Fourier series approximation for the Cauchy singular integral equation

Hamid BOULARES$^\ast $, Hamza GUEBBAI$^\ast $ Amira ARBAOUI$^{\ast }$

August 18, 2017. Accepted: December 19, 2017. Published online: August 6, 2018.

$^\ast $Department of Mathematics, University 8 May 1945 Guelma. B.P. 401 Guelma 24000 Algeria. e-mail: boulareshamid@gmail.com, guebaihamza@yahoo.fr, arbaoui.amira.93@gmail.com.

Using the Fourier series as a projection in the Galerkin method, we approach the solution of the Cauchy singular integral equation. This study is carried in $L^2$. Numerical examples are developped to show the effectiveness of this method.

MSC. 45B05, 65F10, 65J10, 42A16

Keywords. Singular integral equation, Cauchy operator, Galerkin method, Fourier series

1 Introduction

Numerical solution methods of integral equations play a very important role in various scientific fields. With the advantage of numerical calculation machines, including computers, these methods have now become an essential tool for investigation in various fundamental problems of our assimilation of scientific phenomena that are difficult, i.e. impossible to solve in the past.

Integral equations with singular kernel represent a great numerical challenge. In addition, if the approximation is performed in a space with weak property as $L^2$, it accentuates the difficulty.

In this article, we are interested in the following problem: given $f\in L^{2}\left(0,1\right)$, find a function $u\in L^{2}\left(0,1\right)$ such that

\begin{eqnarray} u=Cu+f, \end{eqnarray}

where, $C$ is the Cauchy operator defined by:

\begin{eqnarray*} \textrm{For\ all}\ s\in \left]0,1\right[,\ Cu(s)=\lim _{\varepsilon \rightarrow 0}\left( \int _{0}^{s-\varepsilon }\tfrac {u\left( t\right) }{s-t}dt+\int _{s+\varepsilon }^{1}\tfrac {u\left( t\right) }{s-t}dt\right). \end{eqnarray*}

$C$ is an integral operator with singular kernel. But, in [ 1 ] the authors show that $C$ is a bounded operator on $L^{2}\left(0,1\right)$ to itself with $\left\| C\right\| \leq \pi $. In addition, they show that it is a skew-hermitian operator, i.e., $C^{*}=-C$. Which gives that $\left(I-C\right)^{-1}$ exists and that $\big\| \left(I-C\right)^{-1}\big\| \leq 1$ [ 1 ] .

Then, for all given $f\in L^{2}\left(0,1\right)$, the solution $u$ of $(1)$ exists and is unique.

Our goal is to construct an approximate solution for $(1)$, using Galerkin method. This method is well studied and very known [ 2 , 3 , 4 ] , but for compact operator only. Recently, this method is developed for the Cauchy operator using the piecewise constant projection [ 5 , 6 ] . Our aim is to use Fourier series as projection to make the Galerkin method more efficient.

2 Numerical study

Let $\left\{ \pi _{n}\right\} _{n\geq 0}$ be the set of operators defined on $L^{2}\left(0,1\right)$ to itself by: For all $x\in L^{2}\left(0,1\right)$, $n\geq 0$:

\begin{eqnarray*} \pi _{n}x\left(t\right)=\sum _{k=0}^{n} a_{k}(x)\cos \left( k\pi t\right),\ t\in \left[0,1\right], \end{eqnarray*}

where,

\begin{eqnarray*} a_{0}(x)& =& \int _{0}^{1}x\left( t\right)dt,\\ a_{k}(x)& =& 2\int _{0}^{1}x\left( t\right) \cos \left( k\pi t\right) dt,\ \ k\geq 1. \end{eqnarray*}

It is clear that for all $n\geq 0$, $\pi _{n}$ is a projection and $\left\| \pi _{n}\right\| \leq 1$. $\pi _{n}$ is the Fourier series, truncated at $n$, of $x$ extended over $\mathop\mathbb {R}\nolimits $ in the following sense:

\begin{eqnarray*} x(t)& :=& x(2-t),\ t\in \left[1,2\right],\\ x & \textrm{is}& \ 2\textrm{-periodic\ over}\ \mathop\mathbb {R}\nolimits . \end{eqnarray*}

We choose this approach to weaken the effect of the Gibbs phenomenon [ 7 ] , [ 8 ] .

Proposition 1

For all $n\geq 0$, $\pi _{n}$ is a selfadjoint operator and

\begin{eqnarray*} \lim _{n\rightarrow \infty }\left\Vert \left( I-\pi _{n}\right)x\right\Vert _{L^{2}\left(0,1\right)}=0. \end{eqnarray*}

Proof â–¼

For all $n\geq 0$, for all $x,y\in L^{2}\left(0,1\right)$, we have,

\begin{align*} \langle \pi _{n}x,y \rangle =& \displaystyle \int ^{1}_{0}\pi _{n}x(t)y(t)dt \displaystyle \sum ^{n}_{k=0}a_{k}(x)\int ^{1}_{0}\cos (k\pi t)y(t)dt \\ =& a_{0}(x)a_{0}(y)+\displaystyle \tfrac {1}{2}\displaystyle \sum ^{n}_{k=0}a_{k}(x)a_{k}(y) =\displaystyle \sum ^{n}_{k=0}\left(\int ^{1}_{0}\cos (k\pi t)x(t)dt\right)a_{k}(y) \\ =& \displaystyle \int ^{1}_{0}x(t)\pi _{n}y(t)dt = \langle x,\pi _{n}y \rangle . \end{align*}

For the pointwise convergence of $\pi _{n}$ to the identity $I$, notice that using the previous extension of $x$, we obtain [ 7 ] , [ 8 ]

\begin{eqnarray*} \left\Vert \left( I-\pi _{n}\right)x\right\Vert _{L^{2}\left(0,1\right)}=\tfrac {1}{2}\left\Vert \left( I-\pi _{n}\right)x\right\Vert _{L^{2}\left(-1,1\right)}\rightarrow _{n\rightarrow \infty }0. \end{eqnarray*}

Proof â–¼

Galerkin’s method is to approximate $C$ by $C_{n}=\pi _{n}C\pi _{n}$, for $n\geq 0$.

Theorem 2

For all $n\geq 1$, $C_{n}^{\ast }\! =\! -C_{n}$, $I-C_{n}$ is invertible and

\[ \| \left(I - C_{n}\right)^{-1} \| \leq 1. \]

Proof â–¼

We have,

\begin{eqnarray*} C_{n}^{\ast } & =& (\pi _{n}C\pi _{n})^{\ast } =\pi _{n}^{\ast }C^{\ast }\pi _{n}^{\ast }\\ & =& \pi _{n}\left(-C\right)\pi _{n}=-C_{n}. \end{eqnarray*}

In the same way as in [ 1 ] , we define the operators $A_{n}=iC_{n}$ for all $n\geq 0$. Then, $A_{n}$ is selfadjoint and its spectrum is real. Therefore,

\[ \textrm{sp}\left(C_{n}\right)\subset \left\{ i\alpha \; :\; \alpha \in \mathbb {R}\right\} ,\ n\geq 0. \]

This means that $\left(I-C_{n}\right)$ is invertible. But, for all $x\in L^{2}\left(0,1\right)$,

\begin{eqnarray*} \operatorname {Re}\left( \langle (I-C_{n})x,x \rangle \right)=\tfrac {1}{2}\left( \langle (I-C_{n})x,x \rangle +\overline{ \langle (I-C_{n})x,x \rangle }\right)= \langle x,x \rangle . \end{eqnarray*}

Hence,

\begin{eqnarray*} \left\| x\right\| _{L^{2}\left(0,1\right)}^{2}\leq \left| \langle (I-C_{n})x,x \rangle \right|\leq \left\| (I-C_{n})x\right\| _{L^{2}\left(0,1\right)}\left\| x\right\| _{L^{2}\left(0,1\right)}, \end{eqnarray*}

and $\| \left(I-C_{n}\right)^{-1}\| \leq 1$.

Proof â–¼

The last theorem ensures, for all $n\geq 0$ and all $f\in L^{2}\left(0,1\right)$, the existence and the unicity of $u_{n}$ the solution of the following approximate equation:

\[ u_{n}=C_{n}u_{n}+f. \]

However, the following theorem ensures the convergence of $u_{n}$ to $u$.

Theorem 3

For all $n\geq 0$,

\begin{eqnarray*} \left\Vert u-u_{n}\right\Vert _{L^{2}\left(0,1\right)}\leq \pi \left\Vert \left( I-\pi _{n}\right)u\right\Vert _{L^{2}\left(0,1\right)}+\left\Vert \left( I-\pi _{n}\right)Cu\right\Vert _{L^{2}\left(0,1\right)}. \end{eqnarray*}

Proof â–¼

We have,

\begin{eqnarray*} u-u_{n} & =& Cu-C_{n}u_{n}\\ & =& Cu-C_{n}u+C_{n}\left( u-u_{n}\right), \\ \left( I-C_{n}\right) \left( u-u_{n}\right) & =& Cu-C_{n}u. \end{eqnarray*}

Then,

\begin{eqnarray*} \Vert u-u_{n}\Vert _{L^{2}\left(0,1\right)} & \leq & \Vert \left( I-C_{n}\right) ^{-1}\Vert \left\Vert \left( Cu-C_{n}u\right) \right\Vert _{L^{2}\left(0,1\right)}\\ & \leq & \left\Vert \left( Cu-C_{n}u\right) \right\Vert _{L^{2}\left(0,1\right)}. \end{eqnarray*}

But,

\begin{align*} \left\Vert Cu-C_{n}u\right\Vert _{L^{2}\left(0,1\right)} & =\left\Vert Cu-\pi _{n}C\pi _{n}u\right\Vert _{L^{2}\left(0,1\right)}\nonumber \\ & =\left\Vert Cu-\pi _{n}Cu+\pi _{n}Cu-\pi _{n}C\pi _{n}u\right\Vert _{L^{2}\left(0,1\right)}\nonumber \\ & =\left\Vert \left( I-\pi _{n}\right) Cu+\pi _{n}C\left( I-\pi _{n}\right) u\right\Vert _{L^{2}\left(0,1\right)}\nonumber \\ & \leq \left\Vert \left( I-\pi _{n}\right) Cu\right\Vert _{L^{2}\left(0,1\right)} +\left\Vert \pi _{n}C\left( I-\pi _{n}\right) u\right\Vert _{L^{2}\left(0,1\right)}. \end{align*}

We use $\left\| \pi _{n}C\right\| \leq \pi $ to conclude.

Proof â–¼

3 Numerical structure

We have for: For all $ n\geq 0$,

\begin{eqnarray} u_{n} & =& \pi _{n}C\pi _{n}u_{n}+f ,\nonumber \\ & =& \pi _{n}C\left( \sum _{k=0}^{n}a_{k}\left(u_{n}\right)\cos \left( k\pi \cdot \right)\right)+f,\nonumber \\ & =& \sum _{k=1}^{n}a_{k}\left(u_{n}\right)\pi _{n}\varphi _{k}+f \end{eqnarray}

where

\begin{eqnarray*} \varphi _{k}=C\left(\cos \left(k\pi \cdot \right)\right),\ k\geq 0. \end{eqnarray*}

We have

\begin{eqnarray*} \varphi _{0}(s)& =& \lim _{\varepsilon \rightarrow 0}\left( \int _{0}^{s-\varepsilon }\tfrac {1}{s-t}dt+\int _{s+\varepsilon }^{1}\tfrac {1}{s-t}dt\right) =\ln \left( \tfrac {s}{1-s}\right). \end{eqnarray*}

For $k\geq 1$,

\begin{eqnarray*} \varphi _{k}(s)& =& \lim _{\varepsilon \rightarrow 0}\left( \int _{0}^{s-\varepsilon }\tfrac {\cos \left( k\pi t\right) }{s-t}dt+\int _{s+\varepsilon }^{1}\tfrac {\cos \left( k\pi t\right) }{s-t}dt\right)\\ & =& \lim _{\varepsilon \rightarrow 0}\Big(\sin \left( k\pi s\right) [ \operatorname {Si}\left(k\pi \left( -\varepsilon \right) \right) - \operatorname {Si}\left( k\pi \left(\varepsilon \right) \right) ] \\ & \quad & +\sin \left( k\pi s\right) [ \operatorname {Si}\left(k\pi \left( 1-s\right) \right) - \operatorname {Si}\left( k\pi \left( -\varepsilon \right) \right) ] \\ & \quad & -\cos \left( k\pi s\right) [ \operatorname {Ci}\left( k\pi \left( -\varepsilon \right)\right) - \operatorname {Ci}\left( k\pi \left( \varepsilon \right) \right) ] \\ & \quad & -\cos \left( k\pi s\right) [ \operatorname {Ci}\left( k\pi \left( 1-s\right) \right)- \operatorname {Ci}\left( k\pi \left( -\varepsilon \right) \right) ]\Big), \end{eqnarray*}

where, $ \operatorname {Ci}$ is the cosine integral function and $ \operatorname {Si}$ is the sine integral function. Using the fact that,

\begin{eqnarray*} \operatorname {Ci}\left( -x\right) & =& \operatorname {Ci}\left( x\right)+i\pi ,\\ \operatorname {Si}\left( -x\right) & =& - \operatorname {Si}\left( x\right),\ \operatorname {Si}\left( 0\right) =0, \end{eqnarray*}

we get,

\begin{eqnarray*} \varphi _{k}(s)& =& \sin \left( k\pi s\right) \left[ \operatorname {Si}\left( k\pi \left( 1-s\right) \right) + \operatorname {Si}\left( k\pi s\right) \right] \\ & \quad & -\cos \left( k\pi s\right) \left[ \operatorname {Ci}\left( k\pi \left( 1-s\right) \right) + \operatorname {Ci}\left( k\pi s \right) \right],\ k\geq 1 \end{eqnarray*}

Multiplying $(2)$ by $\cos (i\pi s),\ 0\leq i\leq n$ and integrating over $\left(0,1\right)$, we get the following system:

\begin{eqnarray*} \begin{pmatrix} a_{0}(u_{n}) \\ \vdots \\ a_{n}(u_{n}) \end{pmatrix}= M\begin{pmatrix} a_{0}(u_{n}) \\ \vdots \\ a_{n}(u_{n}) \end{pmatrix}+ \begin{pmatrix} a_{0}(f) \\ \vdots \\ a_{n}(f) \end{pmatrix}, \end{eqnarray*}

where, $\left(a_{i}(u_{n})\right)_{0\leq i\leq n}$ and $\left(a_{i}(f)\right)_{0\leq i\leq n}$ are respectively the Fourier series coefficients of $u_n$ and $f$ and,

\begin{eqnarray*} M\left(i,k\right)=\left(2-\delta \left(i\right)\right)\int _{0}^{1}\varphi _{k}\left( s\right) \cos \left( i\pi s\right)ds, \end{eqnarray*}

where, $\delta \left(0\right)=1$ and $\delta \left(k\right)=0$ if $k\neq 0$. Once, the previous system solved, we construct $u_{n}$ using the following formula:

\begin{eqnarray*} \textrm{For\ all},\ s\in \left[0,1\right],\ u_{n}(s)=\sum _{k=0}^{n}a_{k}\left(u_{n}\right)\cos \left(k\pi s\right). \end{eqnarray*}

The coefficients $\left(a_{i}(f)\right)_{1\leq i\leq n}$ and those of the matrices $M$ are approximated using the numerical integration method, mid-point, with the subdivision: $M\geq 2,\ h=\frac{1}{M},\ t_{p+\frac{1}{2}}=\left(p+\frac{1}{2}\right)h$, for $0\leq p\leq M-1$.

4 Numerical example

To show the effectiveness of our approximation method, we will compare it with the one developed in [ 5 ] , [ 6 ] . The latter consists in using the piecewise constant projection denoted $\left\{ \tilde{\pi }_{n}\right\} _{n\geq 2}$. In what follows, $u_{n,f}$ denotes the approximate solution calculated by our method and $u_{n,pc}$ denotes the one calculated by the method developed in [ 5 ] , [ 6 ] . The theoretical error bounds obtained for the two methods are denoted by:

\begin{eqnarray*} TEBF_{n}& :=& \pi \left\Vert \left( I-\pi _{n}\right)u\right\Vert _{L^{2}\left(0,1\right)}+\left\Vert \left( I-\pi _{n}\right)Cu\right\Vert _{L^{2}\left(0,1\right)},\\ TEBPC_{n}& :=& \pi \left\Vert \left( I-\tilde{\pi }_{n}\right)u\right\Vert _{L^{2}\left(0,1\right)}+\left\Vert \left( I-\tilde{\pi }_{n}\right)Cu\right\Vert _{L^{2}\left(0,1\right)}, \end{eqnarray*}

where, $\left\| \cdot \right\| _{L^{2}\left(0,1\right)}$ is approximated using the mid-point method described below with $M=1000$.

Example 4

If we take

\begin{eqnarray*} f\left(s\right)=s+1+s\ln \left(\tfrac {1-s}{s}\right), \end{eqnarray*}

then the equation $(1)$ admits the following unique solution:

\begin{eqnarray*} u\left( s\right)=s. \end{eqnarray*}

$n$	$M$	$\displaystyle \left\\| u-u_{n,f}\right\\| _{L^{2}\left(0,1\right)}$	$TEBF_{n}$	$\displaystyle \left\\| u-u_{n,pc}\right\\| _{L^{2}\left(0,1\right)}$	$TEBPC_{n}$
10	200	7.41E-2	2.30E-1	2.06E-1	4.55E-1
50	300	2.85E-2	8.89E-2	7.49E-2	1.65E-1
100	500	1.79E-2	5.60E-2	4.82E-2	1.06E-1
200	1000	1.03E-2	3.21E-2	2.99E-2	6.62E-2
300	1000	6.83E-3	2.13E-2	2.16E-2	4.79E-2
400	1000	4.71E-3	1.46E-2	1.67E-2	3.71E-2

Table 1 Fourier projection vs. piecewise constant projection for Ex. 4.

Example 5

If we take

\begin{eqnarray*} f\left( s\right)=s^{-\frac{1}{4}}-s^{-\frac{1}{4}}\left(\ln \left(\tfrac {1+s^{\frac{1}{4}}}{1-s^{\frac{1}{4}}}\right)-2\tan ^{-1}\left(s^{-\frac{1}{4}}\right)\right), \end{eqnarray*}

then the equation $(1)$ admits the following unique solution:

\begin{eqnarray*} u\left( s\right)=s^{-\frac{1}{4}}. \end{eqnarray*}

$n$	$M$	$\displaystyle \left\\| u-u_{n,f}\right\\| _{L^{2}\left(0,1\right)}$	$TEBF_{n}$	$\displaystyle \left\\| u-u_{n,pc}\right\\| _{L^{2}\left(0,1\right)}$	$TEBPC_{n}$
10	200	3.89E-1	1.21E+0	7.41E-1	1.64E+0
50	300	2.09E-1	6.52E-1	4.23E-1	9.36E-1
100	500	1.47E-1	4.59E-1	3.17E-1	7.01E-1
200	1000	9.33E-2	2.90E-1	2.23E-1	4.94E-1
300	1000	6.48E-2	2.01E-1	1.70E-1	3.78E-1
400	1000	4.60E-2	1.43E-1	1.52E-1	3.33E-1

Table 2 Fourier projection vs. piecewise constant projection for Ex. 5.

5 Conclusion

We have constructed a Galerkin-type approximation method for the Cauchy singular integral equation, using the Fourier series as projection in $L^{2}\left(0,1\right)$.

Based on the fact that the Cauchy operator is skew-hermitian and that the Fourier series is a selfadjoint projection, we show the convergence of our method.

The numerical examples developed show that, regarding the approximation error, our method based on Fourier series is more efficient than the procedure that uses piecewise constant functions.

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\(n\)	\(M\)	\(\displaystyle \left\\| u-u_{n,f}\right\\| _{L^{2}\left(0,1\right)}\)	\(TEBF_{n}\)	\(\displaystyle \left\\| u-u_{n,pc}\right\\| _{L^{2}\left(0,1\right)}\)	\(TEBPC_{n}\)
10	200	7.41E-2	2.30E-1	2.06E-1	4.55E-1
50	300	2.85E-2	8.89E-2	7.49E-2	1.65E-1
100	500	1.79E-2	5.60E-2	4.82E-2	1.06E-1
200	1000	1.03E-2	3.21E-2	2.99E-2	6.62E-2
300	1000	6.83E-3	2.13E-2	2.16E-2	4.79E-2
400	1000	4.71E-3	1.46E-2	1.67E-2	3.71E-2