JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 365904 10.1155/2012/365904 365904 Research Article Numerical Solution of Weakly Singular Integrodifferential Equations on Closed Smooth Contour in Lebesgue Spaces Al Faqih Feras M. 1, 2 Li Yongkun 1 Department of Mathematics and Statistics King Faisal University Saudi Arabia kfu.edu.sa 2 Department of Mathematics, Al-Hussein Bin Talal University, P. O. Box 20 Ma'an Jordan ahu.edu.jo 2012 04 10 2012 2012 18 06 2012 03 09 2012 2012 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.

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  and the literature cited therein). The general theory of SIE and SIDE has been widely investigated over the last decades . 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 . 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 . 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=0F+, 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 (2.1)ψ()=,ψ()()=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 Lp(Γ)  (1<p<) be the space of complex functions with norm (2.2)gp=(1lΓ|g|p|dτ|)1/p, where l is the length of Γ.

Let Un be the Lagrange interpolating polynomial (2.3)(Ung)(t)=s=02ng(ts)·ls(t),(2.4)lj(t)=k=0,kj2nt-tktj-tk(tjt)nk=-nnΛk(j)tk,  tΓ,j=0,,2n.

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

In the complex space Lp(Γ)  (1<p<) we consider the weakly singular integro-differential equation (SIDE): (3.1)(Mx)r=0ν[A~r(t)x(r)(t)+B~r(t)  1πiΓ  x(r)(τ)τ-t  dτ  +12πiΓKr(t,τ)|t-τ|γ·x(r)(τ)dτ]=f(t),tΓ, where 0<γ<1, A~r(t), B~r(t), Kr(t,τ)(r=0,,ν) and f(t) are known functions; x(0)(t)=x(t) is an unknown function; x(r)(t)=((drx(t))/dtr)(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)(Mx)r=0ν[ΓAr(t)(Px(r))(t)+Br(t)(Qx(r))(t)  +12πiΓKr(t,τ)|t-τ|γ·x(r)(τ)dτ]=f(t),tΓ, where Ar(t)=A~r(t)+B~r(t), Br(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(3.3)12πiΓx(τ)τ-k-1dτ=0,k=0,,ν-1. 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 (3.4)(Mρ(x))(M0x)(t)+12πir=0νΓKr,ρ(t,τ)x(r)(τ)dτ=f(t),tΓ, where (3.5)Kr,ρ(t,τ)={Kr(t,τ)|t-τ|γ,when|t-τ|ρ,Kr(t,τ)ργ,when|t-τ|<ρ.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 (3.6)xn,ρ(t)=k=0nξk,ρ(n)tk+ν+k=-n-1ξk,ρ(n)tk,tΓ, where ξk,ρ(n)=ξk,ρ(k=-n,,n) are unknown complex numbers. We note that the function xn,ρ(t), constructed by formula, obviously satisfies the condition (3.3). Let Rn(t)=(Mρxn)(t)-f(t) be residual of SIDE. The collocation method consists in setting it equal to zero at some chosen points tj,  j=0,,2n on Γ and thus obtaining a linear algebraic system for unknowns ξk,ρ which is determined by solving it: (3.7)Rn(tj)=0,  j=0,,2n. Using the (3.7) we obtain a system of linear algebraic equations (SLAE) for collocation method: (3.8)r=0ν  Ar(tj)k=0n  (k+ν)!  (k+ν-r)!tjk+ν-rξk,ρ  +Br(tj)k=1n(-1)r(k+r-1)!(k-1)!tj-k-r×ξ-k,ρ    +12πi·k=0n(k+ν)!(k+ν-r)!ΓKr,ρ  (tj,τ)τk+ν-rdτ·ξk,ρ+k=1n(-1)r(k+r-1)!(k-1)!·12πiΓKr,ρ(tj,τ)τ-k-rdτ·ξ-k,ρ=f(tj),j=0,,2n, where tj, (j=0,,2n) are distinct points on Γ and Ar(t)=A~r(t)+B~r(t), Br(t)=A~r(t)-B~r(t). We approximate the integrals in SLAE (3.8) by quadrature formula: (3.9)12πiΓg(τ)τl+kdτ12πiΓUn(τl+1·g(τ))τk-1dτ, where k=0,,n, at l=0,1,2, and k=-1,,-n, for l=-1,-2,, and Un is the Lagrange interpolation operator defined by formula (2.3).

Thus, we obtain the following SLAE from (3.8): (3.10)r=0νAr(tj)k=0n(k+ν)!(k+ν-r)!tjk+ν-rξk,ρ+Br(tj)k=1n(-1)r(k+r-1)!(k-1)!tj-k-r×ξ-k,ρ+k=0n(k+ν)!(k+ν-r)!s=02nKr,ρ(tj,ts)ts1+k-rΛ-k(s)ξk,ρ+k=1n(-1)r(k+r-1)!(k-1)!s=02nKr,ρ(tj,ts)ts-k-rΛk(s)ξ-k,ρ=f(tj),j=0,,2n.

4. Auxiliary Results

We formulate one result from , 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 dν(Px)(t)/dtν and dν(Qx)(t)/dtν can be represented by integrals of Cauchy type with the same density v(t): (4.1)dν(Px)(t)dtν=12πiΓv(τ)τ-tdτ,tF+,dν(Qx)(t)dtν=t-ν2πiΓv(τ)τ-tdτ,tF-. Using the integral representation (4.1) we reduce the problem (3.1)–(3.3) to the equivalent (in sense of solvability) of SIE (4.2)(Υv)C(t)v(t)+D(t)πiΓv(τ)τ-tdτ+12πiΓh(t,τ)|τ-t|γv(τ)dτ=f(t),tΓ, for unknown v(t) where (4.3)C(t)=12[Aν(t)+t-νBν(t)],D(t)=12[Aν(t)-t-νBν(t)],(4.4)h(t,τ)=12[Kν(t,τ)+Kν(t,τ)τ-n]-12πiΓ[Kν(t,t1)-Kν(t,t1)t1-n]dt1t1-τ+j=0ν-1[Aj(t)M~j(t,τ)+ΓKj(t,t1)M~j(t1,τ)dt1]-j=0ν-1[Bj(t)N~j(t,τ)+ΓKj(t,t1)N~j(t1,τ)dt1], where M~j(t,τ), N~j(t,τ)j=0,,ν are Hölder functions. An obvious form for these functions are given in . By virtue of the properties of the functions M~j(t,τ), N~j(t,τ), Kj(t,τ), Aj(t), Bj(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 (4.5)(Px)(t)=(-1)ν2πi(ν-1)!Γv(τ)[(τ-t)ν-1log(1-tτ)+k=1ν-1α~kτν-k-1tk]dτ,(4.6)(Qx)(t)=(-1)ν2πi(ν-1)!Γv(τ)τ-ν[(τ-t)ν-1log(1-τt)+k=1ν-2β~kτν-k-1tk]dτ, where (α~k=j=0k-1((-1)jCν-1j/(k-j)), k=1,,ν-1, β~k=j=k+1ν-1((-1)jCν-1j/(j-k)), k=1,,ν-2 and Cν-1j 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)(4.7)v(t)=dν(Px)(t)dtν+tνdν(Qx)(t)dtν, 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)Lp(Γ), 1<p<. Then the function H(t)=(1/2πi)Γ(h(t,τ)/|τ-t|γ)ψ(τ)dτ, satisfies the inequality (4.8)Hpd1ψp,1p+1q=1,(·)p=|1lΓ|(·)(τ)|pdτ|1/p. By d1, d2,…, we denote the constants.

The proof can be found in .

Lemma 4.3.

Let the assumptions of Lemma 4.2 be satisfied; then ||χρ||pd2ρ(1-γ)/q||ψ||p, where χρ=(1/2πi)Γ[(h(t,τ)/|τ-t|γ)-hρ(t,τ)]ψ(τ)dτ,  1/p+1/q=1.

The proof of this lemma can be found in .

5. Convergence Theorems

Define Wp(ν) as (5.1)Wp(ν)={gLp(Γ):g(ν)Lp(Γ),12πiΓg(τ)τ-k-1dτ=0,k=0,,ν-1}. The norm in Wp(ν) is determined by the equality (5.2)gp,ν=g(ν)Lp. We denote by Lp,ν the image of the space Lp with respect to the map P+t-νQ equipped with the norm of Lp. We formulate Lemmas 5.1 and 5.2 from . We use these lemmas to prove the convergence theorems.

Lemma 5.1.

The differential operator Dν:Wp(ν)Lp,ν, (Dνg)(t)=g(ν)(t) is continuously invertible and its inverse operator D-ν:Lp,νWp(ν) is determined by the equality (5.3)(D-νg)(t)=(N+g)(t)+(N-g)(t),(N+g)(t)=(-1)ν2πi(ν-1)!Γ(Pg)(τ)(τ-t)ν-1log(1-tτ)dτ,(N-g)(t)=(-1)ν-12πi(ν-1)!Γ(Qg)(τ)(τ-t)ν-1log(1-τt)dτ.

From Lemma 5.1   Lemma 5.2 follows.

Lemma 5.2.

The operator B:Wp(ν)Lp,B=(P+tνQ)Dν is invertible and (5.4)B-1=D-ν(P+t-νQ).

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

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

Theorem 5.3.

Let the following conditions be satisfied:

ΓC(2,μ), 0<μ<1;

the functions Ar(t) and Br(t) belong to the space Hα(Γ), 0<α<1;

Aν(t)Bν(t)0, tΓ;

the index of the function tνBν-1(t)Aν(t) is equal to zero;

Kr(t,τ)(r=0,,ν)Hβ(Γ×Γ), 0<β1, function f(t)C(Γ);

the operator M:Wp(ν)Lp(Γ) is linear and invertible;

the points tj(j=0,2n) form a system of Fejér knots on Γ [24, 25]: (5.5)tj=ψ[exp(2πi2n+1(j-n))],j=0,,2n,i2=-1.

Then, the SLAE (3.8) of collocation method has the unique solution ξk(k=-n,,n), for numbers nn1 that are large enough and for numbers ρ small enough. The ρ satisfies the following inequality: (5.6)ερ=d3ρ(1-γ)/qM-1p<q8<1. The approximate solutions xn,ρ(t), constructed by formula (3.6), converge when n in the norm of space Wp(ν) to the exact solution x(t) of the problem (3.1)–(3.3) in sense of (5.7)limρ0limnx-xn,ρp,ν=0, and the following estimation for convergence holds: (5.8)x-xn,ρp,ν=O(ρ(1-γ)/q)+O(1nα)+O(ω(f;1n))+O(ωt(hρ;1n))δn,(1p+1q=1).

The ω(f;1/n) and ωt(h;1/n) are modules of continuity, where (5.9)ω(f;1n)=sup|t-t′′|1/n|f(t)-f(t′′)|,ωt(h;1n)=sup|t-t′′|1/n|h(t;τ)-h(t′′;τ)|,t,t′′Γ.

Proof.

Using the conditions of Theorem 5.3 we have that the operator M:Wop,νLp(Γ) is invertible. We estimate the perturbation of M depending on ρ. Using Lemma 4.3 and the relation Mρ=M0+Kρ we obtain (5.10)M-Mρ=O(ρ(1-γ)/q). Let us show that the operator Mρ is invertible for sufficiently small values ρ such that the inequality (5.6) is valid. Using the representation Mρ=M[I-M-1(M-Mρ)] and (5.10), we obtain from Banach theorem that the inverse operator Mρ-1=[I-M-1(M-Mρ)]-1M-1 exists. The following inequalities hold: (5.11)Mρ-1M-11-q,M-1-Mρ-1d11ρ(1-γ)/qM-1. The SLAE (3.8) of the collocation method for SIDE (3.1) for γ(0;1) is equivalent to the operator equation (5.12)UnMρUnxn,ρUnM0Unxn,ρ+Unr=0ν{12πiΓKr,ρ(t,τ)xn,ρ(r)(τ)dτ}=Unf, where Kr,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 (5.13)UnΥρUnvn,ρ=Unf, where operator Υρ is defined in (4.2), substituting Υ by Υρ and (h(t,τ)/|τ-t|γ) by hρ(t,τ) (where hρ(t,τ) is calculated by formula (3.5)). Equation (5.13) represents the collocation method for SIE (5.14)Υρvρ=f,vρ(t)Lp(Γ). We should show that if n(n1) is large enough and ρ satisfies the relation (5.6) the operator UnMρUn is invertible. The operator acts from the subspace Xn={tνk=0nξk,ρtk+k=-n-1ξk,ρtk} (the norm as in Wp(ν)) to the subspace (5.15)Xn=k=-nnrktk,tΓ. (the norm as in Lp(Γ).)

Using formulas (4.1) the dν(Pxn,ρ)(t)/dtν and dν(Qxn,ρ)(t)/dtν can be represented by Cauchy-type integrals with the same density vn,ρ(t): (5.16)dν(Pxn,ρ)(t)dtν=12πiΓvn,ρ(τ)τ-tdτ,tF+,dν(Qxn,ρ)(t)dtν=t-ν2πiΓvn,ρ(τ)τ-tdτ,tF-. Using the formulas (5.17)(Px)(r)(t)=P(x(r))(t),(Qx)(r)(t)=Q(x(r))(t), and relations (4.1) we obtain from (5.16) (5.18)vn,ρ(t)=k=0n(k+ν)!k!tkξk,ρ+(-1)νk=1n(k+ν-1)!(k-1)!t-kξ-k,ρ. We obtain from previous relation that vn,ρ(t)Xn,tΓ.

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 vn,ρ(t)Xn we conclude that if function vn,ρ(t) is the solution of (5.13) then the function xn,ρ(t) is the discrete solution for the system UnMUnxn,ρ=Unf and vice versa. We can determine the function vn,ρ(t) from relations (4.6): (5.19)(Pxn,ρ)(t)=(-1)ν2πi(ν-1)!Γvn,ρ(τ)[(τ-t)ν-1log(1-tτ)+k=1ν-1α~kτν-k-1tk]dτ,(Qxn,ρ)(t)=(-1)ν2πi(ν-1)!Γvn,ρ(τ)τ-ν[(τ-t)ν-1log(1-τt)+k=1ν-1β~kτν-k-1tk]dτ. From the conditions (3), (4), and (6) of Theorem 5.3 and Lemmas 5.1 and 5.2, the invertibility of operator Υ:Lp(Γ)Lp(Γ) follows. From Banach theorem and Lemma 4.3 for small numbers ρ (ρ satisfies the relation (5.6)) we have that the operator Υρ:Lp(Γ)Lp(Γ) is invertible. We should show that for (5.13) all conditions of the Theorem 1 are satisfied from [19, 20]. Theorem 1  gives the convergence of the collocation method for SIE in spaces Lp(Γ). From condition 3 of Theorem 1  and from (4.3) we obtain the condition 3 of Theorem 5.3. From the equality (5.20)[C(t)-D(t)]-1[C(t)+D(t)]=tνBν-1Aq(t), 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 . Conditions (1)(6) in Theorem 5.3 provide the validity of all conditions of Theorem 1 . Therefore, beginning with numbers nn1 (5.13) is uniquely solvable for numbers ρ small enough where ρ satisfies the relation (5.6). The approximate solutions vn,ρ(t) of (5.13) converge to the exact solution of (4.2) in the norm of the space Lp(Γ) as n. Therefore (5.12) and the SLAE (3.10) have the unique solutions for (nn1). From Theorem 1  the following estimation holds: (5.21)vρ-vn,ρpO(1nα)+O(ω(f;1n))+O(ωt(h;1n)), where O(ωt(h;1/n) and O(ω(f;1/n) are modulus of continuity. From (4.1) and (5.19) we obtain (5.22)(Pxρ)(ν)(t)=(Pvρ)(t),(Qxρ)(ν)(t)=t-ν(Qvρ)(t). Therefore we have (5.23)(Pxn,ρ)(ν)(t)=(Pvn,ρ)(t),(Qxn,ρ)(ν)(t)=t-ν(Qvn,ρ)(t). We proceed to get an error estimate (5.24)xρ-xn,ρp,ν=xρ(ν)-xn,ρ(ν)[Lp]P(vρ-vn,ρ)[Lp]+t-νQ(vρ-vn,ρ)[Lp]P·vρ-vn,ρ[Lp]+t-νQ·vρ-vn,ρ[Lp](P+t-νQ)vρ-vn,ρ[Lp]. Using the inequality (5.25)t-νLp=(1lΓ|t-ν|pdt)1/p=(1lΓ|t-νp|dt)1/p(1l1mintΓ|t|pνl)1/p=(1mintΓ|t|pν)1/p=c1. From (5.21), (5.24), and (5.11), and from the inequality (5.26)x-xn,ρp,νM-1f-Mρ-1p,ν+xρ-xn,ρp,ν 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 ξk,ρ, k=-n,,n for numbers nn2(n1) large enough and for numbers ρ small enough (ρ satisfies the relation (5.6)). The approximate solutions xn,ρ(t) converge when n and ρ0 in the norm Wp(ν) to the exact solution x(t) of the problem (3.1)–(3.3) and the following estimation for the convergence is true: (5.27)x-xn,ρp,ν=δn+O(ωτ(h;1n)).

Proof.

It is easy to verify that SLAE (3.10) is equivalent to the operational equation (5.28)Un{r=0ν[ΓAr(t)(Pxn,ρ(r))(t)+Br(t)(Qxn,ρ(r))(t)  +12πiΓ1τUn(τ)[τν+1-rKρ(t,τ)](Pxn,ρ(r))(τ)dτ+12πiΓ1τUn(τ)[τ-r-1Kρ(t,τ)](Qxn,ρ(r))(τ)dτ]r=0ν[ΓAr(t)(Pxn,ρ(r))(t)+Br(t)(Qxn,ρ(r))(t)}=Unf, which after the application of integral representation (5.19) is equivalent (in the same sense of solvability) to the operator equation (5.29)Un{C(t)vn,ρ(t)+D(t)(Svn,ρ)(t)+12πiΓ1τUn(τ)[τhρ(t,τ)]·vn,ρ(τ)dτ}=Unf, where the functions C(t), D(t), and hρ(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 nn2 and ρ small enough. Moreover, the approximate solutions vn,ρ(t)Xn of this equation converge to the exact solution vρ(t) of SIE (4.2) in the norm Lp(Γ) as n and the following estimation is true: (5.30)vρ-vn,ρp=O(1nα)+O(ω(f;1n))+O(ωτ(h;1n))+O(ωt(h;1n)). The function xn,ρ(t) can be expressed via the function vn,ρ(t) by formula (5.19). Using the definition of the norm in the space Lp(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.

Cohen J. W. Boxma O. J. Boundary Value Problems in Queueing System Analysis 1983 79 Amsterdam, The Netherlands North-Holland 703000 10.1016/S0304-0208(08)71863-2 ZBL0588.76005 Kalandiya A. I. Mathematical Methods of Two-Dimensional Elasticity 1975 Moscow, Russia Mir Publishers 0400846 ZBL0412.73008 Linkov A. Boundary Integral Equations in Elasticity Theory 2002 Dordrecht, The Netherlands Kluwer Academic ZBL1046.74001 Muskhelishvili N. I. Some Basic Problems of the Mathematical Theory of Elasticity. Fundamental Equations, Plane Theory of Elasticity, Torsion and Bending 1953 Groningen, The Netherlands Noordhoff 0058417 ZBL0052.41402 Brockman D. Hufnagel L. Fronnt propagation in reaction—supper diffusion dynamics: taming lavy flights with fluctuations Physical Review Letters 2007 98 178311 178314 Ladopoulos E. G. Singular Integral Equations: Linear and Non-Linear Theory and Its Applications in Science and Engineering 2000 New York, NY, USA Springer 1785305 Ivanov V. V. The Theory of Approximate Methods and Their Application to the Numerical Solution of Singular Integral Equations 1976 Leyden, The Netherlands Noordhoff 0405045 Gakhov F. D. Boundary Value Problems 1966 Reading, Mass, USA Pergamon, Oxford, UK, Addison-Wesley 0198152 Muskhelishvili N. I. Singular Integral Equations: Boundary Problems of Functions Theory and Their Applications to Mathematical Physics 1977 Leyden, The Netherlands Noordhoff Vekua N. P. Systems of Singular Integral Equations 1967 Groningen, The Netherlands Noordhoff 216 Translated from the Russian by A. G. Gibbs and G. M. Simmons 0211220 Gohberg I. Krupnik N. Introduction to the Theory of One-Dimensional Singular Integral Operators 1973 Kishinev, Moldova Stiintsa German translation: Birkhause, Basel, Germany, 1979 Prössdorf S. Silbermann B. Numerical Analysis for Integral and Related Operator Equations 1991 Basel, Switzerland Akademie, Berlin, Germany, Birkhauser 1206476 Mikhlin S. G. Prössdorf S. Singular Integral Operators 1986 68 Berlin, Germany Springer 10.1007/978-3-642-61631-0 867687 Prössdorf S. Some Classes of Singular Equations 1978 17 North-Holland, The Netherlands Elsevier 506681 Gabdulalhaev B. The polynomial approximations of solution of singular integral and integro-differential equations by Dzyadik Izvestia Visshih Ucebhih Zavedenii Mathematics 1978 6 193 51 62 Zolotarevskiĭ V. A. Li Z. Caraus I. Approximate solution of singular integrodifferential equations by the method of reduction over Faber-Laurent polynomials Differential Equation 2004 40 12 1764 1769 translated from Differentsial'nye Uravneniya, vol. 40, no.12, pp. 1682–1686, 2004 Caraus I. The numerical solution for systems of singular integro-differential equations by Faber-Laurent polynomials 3401 Proceedings of the 3rd international conference on Numerical Analysis and its Applications (NAA '04) 2005 New York, NY, USA Springer 219 223 Lecture notes in Computer Science Caraus I. Al Faqih F. M. Approximate solution of singular integro-differential equations in generalized Holder spaces Numerical Algorithms 2007 45 205 215 10.1007/s11075-007-9079-1 Zolotarevski V. Finite-Dimensional Methods For Solving of Singular Integral Equations on the Closed Contours of Integration 1991 Chisinau, Moldova Stiinta Zolotarevskiĭ V. A. Approximate solution of systems of singular integral equations on some smooth contours in Lp spaces Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika 1989 2 79 82 1009294 Krikunov Y. The general boundary Riemann problem and linear singular integrodifferential equation The Scientific Notes of the Kazani University 1956 116 4 3 29 Seĭchuk V. N. Estimates for weakly singular integral operators defined on closed integration contours and their applications to the approximate solution of singular integral equations Differential Equations 2005 41 9 1311 1322 10.1007/s10625-005-0280-5 2247262 ZBL1128.45002 Saks R. Boundary-Value Problems For Elliptic Systems of Differential Equations 1975 Novosibirsk, Russia University of Novosibirsk Smirnov V. I. Lebedev N. A. Functions of a Complex Variable: Constructive Theory 1968 London, UK IIlife Translated by Scripta Technica Novati P. A polynomial method based on Fejèr points for the computation of functions of unsymmetric matrices Applied Numerical Mathematics 2003 44 1-2 201 224 10.1016/S0168-9274(02)00139-3 1951294 ZBL1016.65023 Zolotarevski V. A. Direct methods for solving singular integral quations on closed smooth contour in spaces Lp Revue d'Analyse Numérique et de Théorie de l'Approximation 1996 25 1-2 257 265 1607330