Boundary Galerkin Method of a Skew-Derivative Problem in the Exterior of an Open Arc Based on Chebyshev Polynomials

Abstract

A problem modeling Hall effect in a semiconductor film from an electrode of arbitrary shape is considered, which is a skew-derivative problem. Boundary Galerkin method for solving the problem in Sobolev spaces is developed firstly. The solution is represented in the form of the combined angular potential and single-layer potential. The final integral equations do not contain hypersingular integrals. Uniqueness and existence of the solution to the equations are proved. The weakly singular and Cauchy singular integral arising in these equations can be computed directly by truncated series of Chebyshev polynomials with their weighting function without approximation. The numerical simulation showing the high accuracy of the scheme is presented.

1. Introduction

Hall effect is the production of a voltage difference across a semiconductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current. It is widely used in industrial automation, detection, and information processing, which is the basic method for the study of semiconductor material properties. In this paper, we concern an electric current in a semiconductor coming from an electrode, which can be modeled by an open arc. A magnetic is orthogonal to the film. The directions of electric current and electric intensity are not the same because of Hall effect. From a mathematical standpoint, the problem is a skew-derivative problem in the exterior of an open arc in a plane.

The boundary integral has been a means of solving problems in the exterior of an open arc. The difficulty of the problem is that the solution is continuous at the tips, but its gradient may have weak singularity. Numerical methods of problems are considered in Hölder space in a series of papers [1–3]. The analysis in [4] is performed within a Sobolev space setting and a weak solution concept, but no numerical method is presented. These authors construct the solution, via a combination of a single- and a double-layer potential, which can not describe the singularity at the tips directly. Thus, another more difficult problem is introduced, that is, hypersingularity.

In [5], the author constructs the solution by single-layer potential and nonclassical angular potential [6–10] and proves the existence and uniqueness of the solution. An angular potential can be seen as a double layer potential under certain conditions. It has the same order of singularity as a single-layer potential, which is different from a double-layer potential. This makes the final integral system much simpler. The solution which is a combination of a single-layer and angular potential may have weak singularity at the tips of arc if the density function has the form 𝜇/√1−𝜎².

In [11], a numerical method for solving this problem is first considered. It numerically solved a Fredholm integral equation of the second kind in Hölder space. The method can avoid the hypersingularity caused from the double-layer potential. But due to Cauchy singular integral of the system and the special form of density function, approximate calculation of singular integral can not be avoided. In recent years, the inverse problems for arc have been considered by some researchers. In [12–14], the authors obtain asymptotic analysis in terms of the size of the solutions to scattering problems by cracks. Such asymptotics are useful for solving inverse and imaging problems.

In the present paper, we consider the problem in Sobolev spaces firstly. The equations which consist of a Cauchy integral equation of the first kind with additional condition are derived from summing up a single-layer potential and an angular potential. We discuss the uniqueness and existence of the solution to boundary Galerkin method. The purpose of this paper is to develop an efficient solution method. Due to the properties of Chebyshev polynomials, several researchers [1, 3] use them for solving integral equations. In this paper, we approximate the unknown function by means of truncated series of Chebyshev polynomials with their weighting function. Thus, the numerical solution may have weak singularity at the tips of the arc and the singular integral can be computed directly. The convergence is proved. Some numerical examples are presented in order to test the method by comparison with explicit solutions and the numerical solutions of [11].

The plan of the paper is as follows: in Section 2, model problem is listed. We introduce our formulation of skew-derivative problem and proceed by establishing the existence and uniqueness of a solution to integral equations. Then in Section 3, we develop a spectral boundary element method for solving the integral equations numerically and discuss some numerical techniques. Section 4 presents several numerical experiments to illustrate the accuracy.

2. Uniqueness and Existence

In this section, we introduce the model problem firstly. Thus the problem is reduced to an integral equation with some additional conditions. Then we establish the uniqueness and existence of the solution to the integral equations in Sobolev spaces, by inversion of the integral operator.

Consider a constant magnetic field which is perpendicular to a plane semiconductor film 𝑥=(𝑥₁,𝑥₂)∈ℝ². An electrode is placed in the unbounded film. The electrode is modeled by an open curve Γ∈𝐶^2,𝜆, 𝜆∈(0,1]. Set Γ∶=𝑦=𝑦(𝑠)=𝑦₁(𝑠),𝑦₂(𝑠)∶𝑠∈[−1,1]. (1) By 𝑧₋₁,𝑧₁ we denote the two end points 𝑧₁∶=𝑦(1),𝑧₋₁∶=𝑦(−1) of Γ. Denote the left-hand of Γ by Γ⁺ when 𝑠 increases, while the opposite side is called Γ⁻. The outward normal of Γ⁺ is denoted by 𝑛. The tangent direction 𝜏 of Γ is directed into the direction of 𝑠 increasing. Obviously, 𝑛(𝑥(𝑠))=(𝑥̇₂(𝑠),−𝑥̇₁(𝑠))/|𝑥̇(𝑠)|=𝑥̇(𝑠)^⟂/|𝑥̇(𝑠)|, where |𝑥̇(𝑠)|∶=𝑥̇₁(𝑠)²+𝑥̇₂(𝑠)².

The electric-field potential 𝑢 is the solution of the skew-derivative problem as follows: Δ𝑢=0,ℝ²𝜕⧵Γ,𝜕𝑛_𝑥𝑢(𝑥(𝑠𝜕))+𝛽𝜕𝜏_𝑥𝑢(𝑥(𝑠(𝑠)|𝑢|(1),|𝑥||∇(𝑢|𝑥|))=𝑓,Γ,=𝑂→∞,)|=𝑜⁻¹,|𝑥|→∞, (2) where 𝛽=𝛼𝑀, 𝛼 is the mobility of the carriers, and 𝑀 is the projection of the magnetic induction onto 𝑥₃ axis. Suppose 𝛽 is a real constant. The electric current satisfies the equations as follows: div𝐉=0,𝐉=Λ𝐄,𝐄=−∇𝑢, (3) where 𝜂Λ=1+𝛽²−1𝛽−𝛽1, (4) and 𝜂 is a constant. It is easy to see that the direction of current density 𝐉 and the intensity of the electric field 𝐄 do not coincide if 𝛽≠0. These equations lead to the first formula of (2). The boundary condition on Γ in (2) is deduced by (𝐉,𝑛_𝑥)_𝑥∈Γ=𝜂1+𝛽²𝑓(𝑠). (5)

Problem (2) is considered by [5]. The author establishes the existence of a solution by a combined single-layer potential and angular potential approach in Hölder space and proves uniqueness up to arbitrary constant.

Consider the angular potential introduced (see [6]): 𝑣𝜇𝑥)1(=−2𝜋_Γ𝜇(𝜎)𝑉(𝑥,𝜎)𝑑𝑠_𝜎, (6) where 𝑉(𝑥,𝜎) is defined by ()=𝑥cos𝑉𝑥,𝜎¹−𝑦₁(𝜎)(𝜎()=𝑥𝑥−𝑦),sin𝑉𝑥,𝜎²−𝑦₂(𝜎)(𝜎.𝑥−𝑦) (7) From the definition, it is easy to see that an angular potential is a multivalued function. In order to make it single-valued, it is necessary to require the following additional condition (see [6]): _Γ𝜇𝑑𝑠=0. (8) Integrating it by parts, 𝑣[𝜇](𝑥) becomes a double-layer potential as follows: 𝑣𝜇𝑥)=1(2𝜋_Γ𝜌𝜇𝜕𝜕𝑛_𝑦ln𝑥−𝑦𝑑𝑠_𝑦, (9) where 𝜌𝜇=_Γ𝜎𝜇𝑑𝑠, (10) and Γ_𝜎∶={𝑦(𝑠)=(𝑦₁(𝑠),𝑦₂(𝑠)),𝑠∈[−1,𝜎]}.

In [5], the solution of (2) is constructed in the following form: 𝑢𝜇(𝑥)=𝑣𝜇(𝑥)−𝛽𝑤𝜇(𝑥)+𝐶, (11) where 𝐶 is an arbitrary constant. 𝑤 is the single-layer potential 𝑤𝜇𝑥)=1(2𝜋_Γ𝜇(𝜎)ln𝑥−𝑦𝑑𝑠_𝑦. (12) If 𝜇(𝜎) in (11) behaves as 𝜇/√1−𝜎², where 𝜇 is a continuous function, the solution 𝑢[𝜇](𝑥) has singularity at the end-points of Γ, that is, |∇𝑢(𝑥)|≤𝐶|𝑥−𝑥(𝑑)|^𝜖, (13) with 𝑑=±1 and 𝜖∈(−1,0].

We put (11) in the boundary condition of (2) and get the following integral equation: −1+𝛽²2𝜋_Γ𝜇(𝜎)sin𝜑₀𝑦(𝑠),𝑦(𝜎)𝑦(𝑠)−𝑦(𝜎)𝑑𝑠_𝜎=𝑓(𝑠), (14) where 𝜑₀(𝑥,𝑦) is the angle between the vector 𝑥^→𝑦 and the direction of the normal 𝑛_𝑦. If 𝜇 in (11) satisfies (14), it is the solution of problem (2).

First, we extend the arc Γ to a piecewise smooth, simple connected, closed curve 𝜕𝐷 and define the following spaces (see [15]): 𝐿²(Γ)∶=𝑢|_Γ∶𝑢∈𝐿²(,𝐻𝜕𝐷)_1/2(Γ)∶=𝑢|_Γ∶𝑢∈𝐻^1/2(,𝐻𝜕𝐷)_1/2(Γ)∶=𝑢|_Γ∶𝑢∈𝐻^1/2(𝜕𝐷)∶supp𝑢⊆Γ, (15) where 𝐿²(𝜕𝐷),𝐻^1/2(𝜕𝐷) denote the usual Sobolev spaces. That is to say, 𝐻^1/2(Γ) contains functions 𝑢∈𝐻^1/2(Γ) such that their extension by zero to 𝜕𝐷 is in 𝐻^1/2(𝜕𝐷). In [16], the author characterizes 𝐻^1/2(Γ) by using the norm ‖𝑢‖_𝐻^1/2₀₀={‖𝑢‖²_𝐻1/2(Γ)+𝜌^−1/2𝑢²_𝐿2(Γ)}^1/2, (16) where 𝜌=dist(𝑥,𝑧_±1). It is shown that the norms ‖𝑢‖_𝐻^1/2_00(Γ),‖𝑢‖_𝐻^1/2_(𝜕𝐷)with𝑢=𝑢for𝑥∈Γ,0otherwise. (17) are equivalent.

Now define the dual space of 𝐻^1/2(Γ) by 𝐻^−1/2(Γ) and the dual space of 𝐻^1/2(Γ) by 𝐻^−1/2(Γ). Hence we have the chain 𝐻^1/2(Γ)⊂𝐻^1/2(Γ)⊂𝐿²(Γ)⊂𝐻^−1/2(Γ)⊂𝐻^−1/2(Γ). (18) Note that 𝐻^−1/2(Γ) also coincides with 𝐻^−1/2_Γ(𝜕𝐷)∶={𝑢∈𝐻^−1/2(𝜕𝐷)∶supp𝑢⊂Γ}.

In order to prove the existence and uniqueness of solutions for (8) and (14), we introduce two boundary integral operators 𝑇∶𝐻^1/2(Γ)→𝐻^−1/2(Γ),𝐼∶𝑉→𝐻^1/2(Γ), (19) which are defined by 𝜕𝑇𝜓=𝜕𝑛_𝑥Γ𝜓𝑦𝜕𝜕𝑛_𝑦ln𝑥−𝑦𝑑𝑠_𝑦,𝐼𝜇=_Γ𝜎𝜇𝑑𝑠, (20) with 𝑉={𝜇∈𝐻^−1/2(Γ)∶∫_Γ𝜇𝑑𝑠=0}. It is easy to see 𝑇∘𝐼 is the limit of normal derivative of angular potential on Γ.

Lemma 1. Under the condition of (8) lim_{(𝑠)𝑥→𝑥∈Γ}𝜕𝑣𝜇(𝑥)𝜕𝑛_𝑥1=−2𝜋¹₋₁𝜇(𝜎)sin𝜑₀𝑥(𝑠),𝑦(𝜎)𝑥(𝑠)(𝜎𝑦(̇𝜎−𝑦))𝑑𝜎. (21) One rewrites (14) with (8) in operator form 1+𝛽²𝑇𝐼𝜇=𝑓. (22)

Lemma 2. 𝐼 has a bounded inverse operator.

Proof. Let 𝜇𝑑𝑠=𝑓𝑑𝑡. It is easy to see 𝑓∈𝐻^−1/2[−1,1] and ∫¹₋₁𝑓𝑑𝑡=0. Let 𝐹=𝐼𝑓. Obviously, 𝐹∈𝐻^1/2(Γ) and 𝐹(𝑧_±1)=0. There is a homeomorphism from 𝐻^1/2(Γ) to 𝐻^1/2[0,2𝜋]. Define an operator 𝐷∶𝐷(𝑒^imt)=ime^imt. Thus 𝐷 is the bounded inverse operator of 𝐼.
It is known that −𝑇 is positive and bounded below up to a compact perturbation, that is, there exists a compact operator 𝐿∶𝐻^1/2(Γ)→𝐻^−1/2(Γ) (23) such that Re−(𝑇+𝐿)𝜓,𝜓≥𝐶‖𝜓‖²,for𝜓∈𝐻^1/2(Γ), (24) where ⟨⋅,⋅⟩ denotes the duality pairing between 𝐻^−1/2(Γ) and 𝐻^1/2(Γ). In other words, 𝑇 is a Fredholm operator with zero index. It can be easily proved that the homogeneous equation (14) has only a trivial solution. As noted above, we arrive at the following assertion.

Theorem 3. Equation (22) has a unique solution 𝜇∈𝐻^−1/2(Γ) for any 𝑓∈𝐻^−1/2(Γ).

3. Spectral Boundary Galerkin Method

We approximate the unknown function 𝜇(𝜎) in (22) by truncated series of Chebyshev polynomials with weighting function 1/√1−𝜎². As mentioned above, the solution may have weak singularity. Let 𝑇_𝑛(𝜎) and 𝑈_𝑛(𝜎) denote the Chebyshev polynomials of first and second kinds 𝑇_𝑛(𝜎)(),𝑈=cos𝑛arccos𝜎_𝑛(𝜎)=sin((𝑛+1)arccos𝜎)sin(arccos𝜎) (25) with real values over [−1,1]. Then Chebyshev polynomials are orthogonal with respect to the weighting functions ¹₋₁𝑇_𝑛(𝜎)𝑇_𝑚(𝜎)1√1−𝜎²𝜋𝑑𝜎=0𝑛≠𝑚,2𝑛=𝑚=0,𝜋𝑛=𝑚≠0,₁₋₁𝑈_𝑛(𝜎)𝑈_𝑚(𝜎)√1−𝜎²𝑑𝜎=𝜋0𝑛≠𝑚,2𝑛=𝑚=0. (26)

Next we construct the approximation spaces: ℚ_𝑁1=span√1−𝜎²𝑇_𝑛(𝜎)^𝑁_𝑛=1,ℙ_𝑁=span√1−𝜎²𝑈_𝑛(𝜎)^𝑁_𝑛=0. (27) Let us show approximation property over the associated Sobolev space.

Lemma 4. (1) The space ℙ_𝑁 is a closed subspace of 𝐻^1/2[−1,1].
(2)ℙ_∞=lim_𝑁→∞ℙ_𝑁 is dense in 𝐻^1/2[−1,1].

Proof. Obviously, √1−𝜎²𝑈_𝑛(𝜎) belongs to 𝐻^1/2(Γ). For 𝑓∈𝐻^1/2[−1,1], ∃𝜓∈𝐶^∞₀(−1,1), s.t. 𝑓−𝜓≤𝜀. (28)
According to the Weierstrass approximation theorem, there is an integer 𝑀>0 such that max_{𝑥∈[−1,1]}𝜓−𝑝_𝑀<𝜀, (29) where 𝑝_𝑀 is the Bernstein polynomial of order 𝑀. On the basis of the definition of 𝑝_𝑀 and 𝜓(−1)=𝜓(1)=0, we have 𝑝_𝑀(−1)=𝑝_𝑀(1)=0. (30) Take the transform 𝑝_𝑀=√1−𝑥²𝑝^′. (31) Obviously, 𝑝^′∈𝐶[−1,1]. So there is a polynomial 𝑃 such that max_{𝑥∈[−1,1]}𝑝^′−𝑃<𝜀. (32) Thus 𝑓−√1−𝑥²𝑃≤𝑓−𝜓+𝜓−√1−𝑥²𝑃≤𝑓−𝜓+max𝜓−𝑝_𝑀+√1−𝑥²𝑝^′−𝑃≤3𝜀. (33) The proof is complete.

Obviously, the next lemma follows.

Lemma 5. (1) The space ℚ_𝑁 is a closed subspace of 𝐻^−1/2[−1,1].
(2)ℚ_∞=lim_𝑁→∞ℚ_𝑁 is dense in 𝐻^−1/2[−1,1].
The idea of constructing the finite element space is to describe 𝜇 through the truncated expansion: 𝜑_𝑁(𝜎)=^𝑁_𝑛=1𝑐_𝑛1√1−𝜎²𝑇_𝑛(𝜎). (34)

The spectral Galerkin variational formulation of problem (22) is to find 𝜑_𝑁∈ℚ_𝑁, and ∫¹₋₁𝜑_𝑁(𝜎)|𝑦(̇𝜎)|𝑑𝜎=0, 1+𝛽²𝑇𝐼𝜑_𝑁,𝜓=𝑓,𝜓,∀𝜓∈ℙ_𝑁−1. (35) Thanking [17], the following estimation holds: ‖𝜇−𝜑_𝑁‖𝐻^−1/2≤𝐶inf_{𝜓𝑁∈ℚ𝑁}‖𝜇−𝜓_𝑁‖𝐻^−1/2. (36)

By using of the relation (22) and (14) and the finite dimensional subspace introduced above, we can rewrite the spectral Galerkin method as to find 𝑐_𝑛,𝑛=1,…,𝑁, such that −1+𝛽²2𝜋^𝑁_𝑛=1𝑐_𝑛¹₋₁√1−𝑠²𝑈_𝑚(𝑠𝑦(̇𝑠×))₁₋₁𝑇_𝑛(𝜎)√1−𝜎²sin𝜑₀𝑦(𝑠),𝑦(𝜎)𝑦(𝑠)(𝜎𝑦(̇𝜎=−𝑦))𝑑𝜎𝑑𝑠₁₋₁√1−𝑠²𝑈_𝑚(𝑠)𝑓(𝑠)𝑦(̇𝑠)𝑑𝑠,𝑚=0,1…,𝑀,_𝑁𝑛=1𝑐_𝑛¹₋₁𝑇_𝑛(𝜎)√1−𝜎²𝑦(̇𝜎)𝑑𝜎=0. (37)(38) It can be shown that sin𝜑₀𝑦(𝑠),𝑦(𝜎)𝑦(𝑠)(𝜎−1−𝑦)(𝑦(̇𝜎]×[.𝜎−𝑠))∈𝐶([−1,1−1,1]) (39) Rewrite (37) in the form ^𝑁_𝑛=1𝑐_𝑛¹₋₁√1−𝑠²𝑈_𝑚(𝑠𝑦(̇𝑠×))₁₋₁𝑇_𝑛(𝜎)√1−𝜎²()+𝜎−𝑠𝑑𝜎𝑑𝑠_𝑁𝑛=1𝑐_𝑛¹₋₁√1−𝑠²𝑈_𝑚(𝑠𝑦(̇𝑠×))₁₋₁sin𝜑₀𝑦(𝑠),𝑦(𝜎)𝑦(𝑠)(𝜎−1−𝑦)(𝑦(̇𝜎×𝑇𝜎−𝑠))_𝑛(𝜎)√1−𝜎²𝑦(̇𝜎)𝑑𝜎=−2𝜋1+𝛽²¹⁻¹√1−𝑠²𝑈_𝑚(𝑠)𝑓(𝑠)𝑦(̇𝑠)𝑑𝑠. (40) It is convenient to evaluated integrals in (40) by using the Gauss-Chebyshev integration rule of the first kind ¹₋₁1√1−𝜎²𝑓(𝜎)𝜋𝑑𝜎≈𝐽^𝐽_𝑗=1𝑓𝛼_𝑗, (41) with 𝛼_𝑗=cos{(2𝑗−1)𝜋/2𝐽}, 𝑗=1,…,𝐽, and the second kind ¹₋₁√1−𝜎²𝑓(𝜎)≈𝜋𝐽+1^𝐽_𝑗=1𝑓𝛽_𝑗sin²𝑗𝜋,𝐽+1 (42) where 𝛽_𝑗=cos(𝑗𝜋/(𝐽+1)), 𝑗=1,…,𝐽.

For the first integral in (40), it is Cauchy intergal and the following formula is useful: ¹₋₁𝑇_𝑛(𝜎)(𝜎−𝑠)√1−𝜎²𝑑𝜎=𝜋𝑈_𝑛−1(𝑠),𝑛≥1. (43)

If |𝑦(̇𝑠)|≠const, 𝑀=𝑁−2, the integral can be evaluated by (42): ¹₋₁√1−𝑠²𝑈_𝑚(𝑠)𝑦(̇𝑠)¹₋₁𝑇_𝑛(𝜎)√1−𝜎²()≈𝜋𝜎−𝑠𝑑𝜎𝑑𝑠₂𝐽+1^𝐽_𝑗=1𝑈_𝑚𝛽_𝑗𝑈_𝑛−1𝛽_𝑗𝑦̇𝛽_𝑗sin²𝑗𝜋.𝐽+1 (44) In addition, there is an important condition (8), which can also be computed by (41): 𝜋𝐽^𝑁_𝑛=1𝑐_𝑛^𝐽_𝑗=1𝑇_𝑛𝛼_𝑗𝑦̇𝛼_𝑗=0. (45)

If |𝑦(̇𝑠)|=𝐶, 𝑀=𝑁−1, (8) can be satisfied naturally according to the definition of ℚ_𝑁. We have ¹₋₁√1−𝑠²𝑈_𝑚(𝑠)𝑦(̇𝑠)¹₋₁𝑇_𝑛(𝜎)√1−𝜎²()=𝜋𝜎−𝑠𝑑𝜎𝑑𝑠0,𝑛≠𝑚+1,…₂2𝐶,𝑛=𝑚+1,… (46) which is obtained by directly using of the orthogonal relation of Chebyshev polynomials 𝑈_𝑚.

Assume that the stiffness matrix constructed by the first term in (40) is denoted by 𝐼. In the second case, 𝐼 is a diagonal matrix 𝐼_𝑁×𝑁=𝜋²2𝐶𝜋²2𝐶⋱𝜋²2𝐶. (47)

Consider the computation of the second term of (40). The contribution for stiffness matrix from the second term in (40) is denoted by 𝐴={𝑎_𝑖𝑗}_𝑁×𝑁. According to (41) and (42), the entries of matrix 𝐴𝑎_𝑚+1,𝑛≈𝜋²𝐽(𝐽+1)^𝐽𝐽_𝑖=1𝑗=1𝑈_𝑚𝛽_𝑖𝑇_𝑛𝛼_𝑗𝑦̇𝛼_𝑗×sin𝜑₀𝑦𝛽_𝑖,𝑦𝛼_𝑗𝑦𝛽_𝑖−𝑦𝛼_𝑗−1𝛼_𝑗−𝛽_𝑖𝑦̇𝛼_𝑗×𝑦̇𝛽_𝑖sin²𝑖𝜋.𝐽+1 (48)

Therefore, the stiffness matrix for (40) is 𝐼+𝐴. The third term can also be evaluated by (42): ¹₋₁√1−𝜎²𝑈_𝑚(𝑠)𝑓(𝑠𝑦(̇𝑠≈𝜋))𝑑𝑠𝐽+1^𝐽_𝑗=1𝑓𝛽_𝑗𝑈_𝑚𝛽_𝑗sin²𝑗𝜋𝑦̇𝛽𝐽+1^𝑗. (49)

Finally, we get a linear algebraic system of equations with respect to 𝐶={𝑐₁,𝑐₂,…,𝑐_𝑁}.

Next, we recover single-layer potential 𝑤[𝜇] and angular potential 𝑣[𝜇](𝑥) by (41): 𝑤𝜇𝑥)≈1(2𝐽^𝐽_𝑗=11−𝛼²_𝑗𝜇𝛼_𝑗ln𝑥−𝑦𝛼_𝑗𝑦̇𝛼_𝑗,𝑣𝜇𝑥)≈1(2𝐽^𝐽_𝑗=11−𝛼²_𝑗𝜇𝛼_𝑗𝑉𝑥,𝛼_𝑗𝑦̇𝛼_𝑗. (50)(51)

It is easy to see that we can get approximate electric-field potential 𝑢 by 𝑢(𝑥)=𝑣𝜇(𝑥)−𝛽𝑤𝜇(𝑥)+𝐶. (52)

Although (6) can also be expressed by a double-layer potential, such an expression has more strong singularity near Γ compared with (6). So (51) is deduced by (6).

4. Numerical Examples

In Section 3 we develop a method for finding the numerical solution of the integral equation (22). More precisely, a spectral Galerkin method is used. We reduce it to a linear algebraic system. In this section, we describe the results of the numerical tests mentioned above. We examine the feasibility of the numerical method for solving (22) and compare the numerical solution with these solutions of [11] for certain function 𝑓 on the right-hand side of the integral equation under the assumption that Γ is a straight line or an arc of the unit circle.

Example 6. Let Γ be a segment of the straight line: Γ={(𝑥,0)∶−1≤𝑥≤1}, (53) and the parameters 𝑓≡1, 𝛽≡1. The corresponding exact solution of integral equation (22) can be written as 𝜇(𝑠)2=−1+𝛽²𝑠√1−𝑠². (54) The solution of problem (2) can be expressed by (11). The exact analytical expressions for potentials under the assumption are 𝑤𝜇𝑥)=√(𝑟₁𝑟₂1+𝛽²𝜑cos¹2𝜑cos²2𝜑−sin¹2𝜑sin²2−𝑥₁√𝑟₁𝑟₂1+𝛽²,𝑣𝜇𝑥)=√(𝑟₁𝑟₂1+𝛽²𝜑cos¹2𝜑sin²2𝜑+sin¹2𝜑cos²2−𝑥₂√𝑟₁𝑟₂1+𝛽², (55) where 𝑟₁=(𝑥₁+1)²+𝑥²₂, 𝑟₂=(𝑥₁−1)²+𝑥²₂, 𝜑₁=arctan(𝑥₂/(𝑥₁+1)), 𝜑₂=arctan(𝑥₂/(𝑥₁−1)).

Now, let us consider the numerical solution at this time. Obviously, sin𝜑₀𝑦(𝑠),𝑦(𝜎)𝑦(𝑠)(𝜎−1−𝑦)(𝜎−𝑠)𝑦(̇𝜎)=0. (56) The spectral Galerkin approximate solution of 𝜇 is equal to the exact solution.

This situation allows us to test the accuracy of the numerical method of potentials. We choose 𝑊={(𝑥,𝑑)∶−1≤𝑥≤1} (57) as the points where we compute the potentials. 𝐿²-errors are listed in Table 1, where the numerical solutions 𝑤,𝑣 are compared with analytical solutions 𝑤,𝑣. Table 1 also contains the 𝐿²-errors of the potentials 𝑊,𝑉 which are computed by the methods in [11].

Table 1 

𝐿²-errors for a straight line case with 𝛽≡1, 𝑓≡1 (𝑊, 𝑉 with 𝑀=160, ℎ=0.0125 and 𝑤, 𝑣 with 𝐽=160, 𝑁=10).

It can be seen that the computation errors of the potentials is on the good control. The errors increase as 𝑑 decreases, because logarithmic function in (50) and 𝑉(𝑥,𝑦) in (51) have singularities when 𝑑 is very small.

As shown in Table 1, the numbers of single-potential are superior to those in [11] when 𝑑>10⁻³, but inferior to those when 𝑑=10⁻³. That is because (50) is not as accurate as the approximation formula in [11] when 𝑑 is very small. But it is adequate enough for our purpose and our formula is easier to implement.

According to Table 1, our results of angular potential have higher accuracy than results of [11]. In [11], the author subdivides each interval of length ℎ into some subintervals uniformly in order to improve the accuracy. The values of 𝜇 on subintervals are computed by interpolation. In this paper, we need to increase 𝐽 so that to decrease the errors. The potential can be evaluated easily without interpolation since the numerical result of 𝜇 is a function. The interpolation procedure may deduced the related accuracy.

The straight-line case is too special to show the accuracy of the method for arbitrary arcs. Therefor we consider further numerical tests on a curved arc.

Example 7. Suppose Γ is an arc of the unit circle Γ={(cos𝑥,sin𝑥)∶−1≤𝑥≤1}. (58) If 𝑓≡1, then the density function can be given by 𝜇(𝑥)=−21+𝛽²))))𝑥sin((𝑥+1/2sin((1−𝑥/2sin2. (59) If 𝑓(𝑥)=cos𝑥+sin𝑥, the exact density function is 𝜇(𝑥)=−21+𝛽²))))×𝑥sin((𝑥+1/2sin((1−𝑥/2sin2cos(−1)−sin3𝑥2+cos3𝑥2𝑥−cos2cos(−1). (60) The potentials under these two assumptions cannot be given by explicit expressions. The comparison between numerical solution 𝑀 and the analytical solution 𝜇 is not convenient because of the space 𝐻^−1/2(Γ) they belong to. Let 𝑀_∗=√1−𝑥²𝑀, 𝜇_∗=√1−𝑥²𝜇. 𝐿²-errors between 𝑀_∗ and 𝜇_∗ are listed in Tables 2 and 4. We observe excellent agreement between them. The accuracy of 𝑀_∗ increases if 𝑁 increases. But no significant change can be seen whatever 𝐽 increases or not. The numerical solution 𝜇_∗ of [11] is compared with the analytical solution 𝜇_∗ in Tables 3 and 5. Results in Tables 2 and 4 are smaller than those in Tables 3 and 5. The computational schemes in this paper are stable and efficient.

Table 2 

𝐿²-errors for an arc on the unite circle case with 𝛽≡1, 𝑓≡1.

Table 3 

An arc on the unit circle of [11]: 𝛽≡1, 𝑓≡1.

Table 4 

𝐿²-errors for an arc on the unite circle case with 𝛽≡1, 𝑓≡cos(𝑥)+sin(𝑥).

Table 5 

An arc on the unit circle of [11]: 𝛽≡1, 𝑓≡cos(𝑥)+sin(𝑥).

Let 𝑤[𝜇], 𝑣[𝜇] be the angular potential and single-potential computed by (50) and (51). We use the exact density function to get them. Let 𝑤[𝜇], 𝑣[𝜇] be the angular potential and single-potential used by numerical approximation of the density function.

We compute the potentials in a parallel surface of Γ defined by 𝑊={(𝑑cos𝑥,𝑑sin𝑥),−1≤𝑥≤1}, (61) where 𝑑=1.01. Results of the tests are presented in Tables 6 and 7. Obviously, errors decrease as 𝑁 increases. When 𝑓≡1, it can be seen that the approximation is satisfactory even when 𝑁=3. The simulation of 𝑓=cos𝑥+sin𝑥 is not as good as the simulation of 𝑓≡1. The errors when 𝑁=5 with 𝑓=sin𝑥+cos𝑥 are almost the same as the errors when 𝑁=3 under the condition of 𝑓≡1. But the accuracy for both cases is high, when 𝑁=10.

Table 6 

𝐿²-errors and relative 𝐿²-errors of an arc on the unit circle with 𝑓≡1.

Table 7 

𝐿²-errors and relative 𝐿²-errors of an arc on the unit circle with 𝑓(𝑥)=sin(𝑥)+cos(𝑥).

Example 8. In this example, we take the arc 𝑦=0.2𝑥²−0.1𝑥+0.3 (62) and the parameters 1𝛽=1,𝑓=√×1+0.4𝑥−0.10.16𝑥³+1.08𝑥²−2.94𝑥+2.74. (63) We compute the single-potential and angular potential on a semicircular arc: (2cos(𝑡),2sin(𝑡)),0≤𝑡≤𝜋, (64) see Figures 1 and 2. We choose 𝐽=160 and 𝑁=10 to compute the electric-field potential 𝑢 which is shown in Figure 3.

Figure 1 

Single-layer potential on a semicircular arc.

Figure 2 

angular potential on a semicircular arc.

Figure 3 

Electric-field potential 𝑢.

5. Conclusions

In this paper, we have demonstrated an efficient and highly accurate boundary Galerkin method for a skew-derivative problem. The problem is considered in Sobolev spaces firstly. In our method, approximate calculation of singular integral can be avoided. The density function can be evaluated at any point without interpolation once the series coefficients have been determined. It is direct, simple, and fast.

Acknowledgment

The work of the first author is supported by the Youth Natural Science Foundation of Heilongjiang (Grant no. QC2011c103).