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

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 Γ be an arbitrary smooth closed contour bounding a simply connected region 𝐹⁺ of the complex plane and let 𝑡=0∈𝐹⁺, 𝐹⁻=𝐶⧵{𝐹⁺∪Γ}, where 𝐶 is the complex plane. Let 𝑧=𝜓(𝑤) be a function, mapping conformably the outside of unit circle Γ₀={|𝑤|=1} on the domain 𝐹⁻ so that 𝜓(∞)=∞,𝜓^(′)(∞)=1. (2.1) We assume that the function 𝑧=𝜓(𝑤) has the second derivative, satisfying on Γ₀ the Hölder condition with some parameter 𝜇(0<𝜇<1); the class of such contours is denoted by 𝐶(2;𝜇) [19, 20].

Let 𝐿_𝑝(Γ)(1<𝑝<∞) be the space of complex functions with norm 𝑔_𝑝=1𝑙_Γ𝑔^𝑝|𝑑𝜏|^1/𝑝, (2.2) where 𝑙 is the length of Γ.

Let 𝑈_𝑛 be the Lagrange interpolating polynomial 𝑈_𝑛𝑔(𝑡)=^2𝑛_𝑠=0𝑔𝑡_𝑠⋅𝑙_𝑠(𝑡),𝑙_𝑗(𝑡)=^2𝑛_{𝑘=0,𝑘≠𝑗}𝑡−𝑡_𝑘𝑡_𝑗−𝑡_𝑘𝑡_𝑗𝑡^𝑛≡^𝑛_𝑘=−𝑛Λ_𝑘^(𝑗)𝑡^𝑘,𝑡∈Γ,𝑗=0,…,2𝑛. (2.3)(2.4)

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

In the complex space 𝐿_𝑝(Γ)(1<𝑝<∞) we consider the weakly singular integro-differential equation (SIDE): (𝑀𝑥≡)^𝜈_𝑟=0𝐴_𝑟(𝑡)𝑥^(𝑟)(𝑡)+𝐵_𝑟(𝑡)1𝜋𝑖_Γ𝑥^(𝑟)(𝜏)+1𝜏−𝑡𝑑𝜏2𝜋𝑖_Γ𝐾_𝑟(𝑡,𝜏)|𝑡−𝜏|^𝛾⋅𝑥^(𝑟)(𝜏)𝑑𝜏=𝑓(𝑡),𝑡∈Γ, (3.1) where 0<𝛾<1, 𝐴_𝑟(𝑡), 𝐵_𝑟(𝑡), 𝐾_𝑟(𝑡,𝜏)(𝑟=0,…,𝜈) and 𝑓(𝑡) are known functions; 𝑥⁽⁰⁾(𝑡)=𝑥(𝑡) is an unknown function; 𝑥^(𝑟)(𝑡)=((𝑑^𝑟𝑥(𝑡))/𝑑𝑡^𝑟)(𝑟=1,…,𝜈) (𝜈 is a positive integer). Using the Riesz operators 𝑃=1/2(𝐼+𝑆), 𝑄=𝐼−𝑃, (where 𝐼 is the identity operator, and 𝑆 is the singular operator (with Cauchy kernel)), we rewrite (3.1) in the following form convenient for consideration: (𝑀𝑥≡)^𝜈_𝑟=0𝐴_𝑟(𝑡)𝑃𝑥^(𝑟)(𝑡)+𝐵_𝑟(𝑡)𝑄𝑥^(𝑟)(𝑡)+12𝜋𝑖_Γ𝐾_𝑟(𝑡,𝜏)|𝑡−𝜏|^𝛾⋅𝑥^(𝑟)(𝜏)𝑑𝜏=𝑓(𝑡),𝑡∈Γ, (3.2) where 𝐴_𝑟(𝑡)=𝐴_𝑟(𝑡)+𝐵_𝑟(𝑡), 𝐵_𝑟(𝑡)=𝐴_𝑟(𝑡)−𝐵_𝑟(𝑡), 𝑟=0,…,𝜈.

We search for a solution of (3.1) in the class of functions, satisfying the condition12𝜋𝑖_Γ𝑥(𝜏)𝜏^−𝑘−1𝑑𝜏=0,𝑘=0,…,𝜈−1. (3.3) 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 𝑀_𝜌(𝑥)≡𝑀₀𝑥(𝑡)+12𝜋𝑖^𝜈_𝑟=0Γ𝐾_𝑟,𝜌(𝑡,𝜏)𝑥^(𝑟)(𝜏)𝑑𝜏=𝑓(𝑡),𝑡∈Γ, (3.4) where 𝐾_𝑟,𝜌()=𝐾𝑡,𝜏^𝑟(𝑡,𝜏)|𝑡−𝜏|^𝛾||𝐾,when𝑡−𝜏≥𝜌,_𝑟(𝑡,𝜏)𝜌^𝛾,when|𝑡−𝜏|<𝜌. (3.5)𝜌 is an arbitrary positive number, 𝑀₀ 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 𝑥_𝑛,𝜌(𝑡)=^𝑛_𝑘=0𝜉^(𝑛)_𝑘,𝜌𝑡^𝑘+𝜈+⁻¹_𝑘=−𝑛𝜉^(𝑛)_𝑘,𝜌𝑡^𝑘,𝑡∈Γ, (3.6) where 𝜉^(𝑛)_𝑘,𝜌=𝜉_𝑘,𝜌(𝑘=−𝑛,…,𝑛) are unknown complex numbers. We note that the function 𝑥_𝑛,𝜌(𝑡), constructed by formula, obviously satisfies the condition (3.3). Let 𝑅_𝑛(𝑡)=(𝑀_𝜌𝑥_𝑛)(𝑡)−𝑓(𝑡) be residual of SIDE. The collocation method consists in setting it equal to zero at some chosen points 𝑡_𝑗,  𝑗=0,…,2𝑛 on Γ and thus obtaining a linear algebraic system for unknowns 𝜉_𝑘,𝜌 which is determined by solving it: 𝑅_𝑛𝑡_𝑗=0,𝑗=0,…,2𝑛. (3.7) Using the (3.7) we obtain a system of linear algebraic equations (SLAE) for collocation method: ^𝜈_𝑟=0𝐴_𝑟𝑡_𝑗^𝑛_𝑘=0(𝑘+𝜈)!()!𝑡𝑘+𝜈−𝑟^{𝑗𝑘+𝜈−𝑟}𝜉_𝑘,𝜌+𝐵_𝑟𝑡_𝑗^𝑛_𝑘=1(−1)^𝑟(𝑘+𝑟−1)!()!𝑡𝑘−1^{𝑗−𝑘−𝑟}×𝜉_−𝑘,𝜌+1⋅2𝜋𝑖^𝑛_𝑘=0(𝑘+𝜈)!(𝑘+𝜈−𝑟)!_Γ𝐾_𝑟,𝜌𝑡_𝑗,𝜏𝜏^{𝑘+𝜈−𝑟}𝑑𝜏⋅𝜉_𝑘,𝜌+^𝑛_𝑘=1(−1)^𝑟(𝑘+𝑟−1)!()!⋅1𝑘−12𝜋𝑖_Γ𝐾_𝑟,𝜌𝑡_𝑗,𝜏𝜏^{−𝑘−𝑟}𝑑𝜏⋅𝜉_−𝑘,𝜌=𝑓𝑡_𝑗,𝑗=0,…,2𝑛, (3.8) where 𝑡_𝑗, (𝑗=0,…,2𝑛) are distinct points on Γ and 𝐴_𝑟(𝑡)=𝐴_𝑟(𝑡)+𝐵_𝑟(𝑡), 𝐵_𝑟(𝑡)=𝐴_𝑟(𝑡)−𝐵_𝑟(𝑡). We approximate the integrals in SLAE (3.8) by quadrature formula: 12𝜋𝑖_Γ𝑔(𝜏)𝜏^𝑙+𝑘1𝑑𝜏≅2𝜋𝑖_Γ𝑈_𝑛𝜏^𝑙+1⋅𝑔(𝜏)𝜏^𝑘−1𝑑𝜏, (3.9) where 𝑘=0,…,𝑛, at 𝑙=0,1,2,… and 𝑘=−1,…,−𝑛, for 𝑙=−1,−2,…, and 𝑈_𝑛 is the Lagrange interpolation operator defined by formula (2.3).

Thus, we obtain the following SLAE from (3.8): ^𝜈_𝑟=0𝐴_𝑟𝑡_𝑗^𝑛_𝑘=0(𝑘+𝜈)!()!𝑡𝑘+𝜈−𝑟^{𝑗𝑘+𝜈−𝑟}𝜉_𝑘,𝜌+𝐵_𝑟𝑡_𝑗^𝑛_𝑘=1(−1)^𝑟(𝑘+𝑟−1)!()!𝑡𝑘−1^{𝑗−𝑘−𝑟}×𝜉_−𝑘,𝜌+^𝑛_𝑘=0(𝑘+𝜈)!(𝑘+𝜈−𝑟)!^2𝑛_𝑠=0𝐾_𝑟,𝜌𝑡_𝑗,𝑡_𝑠𝑡_𝑠^{1+𝑘−𝑟}Λ^(𝑠)_−𝑘𝜉_𝑘,𝜌+^𝑛_𝑘=1(−1)^𝑟(𝑘+𝑟−1)!(𝑘−1)!^2𝑛_𝑠=0𝐾_𝑟,𝜌𝑡_𝑗,𝑡_𝑠𝑡_𝑠^{−𝑘−𝑟}Λ_𝑘^(𝑠)𝜉_−𝑘,𝜌=𝑓𝑡_𝑗,𝑗=0,…,2𝑛. (3.10)

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 𝑑^𝜈(𝑃𝑥)(𝑡)/𝑑𝑡^𝜈 and 𝑑^𝜈(𝑄𝑥)(𝑡)/𝑑𝑡^𝜈 can be represented by integrals of Cauchy type with the same density 𝑣(𝑡): 𝑑^𝜈(𝑃𝑥)(𝑡)𝑑𝑡^𝜈=12𝜋𝑖_Γ𝑣(𝜏)𝜏−𝑡𝑑𝜏,𝑡∈𝐹⁺,𝑑^𝜈(𝑄𝑥)(𝑡)𝑑𝑡^𝜈=𝑡^−𝜈2𝜋𝑖_Γ𝑣(𝜏)𝜏−𝑡𝑑𝜏,𝑡∈𝐹⁻. (4.1) Using the integral representation (4.1) we reduce the problem (3.1)–(3.3) to the equivalent (in sense of solvability) of SIE ()𝐶(𝑡)𝑣(𝑡)+𝐷(𝑡)Υ𝑣≡𝜋𝑖_Γ𝑣(𝜏)+1𝜏−𝑡𝑑𝜏2𝜋𝑖_Γℎ(𝑡,𝜏)|𝜏−𝑡|^𝛾𝑣(𝜏)𝑑𝜏=𝑓(𝑡),𝑡∈Γ, (4.2) for unknown 𝑣(𝑡) where 𝐶(𝑡)=12𝐴_𝜈(𝑡)+𝑡^−𝜈𝐵_𝜈(𝑡(𝑡)=1),𝐷2𝐴_𝜈(𝑡)−𝑡^−𝜈𝐵_𝜈(𝑡,ℎ()=1)𝑡,𝜏2𝐾_𝜈(𝑡,𝜏)+𝐾_𝜈(𝑡,𝜏)𝜏^−𝑛−12𝜋𝑖_Γ𝐾_𝜈𝑡,𝑡₁−𝐾_𝜈𝑡,𝑡₁𝑡₁^−𝑛𝑑𝑡₁𝑡₁+−𝜏_{𝜈−1𝑗=0}𝐴_𝑗(𝑡)𝑀_𝑗(𝑡,𝜏)+_Γ𝐾_𝑗𝑡,𝑡₁𝑀_𝑗𝑡₁,𝜏𝑑𝑡₁−^𝜈−1_𝑗=0𝐵_𝑗(𝑡)𝑁_𝑗(𝑡,𝜏)+_Γ𝐾_𝑗𝑡,𝑡₁𝑁_𝑗𝑡₁,𝜏𝑑𝑡₁, (4.3)(4.4) where 𝑀_𝑗(𝑡,𝜏), 𝑁_𝑗(𝑡,𝜏)𝑗=0,…,𝜈 are Hölder functions. An obvious form for these functions are given in [21]. By virtue of the properties of the functions 𝑀_𝑗(𝑡,𝜏), 𝑁_𝑗(𝑡,𝜏), 𝐾_𝑗(𝑡,𝜏), 𝐴_𝑗(𝑡), 𝐵_𝑗(𝑡), 𝑗=0,…,𝜈 the function ℎ(𝑡,𝜏) 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 𝑣(𝑡) of SIE (4.2) there is a solution of problem (3.1)–(3.3), determined by formulae (𝑡)=()𝑃𝑥)(−1^𝜈2𝜋𝑖(𝜈−1)!_Γ𝑣(𝜏)(𝜏−𝑡)^𝜈−1𝑡log1−𝜏+^𝜈−1_𝑘=1𝛼_𝑘𝜏^{𝜈−𝑘−1}𝑡^𝑘(𝑡)=()𝑑𝜏,𝑄𝑥)(−1^𝜈2𝜋𝑖(𝜈−1)!_Γ𝑣(𝜏)𝜏^−𝜈(𝜏−𝑡)^𝜈−1𝜏log1−𝑡+^𝜈−2_𝑘=1𝛽_𝑘𝜏^{𝜈−𝑘−1}𝑡^𝑘𝑑𝜏, (4.5)(4.6) where (𝛼_𝑘=∑^𝑘−1_𝑗=0((−1)^𝑗𝐶^𝑗_𝜈−1/(𝑘−𝑗)), 𝑘=1,…,𝜈−1, 𝛽_𝑘=∑^𝜈−1_𝑗=𝑘+1((−1)^𝑗𝐶^𝑗_𝜈−1/(𝑗−𝑘)), 𝑘=1,…,𝜈−2 and 𝐶^𝑗_𝜈−1 are the binomial coefficients). On the other hand, for each solution 𝑥(𝑡) of the problem (3.1)–(3.3) there is a solution 𝑣(𝑡)𝑣(𝑡)=𝑑^𝜈(𝑃𝑥)(𝑡)𝑑𝑡^𝜈+𝑡^𝜈𝑑^𝜈(𝑄𝑥)(𝑡)𝑑𝑡^𝜈, (4.7) 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 log(1−𝑡/𝜏) we understand the branch which vanishes as 𝑡=0 and by log(1−𝜏/𝑡) the branch which vanishes as 𝑡=∞.

4.1. Estimates for Weakly Singular Integral Operators

Lemma 4.2. Let ℎ(𝑡,𝜏)∈𝐶(Γ×Γ), and 𝜓(𝑡)∈𝐿_𝑝(Γ), 1<𝑝<∞. Then the function 𝐻(𝑡)=(1/2𝜋𝑖)∫_Γ(ℎ(𝑡,𝜏)/|𝜏−𝑡|^𝛾)𝜓(𝜏)𝑑𝜏, satisfies the inequality ‖𝐻‖_𝑝≤𝑑₁𝜓_𝑝,1𝑝+1𝑞=1,‖(⋅)‖_𝑝=1𝑙_Γ|(⋅)(𝜏)|^𝑝𝑑𝜏^1/𝑝. (4.8) By 𝑑₁, 𝑑₂,…, we denote the constants.

The proof can be found in [22].

Lemma 4.3. Let the assumptions of Lemma 4.2 be satisfied; then ||𝜒_𝜌||_𝑝≤𝑑₂𝜌^{(1−𝛾)/𝑞}||𝜓||_𝑝, where 𝜒_𝜌=(1/2𝜋𝑖)∫_Γ[(ℎ(𝑡,𝜏)/|𝜏−𝑡|^𝛾)−ℎ_𝜌(𝑡,𝜏)]𝜓(𝜏)𝑑𝜏,  1/𝑝+1/𝑞=1.

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

5. Convergence Theorems

Define 𝑊^∘_𝑝^(𝜈) as 𝑊^∘_𝑝^(𝜈)=𝑔∈𝐿_𝑝(Γ)∶𝑔^(𝜈)∈𝐿_𝑝(Γ),12𝜋𝑖_Γ𝑔(𝜏)𝜏^−𝑘−1𝑑𝜏=0,𝑘=0,…,𝜈−1. (5.1) The norm in 𝑊^∘_𝑝^(𝜈) is determined by the equality 𝑔_𝑝,𝜈=𝑔^(𝜈)_𝐿𝑝. (5.2) We denote by 𝐿_𝑝,𝜈 the image of the space 𝐿_𝑝 with respect to the map 𝑃+𝑡^−𝜈𝑄 equipped with the norm of 𝐿_𝑝. 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 𝐷^𝜈∶𝑊^∘_𝑝^(𝜈)→𝐿_𝑝,𝜈, (𝐷^𝜈𝑔)(𝑡)=𝑔^(𝜈)(𝑡) is continuously invertible and its inverse operator 𝐷^−𝜈∶𝐿_𝑝,𝜈→𝑊^∘_𝑝^(𝜈) is determined by the equality 𝐷^−𝜈𝑔(𝑡)=𝑁⁺𝑔(𝑡)+𝑁⁻𝑔(𝑡),𝑁⁺𝑔𝑡)=()(−1^𝜈2𝜋𝑖(𝜈−1)!_Γ𝑃𝑔(𝜏)(𝜏−𝑡)^𝜈−1𝑡log1−𝜏𝑁𝑑𝜏,₋𝑔(𝑡)=(−1)^𝜈−12𝜋𝑖(𝜈−1)!_Γ𝑄𝑔(𝜏)(𝜏−𝑡)^𝜈−1𝜏log1−𝑡𝑑𝜏. (5.3)

From Lemma 5.1   Lemma 5.2 follows.

Lemma 5.2. The operator 𝐵∶𝑊^∘_𝑝^(𝜈)→𝐿_𝑝,𝐵=(𝑃+𝑡^𝜈𝑄)𝐷^𝜈 is invertible and 𝐵⁻¹=𝐷^−𝜈𝑃+𝑡^−𝜈𝑄. (5.4)

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)Γ∈𝐶(2,𝜇), 0<𝜇<1;(2)the functions 𝐴_𝑟(𝑡) and 𝐵_𝑟(𝑡) belong to the space 𝐻_𝛼(Γ), 0<𝛼<1;(3)𝐴_𝜈(𝑡)𝐵_𝜈(𝑡)≠0, 𝑡∈Γ;(4)the index of the function 𝑡^𝜈𝐵_𝜈⁻¹(𝑡)𝐴_𝜈(𝑡) is equal to zero; (5)𝐾_𝑟(𝑡,𝜏)(𝑟=0,…,𝜈)∈𝐻_𝛽(Γ×Γ), 0<𝛽≤1, function 𝑓(𝑡)∈𝐶(Γ);(6)the operator 𝑀∶𝑊^∘_𝑝^(𝜈)→𝐿_𝑝(Γ) is linear and invertible; (7)the points 𝑡_𝑗(𝑗=0,…2𝑛) form a system of Fejér knots on Γ [24, 25]: 𝑡_𝑗=𝜓exp2𝜋𝑖2𝑛+1𝑗−𝑛,𝑗=0,…,2𝑛,𝑖²=−1. (5.5)Then, the SLAE (3.8) of collocation method has the unique solution 𝜉_𝑘(𝑘=−𝑛,…,𝑛), for numbers 𝑛≥𝑛₁ that are large enough and for numbers 𝜌 small enough. The 𝜌 satisfies the following inequality: 𝜀_𝜌=𝑑₃𝜌^{(1−𝛾)/𝑞}𝑀⁻¹_𝑝<𝑞₈<1. (5.6) The approximate solutions 𝑥_𝑛,𝜌(𝑡), constructed by formula (3.6), converge when 𝑛→∞ in the norm of space 𝑊^∘_𝑝^(𝜈) to the exact solution 𝑥(𝑡) of the problem (3.1)–(3.3) in sense of lim_𝜌→0lim_𝑛→∞𝑥−𝑥_{𝑛,𝜌𝑝,𝜈}=0, (5.7) and the following estimation for convergence holds: 𝑥−𝑥_{𝑛,𝜌𝑝,𝜈}=𝑂𝜌^{(1−𝛾)/𝑞}1+𝑂𝑛^𝛼𝜔1+𝑂𝑓;𝑛+𝑂𝜔^𝑡ℎ_𝜌;1𝑛^def=𝛿_𝑛,1𝑝+1𝑞=1. (5.8)

The 𝜔(𝑓;1/𝑛) and 𝜔^𝑡(ℎ;1/𝑛) are modules of continuity, where 𝜔1𝑓;𝑛=sup_{|𝑡′−𝑡′′|≤1/𝑛}𝑓𝑡^′−𝑓𝑡^′′,𝜔^𝑡1ℎ;𝑛=sup_{|𝑡′−𝑡′′|≤1/𝑛}ℎ𝑡^′;𝜏−ℎ𝑡^′′;𝜏,𝑡^′,𝑡^′′∈Γ. (5.9)

Proof. Using the conditions of Theorem 5.3 we have that the operator 𝑀∶𝑊^𝑜_𝑝,𝜈→𝐿_𝑝(Γ) is invertible. We estimate the perturbation of 𝑀 depending on 𝜌. Using Lemma 4.3 and the relation 𝑀_𝜌=𝑀₀+𝐾_𝜌 we obtain 𝑀−𝑀_𝜌=𝑂𝜌^{(1−𝛾)/𝑞}. (5.10) Let us show that the operator 𝑀_𝜌 is invertible for sufficiently small values 𝜌 such that the inequality (5.6) is valid. Using the representation 𝑀_𝜌=𝑀[𝐼−𝑀⁻¹(𝑀−𝑀_𝜌)] and (5.10), we obtain from Banach theorem that the inverse operator 𝑀_𝜌⁻¹=[𝐼−𝑀⁻¹(𝑀−𝑀_𝜌)]⁻¹𝑀⁻¹ exists. The following inequalities hold: 𝑀_𝜌⁻¹≤𝑀⁻¹,𝑀1−𝑞⁻¹−𝑀_𝜌⁻¹≤𝑑₁₁𝜌^{(1−𝛾)/𝑞}𝑀⁻¹. (5.11) The SLAE (3.8) of the collocation method for SIDE (3.1) for 𝛾∈(0;1) is equivalent to the operator equation 𝑈_𝑛𝑀_𝜌𝑈_𝑛𝑥_𝑛,𝜌≡𝑈_𝑛𝑀₀𝑈_𝑛𝑥_𝑛,𝜌+𝑈_𝑛^𝜈_𝑟=012𝜋𝑖_Γ𝐾_𝑟,𝜌(𝑡,𝜏)𝑥^(𝑟)_𝑛,𝜌(𝜏)𝑑𝜏=𝑈_𝑛𝑓, (5.12) where 𝐾_𝑟,𝜌(𝑡,𝜏), (𝑟=0,…,𝜈) is defined by formula (3.5). Using the integral presentation (4.1), (5.12) is equivalent to the operator equation 𝑈_𝑛Υ_𝜌𝑈_𝑛𝑣_𝑛,𝜌=𝑈_𝑛𝑓, (5.13) where operator Υ_𝜌 is defined in (4.2), substituting Υ by Υ_𝜌 and (ℎ(𝑡,𝜏)/|𝜏−𝑡|^𝛾) by ℎ_𝜌(𝑡,𝜏) (where ℎ_𝜌(𝑡,𝜏) is calculated by formula (3.5)). Equation (5.13) represents the collocation method for SIE Υ_𝜌𝑣_𝜌=𝑓,𝑣_𝜌(𝑡)∈𝐿_𝑝(Γ). (5.14) We should show that if 𝑛(≥𝑛₁) is large enough and 𝜌 satisfies the relation (5.6) the operator 𝑈_𝑛𝑀_𝜌𝑈_𝑛 is invertible. The operator acts from the subspace 𝑋^∘_𝑛={𝑡^𝜈∑^𝑛_𝑘=0𝜉_𝑘,𝜌𝑡^𝑘+∑⁻¹_𝑘=−𝑛𝜉_𝑘,𝜌𝑡^𝑘} (the norm as in 𝑊^∘_𝑝^(𝜈)) to the subspace 𝑋_𝑛=^𝑛_𝑘=−𝑛𝑟_𝑘𝑡^𝑘,𝑡∈Γ. (5.15) (the norm as in 𝐿_𝑝(Γ).)
Using formulas (4.1) the 𝑑^𝜈(𝑃𝑥_𝑛,𝜌)(𝑡)/𝑑𝑡^𝜈 and 𝑑^𝜈(𝑄𝑥_𝑛,𝜌)(𝑡)/𝑑𝑡^𝜈 can be represented by Cauchy-type integrals with the same density 𝑣_𝑛,𝜌(𝑡): 𝑑^𝜈𝑃𝑥_𝑛,𝜌(𝑡)𝑑𝑡^𝜈=12𝜋𝑖_Γ𝑣_𝑛,𝜌(𝜏)𝜏−𝑡𝑑𝜏,𝑡∈𝐹⁺,𝑑^𝜈𝑄𝑥_𝑛,𝜌(𝑡)𝑑𝑡^𝜈=𝑡^−𝜈2𝜋𝑖_Γ𝑣_𝑛,𝜌(𝜏)𝜏−𝑡𝑑𝜏,𝑡∈𝐹⁻. (5.16) Using the formulas (𝑃𝑥)^(𝑟)(𝑡)=𝑃𝑥^(𝑟)(𝑡),(𝑄𝑥)^(𝑟)(𝑡)=𝑄𝑥^(𝑟)(𝑡), (5.17) and relations (4.1) we obtain from (5.16) 𝑣_𝑛,𝜌(𝑡)=^𝑛_𝑘=0(𝑘+𝜈)!𝑡𝑘!^𝑘𝜉_𝑘,𝜌+(−1)^𝜈𝑛_𝑘=1(𝑘+𝜈−1)!()!𝑡𝑘−1^−𝑘𝜉_−𝑘,𝜌. (5.18) We obtain from previous relation that 𝑣_𝑛,𝜌(𝑡)∈𝑋_𝑛,𝑡∈Γ.
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 𝑣_𝑛,𝜌(𝑡)∈𝑋_𝑛 we conclude that if function 𝑣_𝑛,𝜌(𝑡) is the solution of (5.13) then the function 𝑥_𝑛,𝜌(𝑡) is the discrete solution for the system 𝑈_𝑛𝑀𝑈_𝑛𝑥_𝑛,𝜌=𝑈_𝑛𝑓 and vice versa. We can determine the function 𝑣_𝑛,𝜌(𝑡) from relations (4.6): 𝑃𝑥_𝑛,𝜌(𝑡)=(−1)^𝜈2𝜋𝑖(𝜈−1)!_Γ𝑣_𝑛,𝜌(𝜏)(𝜏−𝑡)^𝜈−1𝑡log1−𝜏+^𝜈−1_𝑘=1𝛼_𝑘𝜏^{𝜈−𝑘−1}𝑡^𝑘𝑑𝜏,𝑄𝑥_𝑛,𝜌(𝑡)=(−1)^𝜈2𝜋𝑖(𝜈−1)!_Γ𝑣_𝑛,𝜌(𝜏)𝜏^−𝜈(𝜏−𝑡)^𝜈−1𝜏log1−𝑡+^𝜈−1_𝑘=1𝛽_𝑘𝜏^{𝜈−𝑘−1}𝑡^𝑘𝑑𝜏. (5.19) From the conditions (3), (4), and (6) of Theorem 5.3 and Lemmas 5.1 and 5.2, the invertibility of operator Υ∶𝐿_𝑝(Γ)→𝐿_𝑝(Γ) follows. From Banach theorem and Lemma 4.3 for small numbers 𝜌 (𝜌 satisfies the relation (5.6)) we have that the operator Υ_𝜌∶𝐿_𝑝(Γ)→𝐿_𝑝(Γ) 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 𝐿_𝑝(Γ). From condition 3 of Theorem 1 [20] and from (4.3) we obtain the condition 3 of Theorem 5.3. From the equality [𝐶(𝑡)−𝐷(𝑡)]⁻¹[𝐶(𝑡)+𝐷(𝑡)]=𝑡^𝜈𝐵_𝜈⁻¹𝐴_𝑞(𝑡), (5.20) we conclude that the index of the function [𝐶(𝑡)−𝐷(𝑡)]⁻¹[𝐶(𝑡)+𝐷(𝑡)] is equal to zero, which coincides with condition (4) of Theorem 5.3. Other conditions of Theorem 5.3 coincide with conditions of Theorem 1 [20]. Conditions (1)–(6) in Theorem 5.3 provide the validity of all conditions of Theorem 1 [20]. Therefore, beginning with numbers 𝑛≥𝑛₁ (5.13) is uniquely solvable for numbers 𝜌 small enough where 𝜌 satisfies the relation (5.6). The approximate solutions 𝑣_𝑛,𝜌(𝑡) of (5.13) converge to the exact solution of (4.2) in the norm of the space 𝐿_𝑝(Γ) as 𝑛→∞. Therefore (5.12) and the SLAE (3.10) have the unique solutions for (𝑛≥𝑛₁). From Theorem 1 [20] the following estimation holds: 𝑣_𝜌−𝑣_{𝑛,𝜌𝑝}1≤𝑂𝑛^𝛼𝜔1+𝑂𝑓;𝑛+𝑂𝜔^𝑡1ℎ;𝑛, (5.21) where 𝑂(𝜔^𝑡(ℎ;1/𝑛) and 𝑂(𝜔(𝑓;1/𝑛) are modulus of continuity. From (4.1) and (5.19) we obtain 𝑃𝑥_𝜌^(𝜈)(𝑡)=𝑃𝑣_𝜌(𝑡),𝑄𝑥_𝜌^(𝜈)(𝑡)=𝑡^−𝜈𝑄𝑣_𝜌(𝑡). (5.22) Therefore we have 𝑃𝑥_𝑛,𝜌^(𝜈)(𝑡)=𝑃𝑣_𝑛,𝜌(𝑡),𝑄𝑥_𝑛,𝜌^(𝜈)(𝑡)=𝑡^−𝜈𝑄𝑣_𝑛,𝜌(𝑡). (5.23) We proceed to get an error estimate 𝑥_𝜌−𝑥_{𝑛,𝜌𝑝,𝜈}=𝑥_𝜌^(𝜈)−𝑥^(𝜈)_{𝑛,𝜌[𝐿𝑝]}≤𝑃(𝑣_𝜌−𝑣_𝑛,𝜌)_[𝐿𝑝]+𝑡^−𝜈𝑄(𝑣_𝜌−𝑣_𝑛,𝜌)_[𝐿𝑝]≤‖𝑃‖⋅𝑣_𝜌−𝑣_{𝑛,𝜌[𝐿𝑝]}+𝑡^−𝜈‖𝑄‖⋅𝑣_𝜌−𝑣_{𝑛,𝜌[𝐿𝑝]}≤‖𝑃‖+𝑡^−𝜈‖𝑄‖𝑣_𝜌−𝑣_{𝑛,𝜌[𝐿𝑝]}. (5.24) Using the inequality 𝑡^−𝜈_𝐿𝑝=1𝑙_Γ𝑡^−𝜈𝑝𝑑𝑡^1/𝑝=1𝑙_Γ𝑡^−𝜈𝑝𝑑𝑡^1/𝑝≤1𝑙1min_𝑡∈Γ|𝑡|^𝑝𝜈𝑙^1/𝑝=1min_𝑡∈Γ|𝑡|^{𝑝𝜈1/𝑝}=𝑐₁. (5.25) From (5.21), (5.24), and (5.11), and from the inequality 𝑥−𝑥_{𝑛,𝜌𝑝,𝜈}≤𝑀⁻¹𝑓−𝑀_𝜌⁻¹_𝑝,𝜈+𝑥_𝜌−𝑥_{𝑛,𝜌𝑝,𝜈} (5.26) 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 𝜉_𝑘,𝜌, 𝑘=−𝑛,…,𝑛 for numbers 𝑛≥𝑛₂(≥𝑛₁) large enough and for numbers 𝜌 small enough (𝜌 satisfies the relation (5.6)). The approximate solutions 𝑥_𝑛,𝜌(𝑡) converge when 𝑛→∞ and 𝜌→0 in the norm 𝑊^∘_𝑝^(𝜈) to the exact solution 𝑥(𝑡) of the problem (3.1)–(3.3) and the following estimation for the convergence is true: 𝑥−𝑥_{𝑛,𝜌𝑝,𝜈}=𝛿_𝑛+𝑂𝜔^𝜏1ℎ;𝑛. (5.27)

Proof. It is easy to verify that SLAE (3.10) is equivalent to the operational equation 𝑈_𝑛^𝜈_𝑟=0𝐴_𝑟(𝑡)𝑃𝑥^(𝑟)_𝑛,𝜌(𝑡)+𝐵_𝑟(𝑡)𝑄𝑥^(𝑟)_𝑛,𝜌(𝑡)+12𝜋𝑖_Γ1𝜏𝑈_𝑛^(𝜏)𝜏^{𝜈+1−𝑟}𝐾_𝜌(𝑡,𝜏)𝑃𝑥^(𝑟)_𝑛,𝜌(𝜏)+1𝑑𝜏2𝜋𝑖_Γ1𝜏𝑈_𝑛^(𝜏)𝜏^−𝑟−1𝐾_𝜌(𝑡,𝜏)𝑄𝑥^(𝑟)_𝑛,𝜌(𝜏)𝑑𝜏=𝑈_𝑛𝑓, (5.28) which after the application of integral representation (5.19) is equivalent (in the same sense of solvability) to the operator equation 𝑈_𝑛𝐶(𝑡)𝑣_𝑛,𝜌(𝑡)+𝐷(𝑡)𝑆𝑣_𝑛,𝜌(𝑡)+12𝜋𝑖_Γ1𝜏𝑈_𝑛^(𝜏)𝜏ℎ_𝜌(𝑡,𝜏)⋅𝑣_𝑛,𝜌(𝜏)𝑑𝜏=𝑈_𝑛𝑓, (5.29) where the functions 𝐶(𝑡), 𝐷(𝑡), and ℎ_𝜌(𝑡,𝜏) 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 𝑛≥𝑛₂ and 𝜌 small enough. Moreover, the approximate solutions 𝑣_𝑛,𝜌(𝑡)∈𝑋_𝑛 of this equation converge to the exact solution 𝑣_𝜌(𝑡) of SIE (4.2) in the norm 𝐿_𝑝(Γ) as 𝑛→∞ and the following estimation is true: 𝑣_𝜌−𝑣_{𝑛,𝜌𝑝}1=𝑂𝑛^𝛼𝜔1+𝑂𝑓;𝑛+𝑂𝜔^𝜏1ℎ;𝑛+𝑂𝜔^𝑡1ℎ;𝑛. (5.30) The function 𝑥_𝑛,𝜌(𝑡) can be expressed via the function 𝑣_𝑛,𝜌(𝑡) by formula (5.19). Using the definition of the norm in the space 𝐿_𝑝(Γ), 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.