On the Convergence Problem of One-Dimensional Hypersingular Integral Equations

Abstract

We develop the expansion method of singular integral equation (SIE) for hypersingular integral equation (HSIE). Relating the hypersingular integrals to Cauchy principal-value integrals, we interpolate the kernel and the density functions to the truncated Chebyshev series of the second kind. The corresponding convergence results for the functions 𝑓∈𝐶^ℓ([−1,1]) and 𝐾(𝑡,𝑥)∈𝐶^ℓ([−1,1]×[−1,1]), ℓ≥1, are derived in an appropriate 𝐿₂[−1,1] norm to the true solution of the weight function. Numerical examples are also presented to validate the theoretical findings.

1. Introduction

The integral equation is defined as an equation with an unknown function that appears under the integral sign. These equations could be classified by the order of singularity [1]. Singular integral has been widely used and well developed [2–4]. A wealth of the literature on applications related to the numerical evaluation of hypersingular integral equations HSIEs could be found in [5–10]. This paper focuses on one-dimensional singular integral equations (SIEs) found in various mixed boundary value problems of mathematical physics and engineering such as isotropic elastic bodies involving cracks, aerodynamics, hydrodynamics, elasticity, and other related areas.

Let us consider the hypersingular integral equation of the following form: 1𝜋¹₋₁𝑄(𝑡)(𝑡−𝑥)²𝑑𝑡+¹₋₁𝐾()𝑄(𝑡)(𝑥),(),𝑡,𝑥𝑑𝑡=𝑓𝑥∈−1,1 (1) where 𝑓(𝑥) is a given function, the unknown function 𝑄 satisfies the boundary conditions 𝑄(±1)=0, and the kernel function 𝐾(𝑡,𝑥) satisfies a Hölder continuous first-derivative condition. The improper integral in the left side ⨎ is defined as the finite part of the strongly singular integral in the sense of Hadamard which is defined as [11] ¹₋₁𝑓(𝑡)𝑑𝑡(𝑡−𝑥)²=lim_𝜀→0^𝑥−𝜀₋₁+¹_𝑥+𝜀𝑓(𝑡)𝑑𝑡(𝑡−𝑥)²−2𝑓(𝑥)𝜀. (2) One of the main concepts in the derivations is that the higher order singularity could be obtained from a lower order singularity by the accepted exchangeability of integration and differentiation [12]: ¹₋₁𝑓(𝑡)(𝑡−𝑥)²𝑑𝑑𝑥=𝑑𝑥¹₋₁𝑓(𝑡)()().𝑡−𝑥𝑑𝑡,𝑥∈−1,1 (3) The objective of this paper is to develop an expansion method of HSIE in (1), for an effective error estimate, that is simple and easy to use. The numerical solution of HSIE of the first kind is given in Section 2. Section 3 presents the convergence of the numerical method in the class of function 𝐻^𝛼([−1,1]) and 𝐿₂([−1,1]). Moreover, the numerical results in Section 4 show that the constructed technique is fast and provides excellent accuracy where the error is almost zero. Section 5 presents the conclusion of this paper.

2. Description of the General Method

Direct numerical treatment of (1) is not simple. Many researchers applied the classical method by approximating the unknown function 𝑄 using the finite sum of Chebyshev polynomial of the second kind of the form: 𝑄_𝑛(𝑡)=1−𝑡^2𝑛_𝑖=0𝐶_𝑖𝑈_𝑖(𝑡),|𝑡|<1, (4) where 𝐶_𝑖, 𝑖=0,1,2,…,𝑛, are the unknown coefficients and 𝑈_𝑛(𝑡) is the Chebyshev polynomial of the second kind: 𝑈_𝑛(𝑡)=sin(𝑛+1)𝜃sin𝜃,𝑡=cos𝜃,0≤𝜃≤𝜋. (5) The second kind Chebyshev polynomial is orthogonal on [−1,1] with the weight 𝑤(𝑡)=√1−𝑡² means ¹₋₁√1−𝑡²𝑈_𝑖(𝑡)𝑈_𝑗(𝑡)𝑑𝑡=𝜋2𝛿_𝑖𝑗, (6) where 𝛿_𝑖𝑗 is the Kronecker delta and the recurrence relation is defined as follows: 𝑈_𝑛(𝑡)=2𝑡𝑈_𝑛−1(𝑡)−𝑈_𝑛−2(𝑡), (7) with the initial values 𝑈₀(𝑡)=1 and 𝑈₁(𝑡)=2𝑡.

One could look for the hypersingular kernel in (1) in the form of the series approximation of the form [5] 1(𝑡−𝑥)²≈−2^𝑚_𝑗=0𝑗+1𝑈_𝑗(𝑥)𝑈_𝑗(𝑡), (8) while the regular kernel 𝐾(𝑡,𝑥)=^𝑝_𝑟=0𝜌_𝑟(𝑥)𝑈_𝑟(𝑡), (9) substituting (4), (8), and (9) into (1) and using the orthogonal property in (6) yield ^𝑛_𝑖=0𝐶_𝑖−(𝑖+1)𝑈_𝑖(𝑥)+𝜋2𝜌_𝑖(𝑥)=𝑓(𝑥). (10) From the approximation (9) and by applying the scalar product to the functions 𝑈_𝑗(𝑡)(𝑗=0,1,…,𝑛) and integrating both sides with respect to 𝑡, we obtain the linear system 𝜌_𝑗(𝑥)=2𝜋¹₋₁√1−𝑡²𝐾(𝑡,𝑥)𝑈_𝑗(𝑡)𝑑𝑡. (11) Choosing the roots of 𝑇_𝑛+1(𝑥) as the collocation points 𝑥_𝑗 along the interval [−1,1], which are 𝑥_𝑗=cos2𝑗−12()𝜋𝑛+1,𝑗=0,1,…,𝑛, (12) one could reduce initial integral equation (1) to the finite linear algebraic system of 𝑛+1 linear equations with 𝑛+1 unknown coefficients 𝐶_𝑖 of the form ^𝑛_𝑖=0𝐶_𝑖−(𝑖+1)𝑈_𝑖𝑥_𝑗+𝜋2𝜌_𝑖𝑥_𝑗=𝑓𝑥_𝑗, (13) where the calculation of 𝐶_𝑖 endorses the evaluation of 𝑄_𝑛(𝑡) in (4).

3. Convergence Rates

In this section, we illustrate the steps of the error estimate for the approximate solution of our interest in HSIE of the form in (1). First, we recap the equation concerning the following operators: (𝑥)=1𝐻𝑄𝜋¹₋₁√1−𝑡²𝑄(𝑡)(𝑡−𝑥)²(𝑥)=𝑑𝑡,𝐾𝑄¹₋₁√1−𝑡²𝐾(𝑡,𝑥)𝑄(𝑡)𝑑𝑡. (14) Taking into consideration (14), the operator equation of (1) could be rewritten as (𝐻+𝐾)𝑄(𝑥)=𝑓(𝑥). (15) To solve a convergence problem of 𝑄_𝑛(𝑡), we should prove that 𝑄_𝑛(𝑡) satisfies the operator equation in (15).

Theorem 1. If 𝑓(𝑥)∈𝐶^ℓ[−1,1] and 𝑘(𝑡,𝑥)∈𝐶^ℓ([−1,1]×[−1,1]), ℓ≥1, then 𝑄−𝑄_{𝑛𝐿2,1−𝑡2}=𝑂𝑛^−ℓ. (16)

Proof. Using the relation [12, 13] ¹₋₁1−𝑡²𝑈_𝑘(𝑡)(𝑡−𝑥)²𝑑𝑡=−𝜋(𝑘+1)𝑈_𝑘(𝑡), (17) along with the approximation in (4), yields 𝐻𝑈_𝑘=−(𝑘+1)𝑈_𝑘. (18) Let us denote the space 𝐿_2,𝑤 of real valued functions square integrable with respect to the weight function 𝑤=√1−𝑥² as 𝐿_2,𝑤=ℎ(𝑥)∣¹₋₁𝑤(𝑥)|ℎ(𝑥)|²𝑑𝑥<∞. (19) Based on (19), the inner product might be defined as follows: 𝑄,𝑔_𝑤=¹₋₁𝑄(𝑥)𝑔(𝑥)𝑤(𝑥)𝑑𝑥,∀𝑄,𝑔∈𝐿_2,𝑤. (20) The set {𝑈_𝑘}^∞_𝑘=0 is the orthogonal basis in 𝐿_2,𝑤, so that if 𝑢∈𝐿_2,𝑤 then 𝑢=^∞_𝑛=0𝑢,𝑈_𝑛𝑤𝑈_𝑛, (21) where the sum converge in 𝐿_2,𝑤. Then we would need to use a subspace of 𝐿_2,𝑤 which contains all 𝑢, such that ^∞_𝑛=0(𝑛+1)²𝑢,𝑈_𝑛²_𝑤<∞. (22) We define this set by 𝐿_1,𝑤, and it is made into Hilbert space by the following inner product: ⟨𝑢,𝑣⟩₁=^∞_𝑛=0(𝑛+1)²𝑢,𝑈_𝑛𝑤𝑣,𝑈_𝑛𝑤, (23) where the norm of 𝑣∈𝐿_1,𝑤 is defined by ‖𝑢‖²₁=^∞_𝑛=0(𝑛+1)²𝑢,𝑈_𝑛²_𝑤. (24) Assume that 𝑣_𝑛=(𝑈_𝑛/(𝑛+1)) and note that ‖𝑣_𝑛‖₁=1, therefore {𝑣_𝑛}^∞_𝑛=0 from an orthonormal basis of 𝐿_1,𝑤, and if 𝑢∈𝐿_1,𝑤 then 𝑢=^∞_𝑛=0𝑢,𝑣_𝑛1𝑣_𝑛. (25) To show that 𝐻⁻¹∶𝐿_2,𝑤→𝐿_1,𝑤 is unitary, let us consider expansions of 𝑓∈𝐿_2,𝑤 and 𝑔∈𝐿_1,𝑤. It is not difficult to see that 𝐻𝑓(𝑥)=^∞_𝑛=0−(𝑛+1)𝐶_𝑛𝑈_𝑛(𝑥)=𝑔(𝑥), (26) where 𝐶_𝑛=⟨𝑓,𝑈_𝑛⟩, and we use the expansion of 𝑔, such that 𝑔(𝑥)=^∞_𝑛=0𝑏_𝑛𝑈_𝑛(𝑥). (27) Then −(𝑛+1)𝐶_𝑛=𝑏_𝑛 which gives 𝐶_𝑛𝑏=−^𝑛.𝑛+1 (28) So, 𝑓(𝑥)=^∞_𝑛=0−𝑏_𝑛𝑈𝑛+1^𝑛(𝑥). (29) We have 𝑓=𝐻⁻¹𝑔, and by using (29), becomes 𝑓=𝐻⁻¹𝑔(𝑥)=^∞_𝑛=0−𝑏_𝑛𝑈𝑛+1^𝑛(𝑥). (30) For the approximation in (4), the coefficients {𝐶_𝑖}^𝑛_𝑖=0 satisfy 𝐻𝑄_𝑛+𝐾𝑄_𝑛−𝑓,𝑄_𝑘(𝑥)=0,𝑘=0,1,2,…. (31) To show that {𝑄_𝑛} converges, it should verify the operator equation in (15).
Let 𝑃_𝑛 be the projection operator to the span of {𝑈₀,𝑈₁,…,𝑈_𝑛}; then, the orthogonal projection space 𝑃_𝑛∶𝐿_2,𝑤→𝐿_2,𝑤, satisfies [14] 𝑃_𝑛𝐻𝑄_𝑛+𝐾𝑄_𝑛−𝑓=0. (32) Since 𝑃_𝑛𝐻𝑄_𝑛=𝐻𝑄_𝑛, (32) becomes 𝐻𝑄_𝑛+𝑃_𝑛𝐾𝑄_𝑛=𝑃_𝑛𝑓,𝐻𝑄+𝑃_𝑛𝐾𝑄=𝑓−𝐾𝑄+𝑃_𝑛𝐾𝑄,𝐻+𝑃_𝑛𝐾𝑄=𝑓−𝐾𝑄+𝑃_𝑛𝐾𝑄. (33) We know that 𝐻+𝑃_𝑛𝐾𝑄_𝑛=𝑃_𝑛𝑓, (34) which gives 𝐻+𝑃_𝑛𝐾𝑄−𝑄_𝑛=𝑓−𝐾𝑄+𝑃_𝑛𝐾𝑄−𝑃_𝑛𝑓=𝐻𝑄−𝑃_𝑛𝐻𝑄. (35)𝑃_𝑛∶𝐿_2,𝑤→Ω=span{𝜙₁,𝜙₂,…,𝜙_𝑛}. Let 𝑔∈𝐿_2,𝑤, then 𝑃_𝑛𝑔=ℎ=𝐶₁𝜙₁+𝐶₂𝜙₂+⋯+𝐶_𝑛𝜙_𝑛. From (18), we could extend 𝐻 as a bounded operator from 𝐿₂(𝑝) to 𝐿_1,𝑤; then using (20), 𝑃_𝑛(𝐻+𝐾)𝑄_𝑛−𝑓=0. (36) Since 𝑃_𝑛(𝐻+𝐾)𝑄_𝑛−𝑓=^𝑛_𝑘=0𝛼_𝑘𝜙_𝑘,𝛼_𝑘≠0, (37) then (𝐻+𝐾)𝑄_𝑛−𝑓,𝜙_𝑘=0,𝑘=0,1,2,…,𝑛. (38) Applying the approximate solution in (4) to the inner product in (20), so that 𝐻𝑄_𝑛+𝐾𝑄_𝑛−𝑓,𝑈_𝑘=0, (39) then the bounded operator 𝐻 could be written as ^𝑛_𝑘=0−(𝑘+1)𝐶_𝑘𝑈_𝑘(𝑥)+^𝑛_𝑘=0𝑔_𝑘(𝑡)𝑈_𝑘(𝑥)=⟨𝑓,𝑈_𝑘⟩,𝑓,𝜙_𝑘=^𝑛_𝑘=0𝑔_𝑘(𝑡)−(𝑘+1)𝐶_𝑘𝑈_𝑘(𝑥),ℎ,𝑈_𝑘=0. (40) We define the projection Ω_𝑛 such that Ω_𝑛=ℎ∣ℎ⟂𝑈_𝑘,𝐿,𝑘=0,1,…,𝑛_2,𝑤=Ω₁⊕Ω₂⟶ℎ=ℎ₁+ℎ₂,Ω₁=ℎ₁∈𝐿_2,𝑤∣ℎ₁=^𝑛_𝑘=0𝐶_𝑘𝜙_𝑘=span𝑄₁,…,𝑄_𝑛,Ω₂=ℎ₂∣ℎ₂⟂ℎ₁;ℎ₁∈Ω₁,𝑃_𝑛∶𝐿_2,𝑤⟶Ω₁, (41) then 𝐻𝑄_𝑛=^𝑛_𝑘=0−(𝑘+1)𝐶_𝑘𝑈_𝑘(𝑥)∈Ω₁. (42) Since 𝐻𝑄∈Ω₁, 𝑃_𝑛 is a linear operator that gives (35).
For 𝐻⁻¹, 𝐻𝑔=−^𝑛_𝑘=0(𝑘+1)𝑔_𝑘𝑈_𝑘(𝑥)⟹𝐻⁻¹ℎ=^𝑛_𝑘=01−()ℎ𝑘+1^𝑘𝑈_𝑘(𝑥). (43) It is known that 𝐻∗𝐻⁻¹=𝐼.
Let 𝜙∈𝐿_2,𝑤 and 𝜙=ℎ₁+ℎ₂; then the projection operator 𝑃_𝑛 is as follows: 𝑃_𝑛𝜙=ℎ₁⊂Ω₁. (44) It is well known that 𝑃_𝑛 is bounded operator: 𝑃_𝑛𝜙≤𝜙,∀𝜙∈𝐿_2,𝑤. (45) Let us consider 𝐻+𝑃_𝑛𝐾 operator in 𝐿_2,𝑤: 𝑃_𝑛𝐾<1. (46) We choose 𝑛₀, such that for all 𝑛≥𝑛₀, one has 𝑃_𝑛𝜙<1. (47) Let us fix the element, 𝜙∈𝐿_2,𝑤, and by considering the operator: 𝐵𝑔=𝐻⁻¹𝜙−𝑃_𝑛𝐾𝑔. (48) This operator is contraction: 𝐵𝑔₁−𝐵𝑔₂<𝛼𝑔₁−𝑔₂,0<𝛼<1. (49) Since 𝐻⁻¹𝜙−𝐻⁻¹𝑃_𝑛𝐾𝑔₁−𝐻⁻¹𝜙+𝐻⁻¹𝑃_𝑛𝐾𝑔₂=𝐻⁻¹𝑃_𝑛𝐾𝑔₂−𝑔₁≤𝐻⁻¹𝑃_𝑛𝐾𝑔₂−𝑔₁,𝛼=𝐻⁻¹𝑃_𝑛𝐾<1. (50) Since 𝐵-contraction operator, then it has an unique fixed point 𝑔: 𝑔=𝐻⁻¹𝜙−𝑃_𝑛𝐾𝑔. (51) By applying 𝐻 to both sides, we obtain 𝐻𝑔=𝜙−𝑃_𝑛𝐾𝑔 (52) or 𝜙=𝐻𝑔+𝑃_𝑛𝐾𝑔=𝐻+𝑃_𝑛𝐾𝑔. (53) For any 𝜙∈𝐿_2,𝑤, there exists an unique 𝑔∈𝐿_2,𝑤, such that 𝜙=(𝐻+𝑃_𝑛𝐾)𝑔, which means that (𝐻+𝑃_𝑛𝐾)⁻¹ exists: 𝐻+𝑃_𝑛𝐾𝑄−𝑄_𝑛=𝐻𝑄−𝑃_𝑛𝐻𝑄,𝑄−𝑄_𝑛=𝐻+𝑃_𝑛𝐾⁻¹𝐻𝑄−𝑃_𝑛𝐻𝑄. (54) To show that our operator 𝐻+𝑃_𝑛𝐾 is bounded, we apply the following theorem [15].

Theorem 2. Let 𝐿 be a Banach space, and let 𝐼 be the identity operator on 𝐿. Suppose that 𝐴 is a bounded linear operator mapping 𝐿 into itself, such that ‖𝐴‖<1. (55) Then the operator (𝐼−𝐴)⁻¹ that exists is bounded and could be represented in the form (𝐼−𝐴)⁻¹=^∞_𝑘=0𝐴^𝑘. (56)

For the details of Theorem 2, see [15].

Let us consider that 𝐻+𝑃_𝑛𝐾=𝐻𝐼+𝐻⁻¹𝑃_𝑛𝐾. (57) Obviously we have 𝐻⁻¹𝑃_𝑛𝐾<1. (58) It follows that 𝐻+𝑃_𝑛𝐾⁻¹=𝐻𝐼+𝐻⁻¹𝑃_𝑛𝐾⁻¹=𝐼+𝐻⁻¹𝑃_𝑛𝐾⁻¹𝐻⁻¹=^∞_𝑗=0(−1)^𝑗𝐻⁻¹𝑃_𝑛𝐾^𝑗𝐻⁻¹. (59) Applying Theorem 2 yields 𝐻+𝑃_𝑛𝐾⁻¹=^∞_𝑗=0𝐻^−𝑗−1𝑃_𝑛𝐾^𝑗. (60) The norm would be 𝐻+𝑃_𝑛𝐾⁻¹≤^∞_𝑗=0𝐻^−𝑗−1𝑃_𝑛𝐾^𝑗𝜙≤^∞_𝑗=0𝐻⁻¹⋅𝐻⁻¹𝑃_𝑛𝐾^𝑗. (61) Since 𝐻⁻¹𝑃_𝑛𝐾=𝑞<1, (62) then 𝐻+𝑃_𝑛𝐾⁻¹=𝐻^−1∞_𝑗=0𝑞^𝑗=𝐻⁻¹1,1−𝑞𝐻+𝑃_𝑛𝐾⁻¹=𝐻⁻¹1−𝐻⁻¹𝑃_𝑛𝐾=𝐶𝑄−𝑄_𝑛2,𝑤≤𝐻+𝑃_𝑛𝐾⁻¹⋅𝐻𝑄−𝑃_𝑛𝐻𝑄_2,𝑤≤𝐶𝐻𝑄−𝑃_𝑛𝐻𝑄_2,𝑤. (63) Applying (19) to (60) gives 𝑄−𝑄_𝑛2=𝐶¹₋₁𝐻𝑄−𝑃_𝑛𝐻𝑄²𝑤(𝑥)𝑑𝑥^1/2≤𝐶¹₋₁𝐻𝑄(𝑥)−𝜂_𝑛(𝑥)²𝑤(𝑥)𝑑𝑥^1/2≤𝐶𝐻𝑄(𝑥)−𝜂_𝑛(𝑥)_𝐿∞¹₋₁𝑤(𝑥)𝑑𝑥^1/2. (64) Applying Jackson’s theorem [14] yields 𝑄−𝑄_𝑛2=𝐶𝑛^−ℓ. (65) From (64), Theorem 1 is proved.

4. Numerical Example

Here, we illustrate the above method to obtain approximate numerical solution of Fredholm integral equation.

Example 1. We consider Fredholm full equation of the form 1𝜋¹₋₁√1−𝑡²𝑔(𝑡)(𝑡−𝑥)²+𝑑𝑡16𝜋¹₋₁√1−𝑡²𝑥³𝑡³𝑔(𝑡)𝑑𝑡=16𝑥−31𝑥³. (66) The exact solution of (66) is 𝑔(𝑡)=8𝑡³−4𝑡. (67) By using the method in Section 2, if we approximate 𝑔(𝑡) using Chebyshev polynomials of the second kind, 𝑔(𝑡)=^𝑛_𝑖=0𝐶_𝑖𝑈_𝑖(𝑡), (68) and 𝐶_𝑖 satisfies the linear system in (10).
According to (11) and (7), 𝜌_𝑗(𝑥)=4𝑥³𝜋²¹₋₁√1−𝑡²𝑈₃(𝑡)+2𝑈₁(𝑡)𝑈_𝑗(𝑡)𝑑𝑡. (69) Using the orthogonal property in (6), we obtain 𝜌₀(𝑥)=0,𝜌₁(𝑥)=4𝑥³𝜋,𝜌₂(𝑥)=0,𝜌₃(𝑥)=2𝑥³𝜋,𝜌_𝑗(𝑥)=0,𝑗≥4. (70) Substituting the above result in the system of equations into (13) for 𝑛=3 gives ³_𝑖=0𝐶_𝑖−(𝑖+1)𝑈_𝑖𝑥_𝑗+2𝐶₁𝑥³_𝑗+𝐶₃𝑥³_𝑗=16𝑥_𝑗−31𝑥³_𝑗,𝑖=0,1,2,3,−31𝐶₃+2𝐶₁𝑥³_𝑗−12𝐶₂𝑥²_𝑗+16𝐶₃−4𝐶₁𝑥_𝑗+3𝐶₂−𝐶₀=16𝑥_𝑗−31𝑥³_𝑗,𝑗=0,1,2,3. (71) Using the comparison method for solving the above system of equation leads to −31𝐶₃+2𝐶₁=−31,−12𝐶₂=0,16𝐶₃−4𝐶₁=16,3𝐶₂−𝐶₀=0, (72) then the solution of the above system is 𝐶₀=𝐶₁=𝐶₂=0,𝐶₃=1. (73) Substituting the values of (73) into (68) gives the numerical solution of (66) which is identical to the exact solution.

Example 2. We solve Fredholm HSIE of the form 1𝜋¹₋₁𝑄(𝑡)(𝑡−𝑥)²𝑑𝑡+¹₋₁𝑒^𝑥𝑄(𝑡)𝑡−𝑥𝑑𝑡=𝜋𝑒^𝑥𝑥−4𝑥³−12𝑥²+1. (74) Note that, by the help of (3), the exact solution of (74) is 𝑄(𝑥)=√1−𝑥²4𝑥²+1. (75) From (11), 𝜌_𝑖(𝑥)=2𝜋¹₋₁√1−𝑡²𝑒^𝑥𝑈𝑡−𝑥^𝑖(𝑡)𝑑𝑡. (76) It is well known that [12, 13] ¹₋₁√1−𝑡²𝑈_𝑖−1(𝑡)𝑡−𝑥𝑑𝑥=−𝜋𝑇_𝑖(𝑥), (77) so that 𝜌_𝑖(𝑥)=−2𝑒^𝑥𝑇_𝑖+1(𝑥), (78) where 𝑇_𝑖 is the Chebyshev polynomial of the first kind defined: 𝑇_𝑛(𝑡)=cos𝑛𝜃,𝑡=cos𝜃,0≤𝜃≤𝜋. (79) This polynomial is orthogonal on [−1,1] with respect to 𝑤(𝑡)=(1/√1−𝑡²), and the terms could be found using the recurrence relation: 𝑇_𝑛(𝑡)=2𝑡𝑇_𝑛−1(𝑡)−𝑇_𝑛−2(𝑡), (80) where 𝑇₀(𝑡)=1 and 𝑇₁(𝑡)=𝑡 are the starting values of (80), which provides the values of 𝜌_𝑖(𝑥) for a certain 𝑖: 𝜌₀(𝑥)=−2𝑥𝑒^𝑥,𝜌₁(𝑥)=−22𝑥²−1𝑒^𝑥,𝜌₂(𝑥)=−24𝑥³−3𝑥𝑒^𝑥,𝜌₃(𝑥)=−28𝑥⁴−8𝑥²+1𝑒^𝑥,…. (81) Substituting the values of (81) into the system of (13) for 𝑛=4 gives ⁴_𝑖=0𝐶_𝑖−(𝑖+1)𝑈_𝑖𝑥_𝑗+𝜋2𝜌_𝑖𝑥_𝑗=𝜋𝑒^𝑥𝑗𝑥_𝑗−4𝑥³_𝑗−12𝑥²_𝑗+1,𝑗=0,1,2,3,4. (82) Using (12) as a collocation points for 𝑗=4, give the required system of unknown coefficients 𝐶_𝑖, 𝑖=0,…,4. The solution of the system in (82) gives 𝐶₀=2,𝐶₂=1,𝐶₁=𝐶₃=𝐶₄=0. (83) Substituting the above values into (4), we obtain the numerical solution of (74), which is identical to the exact solution. The error of the numerical solution (4) of (74) is given in Table 1.

Table 1 

The error of the numerical solution of (74).

5. Conclusion

We have investigated the numerical solutions for the bounded hypersingular integral equation of the first kind using an efficient expansion method. We are able to prove the convergence of our method for the function 𝑓∈𝐶^ℓ([−1,1]) in the class of functions 𝐻^𝛼([−1,1]) and 𝐿₂([−1,1]). Moreover, the error in (65) decreases very quickly and the convergence is very fast to the exact solution even when 𝑥 is close to the end points. In particular, if ℓ in the relation (65) can be chosen to be any large positive number, then the error decreases rapidly as 𝑛 increases. Then the sequence {𝑄_𝑛} converges uniformly in 𝐿_2,𝑤 norm to {𝑄}. The numerical examples give the exact solution for different kernels 𝐾(𝑥,𝑡) and the unknown function 𝑄. Therefore, referring to the computed numerical solution, we are able to obtain a very good convergence for any singular point 𝑥∈(−1,1). MATLAB codes are developed to obtain the numerical results where the numerical experiments reaffirm the theoretical results.