A new effective weighted modified perturbation technique for solving a class of hypersingular integral equations

TL;DR

This paper introduces a new weighted modified perturbation technique for efficiently solving hypersingular integral equations of the second kind, demonstrating its simplicity and fast computation through practical examples.

Contribution

The paper presents a novel weighted and modified perturbation method that encompasses special cases of the Adomian decomposition, improving solution efficiency for hypersingular integral equations.

Findings

Method is simple and easy to implement.

Demonstrates fast computational performance.

Effective in solving hypersingular integral equations.

Abstract

This paper is an attempt to solve an important class of hypersingular integral equations of the second kind. To this end, we apply a new weighted and modified perturbation method which includes some special cases of the Adomian decomposition method. To justify the efficiency and applicability of the proposed method, we examine some examples. The principal aspects of this method are its simplicity along with fast computations.

Equations

$$u(x)=\sqrt{1-x^2}f(x)+\frac{\alpha }{\pi }\sqrt{1-x^2}{\int }_{-1}^1\frac{u(t)}{(t-x{)}^n}dt,n=2,3,\dots ,-1<x<1,$$

$$\int_{-1}^{1}\frac{u(t)}{(t-x)^{2}}dt=\lim_{\epsilon\rightarrow 0^{+}}\bigg{[}\int_{-1}^{x-\epsilon}\frac{u(t)}{(t-x)^{2}}dt+\int_{x+\epsilon}^{1}\frac{u(t)}{(t-x)^{2}}dt-\frac{u(x+\epsilon)+u(x-\epsilon)}{\epsilon}\bigg{]}.$$

$$u(x)=\sqrt{1-x^2}\psi (x),$$

$$\psi (x)=f(x)+\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}\psi (t)}{t-x}dt.$$

$$I_j(x)={\int }_{-1}^1\frac{\sqrt{1-t^2}}{t-x}{t}^{j}dt,j=0,1,\cdots ,$$

$$I_j(x)=-\pi x^{j+1}+\sum _{i=0}^{j-1}\frac{1+(-1{)}^i}{4}\frac{\Gamma (1/2)\Gamma ((i+1)/2)}{\Gamma ((i+4)/2)}{x}^{j-i-1}j=0,1,\cdots ,$$

$$I_{2k}(x)=x{I}_{2k-1}(x).$$

$$\begin{split}I_{2k}(x)&=-\pi x^{2k+1}+\sum_{i=0}^{2k-1}\frac{1+(-1)^{i}}{4}\frac{\Gamma(1/2)\Gamma((i+1)/2)}{\Gamma((i+4)/2)}x^{2k-i-1}\\ &=x\big{(}-\pi x^{2k}+\sum_{i=0}^{2k-2}\frac{1+(-1)^{i}}{4}\frac{\Gamma(1/2)\Gamma((i+1)/2)}{\Gamma((i+4)/2)}x^{2k-i-2}\big{)}=xI_{2k-1}(x)\end{split}$$

$$L(\psi )=L_1(\psi )+L_2(\psi )=f(x).$$

$$L_1(\psi )=f(x)+\epsilon L_2(\psi ).$$

$$L_1(\psi )=\psi (x),L_2(\psi )=-\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}\psi (t)}{t-x}dt,$$

$$\psi (x)=\underset{\epsilon \to 1}{\lim }\sum _{k=0}^{\infty }{\psi }_k(x){\epsilon }^k.$$

$$\sum _{k=0}^{\infty }{\psi }_k(x){\epsilon }^k=f(x)+\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}\sum _{k=0}^{\infty }{\epsilon }^{k+1}{\int }_{-1}^1\frac{\sqrt{1-t^2}{\psi }_k(t)}{t-x}dt.$$

$$\{\begin{array}{ll}{\epsilon }^0:& {\psi }_0(x)=f(x),\\ {\epsilon }^k:& {\psi }_k(x)=\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}}{t-x}{\psi }_{k-1}(t)dt,k=1,2,\cdots .\end{array}$$

$${\psi }_k(x)=\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}}{t-x}{\psi }_{k-1}(t)dt,k=1,2,\dots ,$$

$$L_1(\psi )=\psi ,L_2(\psi )=-\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}\psi }{t-x}dt+f_2(x).$$

$$\{\begin{array}{l}{\psi }_0(x)=f_1(x),\\ {\psi }_1(x)=f_2(x)+\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}}{t-x}{\psi }_0(t)dt,\\ {\psi }_k(x)=\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}}{t-x}{\psi }_{k-1}(t)dt,k=2,3,\cdots .\end{array}$$

$${\psi }_{n+1}(x)=\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}}{t-x}{\psi }_n(t)dt,n=1,2\dots ,$$

$$f(x)=\sum _{i=0}^m{a}_i{x}^i.$$

$$f_1(x)$$

$$f_2(x)$$

$$\begin{array}{r}u(x)=2\pi \sqrt{1-x^2}+\frac{1}{2}\sqrt{1-x^2}{\int }_{-1}^1\frac{u(t)}{(t-x{)}^2}dt,-1<x<1.\end{array}$$

$$\begin{array}{r}\psi (x)-\frac{1}{2}\frac{d}{dx}{\int }_{-1}^1\frac{\sqrt{1-t^2}\psi (t)}{t-x}dt=2\pi ,-1<x<1.\end{array}$$

$$\begin{array}{r}{\psi }_n(x)=\frac{(-1{)}^n{\pi }^{n+1}}{2^{n-1}},n=0,1,2,\cdots .\end{array}$$

$$\begin{array}{rl}& \psi (x)=\sum _{n=0}^{\infty }\frac{(-1{)}^n{\pi }^{n+1}}{2^{n-1}},\end{array}$$

$$\begin{array}{rl}& {\psi }_0(x)=b_0,{\psi }_1(x)=2\pi -b_0-b_0\frac{\pi }{2}.\end{array}$$

$$\begin{array}{r}{b}_0=\frac{4\pi }{\pi +2}.\end{array}$$

$$\begin{array}{r}\psi (x)=\frac{4\pi }{\pi +2},\end{array}$$

$$\begin{array}{r}u(x)=\frac{4\pi }{\pi +2}\sqrt{1-x^2},\end{array}$$

$$\begin{split}u(x)=\big{(}a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{m}x^{m}\big{)}\sqrt{1-x^{2}}+\frac{\alpha}{\pi}\sqrt{1-x^{2}}\int_{-1}^{1}\frac{u(t)}{(t-x)^{n}}dt,\ \ n\geq 2,\ \ \ -1<x<1.\end{split}$$

$$\begin{array}{r}\psi (x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots +a_m{x}^m+\frac{\alpha }{\pi \Gamma (n)}\frac{d^{n-1}}{d{x}^{n-1}}{\int }_{-1}^1\frac{\sqrt{1-t^2}\psi (t)}{(t-x)}dt=,-1<x<1.\end{array}$$

Full text

Keywords: Hypersingular integral equations, perturbation method, modified perturbation method, weighted perturbation method, Adomian decomposition method.

1 Introduction

Enormous physical and engineering problems, such as crack problems in fracture mechanics [4], water wave scattering problems involving barriers [5], diffraction of electromagnetic wave and aerodynamic problems [6, 7], generalization of the elliptic wing case of Prandtl’s equation [8, 9] are hypersingular integral equations of the second kind. For more applications one can consult [10, 11, 12, 13, 16].

In this work, we consider the following hypersingular integral equation of the second kind

[TABLE]

where $\alpha$ is a real positive number, $f(x)$ is a given function, and $u(x)$ is unknown. Moreover, $u(\pm 1)=0$. Under these assumptions, Eq. (1.1) can be called the generalized case for the oval wing of Prandtl’s equation. In addition, Eq. (1.1) is referred to as Hadamard finite part [7]

[TABLE]

Recently, Mahmoudi introduced an efficient method to solve (1.1) based on the weighted modified Adomian’s decomposition method (WMADM). Motivated by his work, we attempt to solve Eq. (1.1) by applying the weighted modified perturbation method (WMPM), which includes the mentioned method as the special case. The rest of this paper is organized as follows: Section 2 is devoted to the description of the solution method. It contains the classic perturbation method (PM) which does not work even for simple examples here. Therefore, we try to put forward the modified and weighted modified version of the classic PM in such a way that it is possible to find a solution quickly. In other words, the introduced weighted modified version can save many of the computations and prevents the construction of commonly used series solutions which can be considered other virtues for these methods. To support the given method, some applications and illustrations are illustrated in Section 3 . Finally, Section 4 includes some concluding remarks.

2 Description of the solution method

This section deals with applying the perturbation technique for solving hypersingular integral equation of the second kind (1.1). Before describing the whole process, for the sake of simplicity in computations [7], in Eq. (1.1) it is assumed that:

[TABLE]

where $\psi(x)$ is a smooth function. Substituting (2.1) into (1.1) and simplifying, it induces:

[TABLE]

We need the following lemma [8]:

Lemma 2.0.1**.**

If

[TABLE]

then

[TABLE]

where $\Gamma(n)$ stands for the Gamma function.

Also, we have the following lemma:

Lemma 2.0.2**.**

For $k\geq 1$,

[TABLE]

Proof.

By (2.3), we have:

[TABLE]

∎

Here, we try to explain the new method for solving (1.1). For this purpose, based upon the Eq. (2.2), we define the operator:

[TABLE]

Next, we consider the perturbation technique as follows:

[TABLE]

Now, the stage is ready for computing with the perturbation technique. Considering Eq. (2.2), we set:

[TABLE]

where

[TABLE]

The series (2.9) converges to the exact solution of Eq. (2.2) provided that such a solution exists [3, 15]. Substituting (2.8) and (2.9) into (2.7), one obtains

[TABLE]

Finally, equating the terms with like powers of the embedding parameter $\varepsilon$, we can compute $\psi_{n}$ by the following recurrence relation:

[TABLE]

Note that this method generates the Adomian decomposition method (ADM), and we state this fact formally in the following theorem.

Theorem 2.0.3**.**

If $\psi_{0}(x)=f(x)$, then the relation

[TABLE]

generates Adomian decomposition method for solving (1.1).

2.1 Modified perturbation method

In some situations, Formula (2.11) does not work perfectly or the solution series converges slowly. To overcome these difficulties, it is still possible to modify the proposed perturbation technique (2.11) as follows. First, we split $f(x)$ as $f(x)=f_{1}(x)+f_{2}(x)$. Then, we set

[TABLE]

At once, we can modify (2.11) and call it modified perturbation technique given by:

[TABLE]

Similar to the Theorem (2.0.3), we have:

Theorem 2.1.1**.**

If $\psi_{0}(x)=f_{1}(x)$, $\psi_{1}(x)=f_{2}(x)+\frac{\alpha}{\pi\Gamma(n)}\frac{d^{n-1}}{dx^{n-1}}\int_{-1}^{1}\frac{\sqrt{1-t^{2}}}{t-x}\,\psi_{0}(t)\,dt$, then

[TABLE]

generates the modified Adomian decomposition method (MADM) for solving (1.1).

2.2 Weighted modified perturbation method

Choosing functions $f_{1}(x)$ and $f_{2}(x)$ in applying the modified perturbation technique deeply affects the rate of convergence. Generally speaking, they should be chosen so that $\psi_{k}=0$ for $k=1,2,\cdots$. To select these functions effectively, the weighted modified perturbation technique is advised. So, we pay attention to this issue when $f(x)$ is a polynomial and leave the other cases for future research. More details can be found in [1, 2, 14, 16].

Suppose that $f(x)$ in (2.2) is a polynomial with degree $m$. Hence, $\psi(x)$ must be a polynomial of the same degree. In other words, let:

[TABLE]

It is possible to write $f(x)=f_{1}(x)+f_{2}(x)$, where

[TABLE]

The unknowns $b_{i}$, $i=0,1,\cdots,m$ are determined in the modified perturbation technique (2.13)in such a way that $\psi_{1}(x)=0$ . This process is called the weighted modified perturbation technique and will be illustrated in the next section.

3 Applications and illustrations

This section is concerned with applying the modified perturbation technique (2.11) or the weighted modified perturbation technique (2.13) to solve some concretes of the hypersingular integral equation (1.1) or equivalents (2.2).

Example 3.1**.**

[14] Consider:

[TABLE]

Substituting $u(x)=\sqrt{1-x^{2}}\psi(x)$ into (3.1), one obtains

[TABLE]

By perturbation technique (2.11), Lemma (2.0.2), and (3.2), we have:

[TABLE]

Therefore, from (2.9) we have:

[TABLE]

which is a divergent series since $\lim_{n\to\infty}{\psi_{n}}\neq 0$.

So, we apply the weighted modified perturbation technique (2.13). Let $f_{1}(x)=b_{0}$ and $f_{2}(x)=2\pi-b_{0}$. Thus

[TABLE]

Putting $\psi_{1}(x)=0$ leads to:

[TABLE]

Consequently,

[TABLE]

and

[TABLE]

which are the exact solutions of the equivalent hypersingular integral equations (3.2) and (3.1), respectively.

Example 3.2**.**

In this example, we try to solve the general case of the hypersingular integral equation by using the weighted modified perturbation technique. Let:

[TABLE]

Substituting $u(x)=\sqrt{1-x^{2}}\psi(x)$ into (3.8) causes

[TABLE]

Set

[TABLE]

Therefore, to obtain $\psi_{1}(x)=0$ , by Lemma (2.0.2) and the weighted modified perturbation technique (2.13), we acquire

[TABLE]

Consequently, the exact solution of the integral equation (3.8) is given by:

[TABLE]

4 Concluding remarks

This paper has introduced a new method for solving an important class of hypersingular integral equations of the second kind. The main features of the proposed method are avoiding complicated procedures and having simplicity. Using an example, we have shown that the classic perturbation method does not work while its weighted modified version can remedy this issue.

5 Acknowledgments

The authors appreciate Hamedan Branch of Islamic Azad University for the financial support of this research.