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

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.


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

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 Γ 0 {|w| 1} on the domain

2.1
We assume that the function z ψ w has the second derivative, satisfying on Γ 0 the H ölder 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 where l is the length of Γ.Let U n be the Lagrange interpolating polynomial

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 : 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 r x 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:

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

Journal of Applied Mathematics
ρ is an arbitrary positive number, M 0 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 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 ρ x 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 n t j 0, j 0, . . ., 2n.

3.7
Using the 3.7 we obtain a system of linear algebraic equations SLAE for collocation method: 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: 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 : 3.10

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 :

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 where M j t, τ , N j t, τ j 0, . . ., ν are H ölder 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.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 ∞.
The proof can be found in 22 .
Lemma 4.3.Let the assumptions of Lemma 4.2 be satisfied; then The proof of this lemma can be found in 22 .

Convergence Theorems
The norm in .

5.2
We denote by L p,ν the image of the space L p with respect to the map P t −ν Q 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 5.2.The operator B : 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: 2 the functions A r t and B r t belong to the space H α Γ , 0 < α < 1;

5.5
Then, the SLAE 3.8 of collocation method has the unique solution ξ k k −n, . . ., n , for numbers n ≥ n 1 that are large enough and for numbers ρ small enough.The ρ satisfies the following inequality:

5.6
The approximate solutions x n,ρ t , constructed by formula 3.6 , converge when n → ∞ in the norm of space x − x n,ρ p,ν 0, 5.7 and the following estimation for convergence holds:

5.8
The ω f; 1/n and ω t h; 1/n are modules of continuity, where

5.9
Proof.Using the conditions of Theorem 5.3 we have that the operator M : o Wp,ν → L p Γ is invertible.We estimate the perturbation of M depending on ρ.Using Lemma 4.3 and the relation M ρ M 0 K ρ we obtain 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 −1 exists.The following inequalities hold:

5.11
The SLAE 3.8 of the collocation method for SIDE 3.1 for γ ∈ 0; 1 is equivalent to the operator equation 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 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

Journal of Applied Mathematics
We should show that if n ≥ n 1 is large enough and ρ satisfies the relation 5.6 the operator U n M ρ U n is invertible.The operator acts from the subspace to the subspace 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 :

5.16
Using the formulas Px r t P x r t , Qx r t Q x r t , 5.17 and relations 4.1 we obtain from 5.16

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 n MU n x n,ρ U n f and vice versa.We can determine the function v n,ρ t from relations 4.6 :

5.19
From the conditions 3 , 4 , and 6 of Theorem 5. We proceed to get an error estimate

5.24
Using the inequality

5.25
From 5.21 , 5.24 , and 5.11 , and from the inequality 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 ≥ n 2 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:

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.

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.

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

Lemma 5 . 1 .→
The differential operator D ν : L p,ν , D ν g t g ν t is continuously invertible and its inverse operatorD −ν : L p,ν → • W ν p is determined by the equality ν − 2 and C 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 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 ≥ n 1 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 ≥ n 1 .
t h; 1/n and O ω f; 1/n are modulus of continuity.From 4.1 and 5.19 we obtain 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 ≥ n 2 ≥ n 1 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