Journal of Applied Mathematics

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

http://dx.doi.org/10.1155/2012/365904

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

^{1}Department of Mathematics and Statistics, King Faisal University, Saudi Arabia^{2}Department 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.

#### References

- J. W. Cohen and O. J. Boxma,
*Boundary Value Problems in Queueing System Analysis*, vol. 79, North-Holland, Amsterdam, The Netherlands, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. I. Kalandiya,
*Mathematical Methods of Two-Dimensional Elasticity*, Mir Publishers, Moscow, Russia, 1975. View at Zentralblatt MATH - A. Linkov,
*Boundary Integral Equations in Elasticity Theory*, Kluwer Academic, Dordrecht, The Netherlands, 2002. View at Zentralblatt MATH - N. I. Muskhelishvili,
*Some Basic Problems of the Mathematical Theory of Elasticity. Fundamental Equations, Plane Theory of Elasticity, Torsion and Bending*, Noordhoff, Groningen, The Netherlands, 1953. View at Zentralblatt MATH - D. Brockman and L. Hufnagel, “Fronnt propagation in reaction—supper diffusion dynamics: taming lavy flights with fluctuations,”
*Physical Review Letters*, vol. 98, pp. 178311–178314, 2007. View at Google Scholar - E. G. Ladopoulos,
*Singular Integral Equations: Linear and Non-Linear Theory and Its Applications in Science and Engineering*, Springer, New York, NY, USA, 2000. - V. V. Ivanov,
*The Theory of Approximate Methods and Their Application to the Numerical Solution of Singular Integral Equations*, Noordhoff, Leyden, The Netherlands, 1976. - F. D. Gakhov,
*Boundary Value Problems*, Pergamon, Oxford, UK, Addison-Wesley, Reading, Mass, USA, 1966. - N. I. Muskhelishvili,
*Singular Integral Equations: Boundary Problems of Functions Theory and Their Applications to Mathematical Physics*, Noordhoff, Leyden, The Netherlands, 1977. - N. P. Vekua,
*Systems of Singular Integral Equations*, Noordhoff, Groningen, The Netherlands, 1967, Translated from the Russian by A. G. Gibbs and G. M. Simmons. - I. Gohberg and N. Krupnik,
*Introduction to the Theory of One-Dimensional Singular Integral Operators*, Stiintsa, Kishinev, Moldova, 1973, German translation: Birkhause, Basel, Germany, 1979. - S. Prössdorf and B. Silbermann,
*Numerical Analysis for Integral and Related Operator Equations*, Akademie, Berlin, Germany, Birkhauser, Basel, Switzerland, 1991. - S. G. Mikhlin and S. Prössdorf,
*Singular Integral Operators*, vol. 68, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar - S. Prössdorf,
*Some Classes of Singular Equations*, vol. 17, Elsevier, North-Holland, The Netherlands, 1978. - B. Gabdulalhaev, “The polynomial approximations of solution of singular integral and integro-differential equations by Dzyadik,”
*Izvestia Visshih Ucebhih Zavedenii Mathematics*, vol. 6, no. 193, pp. 51–62, 1978 (Russian). View at Google Scholar - V. A. Zolotarevskiĭ, Z. Li, and I. Caraus, “Approximate solution of singular integrodifferential equations by the method of reduction over Faber-Laurent polynomials,”
*Differential Equation*, vol. 40, no. 12, pp. 1764–1769, 2004, translated from*Differentsial'nye Uravneniya*, vol. 40, no.12, pp. 1682–1686, 2004. View at Google Scholar - I. Caraus, “The numerical solution for systems of singular integro-differential equations by Faber-Laurent polynomials,” in
*Proceedings of the 3rd international conference on Numerical Analysis and its Applications (NAA '04)*, vol. 3401 of*Lecture notes in Computer Science*, pp. 219–223, Springer, New York, NY, USA, 2005. - I. Caraus and F. M. Al Faqih, “Approximate solution of singular integro-differential equations in generalized Holder spaces,”
*Numerical Algorithms*, vol. 45, pp. 205–215, 2007. View at Publisher · View at Google Scholar - V. Zolotarevski,
*Finite-Dimensional Methods For Solving of Singular Integral Equations on the Closed Contours of Integration*, Stiinta, Chisinau, Moldova, 1991. - V. A. Zolotarevskiĭ, “Approximate solution of systems of singular integral equations on some smooth contours in ${L}_{p}$ spaces,”
*Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika*, no. 2, pp. 79–82, 1989. View at Google Scholar - Y. Krikunov, “The general boundary Riemann problem and linear singular integrodifferential equation,”
*The Scientific Notes of the Kazani University*, vol. 116, no. 4, pp. 3–29, 1956 (Russian). View at Google Scholar - V. N. Seĭchuk, “Estimates for weakly singular integral operators defined on closed integration contours and their applications to the approximate solution of singular integral equations,”
*Differential Equations*, vol. 41, no. 9, pp. 1311–1322, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Saks,
*Boundary-Value Problems For Elliptic Systems of Differential Equations*, University of Novosibirsk, Novosibirsk, Russia, 1975. - V. I. Smirnov and N. A. Lebedev,
*Functions of a Complex Variable: Constructive Theory*, IIlife, London, UK, 1968, Translated by Scripta Technica. - P. Novati, “A polynomial method based on Fejèr points for the computation of functions of unsymmetric matrices,”
*Applied Numerical Mathematics*, vol. 44, no. 1-2, pp. 201–224, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. A. Zolotarevski, “Direct methods for solving singular integral quations on closed smooth contour in spaces ${L}_{p}$,”
*Revue d'Analyse Numérique et de Théorie de l'Approximation*, vol. 25, no. 1-2, pp. 257–265, 1996. View at Google Scholar