other conditions of Theorem 5.3 coincide with conditions of Theorem 1 [20]. Conditions – in Theorem 5.3 provide the validity of all conditions of Theorem 1 [20]. Therefore, beginning with numbers (5.13) is uniquely solvable for numbers small enough where satisfies the relation (5.6). The approximate solutions of (5.13) converge to the exact solution of (4.2) in the norm of the space as . Therefore (5.12) and the SLAE (3.10) have the unique solutions for . From Theorem 1 [20] the following estimation holds:

where and are modulus of continuity. From (4.1) and (5.19) we obtain

Therefore we have

We proceed to get an error estimate

Using the inequality

From (5.21), (5.24), and (5.11), and from the inequality

we obtain the relation (5.8). Thus Theorem 5.3 is proved.

Theorem 5.4. Let all conditions of Theorem 5.3 be satisfied. Then the SLAE (3.10) has a unique solution , for numbers large enough and for numbers small enough ( satisfies the relation (5.6)). The approximate solutions converge when and in the norm to the exact solution of the problem (3.1)–(3.3) and the following estimation for the convergence is true:

Proof. It is easy to verify that SLAE (3.10) is equivalent to the operational equation

which after the application of integral representation (5.19) is equivalent (in the same sense of solvability) to the operator equation

where the functions , , and are determined above. The equation (5.28) represents an equation of the mechanical quadrature method for (5.14). It is easy to verify (as in the proof of Theorem 5.3), that the conditions of Theorem 5.4 provide the validity of all conditions of Theorem 2 from [19, 26] (for the mechanical quadrature method). It follows that (5.29) is uniquely solvable for and small enough. Moreover, the approximate solutions of this equation converge to the exact solution of SIE (4.2) in the norm as and the following estimation is true:

The function can be expressed via the function by formula (5.19). Using the definition of the norm in the space , and the relations (4.6), (5.30), and equality (5.26) we obtain (5.27). Theorem 5.4 is proved.

6. Conclusion

In this paper, we have proposed the numerical schemes of the collocation method and mechanical quadrature method for solving of weakly SIDE. The equations are defined on an arbitrary smooth closed contour. The convergence of these methods was proved in Lebesgue spaces.

Acknowledgments

The author would like to thank the Deanship of Scientific Research at King Faisal Universit, Saudi Arabia, for supporting this work by Grant no. 120037.

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