# Operational matrices of Chebyshev polynomials for solving singular Volterra integral equations

- 667 Downloads

## Abstract

An effective technique based on fractional calculus in the sense of Riemann–Liouville has been developed for solving weakly singular Volterra integral equations of the first and second kinds. For this purpose, orthogonal Chebyshev polynomials are applied. Properties and some operational matrices of these polynomials are first presented and then the unknown functions of the integral equations are represented by these polynomials in the matrix form. These matrices are then used to reduce the singular integral equations to some linear algebraic system. For solving the obtained system, Galerkin method is utilized via Chebyshev polynomials as weighting functions. The method is computationally attractive, and the validity and accuracy of the presented method are demonstrated through illustrative examples. As shown in the numerical results, operational matrices, even for first kind integral equations, have relatively low condition numbers, and thus, the corresponding matrices are well posed. In addition, it is noteworthy that when the solution of equation is in power series form, the method evaluates the exact solution.

### Keywords

Chebyshev polynomials Singular integral equations Operational matrix Fractional calculus Galerkin method### Mathematics Subject Classification

41A50 26A33 65L60## Introduction

*y*(

*x*) is the unknown function that to be determined, and

*T*is a positive constant.

Abel’s equation is one of the integral equations derived directly from a concrete problem of mechanics or physics (without passing through a differential equation). Historically, Abel’s problem is the first one that led to the study of integral equations. The generalized Abel’s integral equations on a finite segment appeared in the paper of Zeilon [15] for the first time.

A comprehensive reference on Abel-type equations, including an extensive list of applications, can be found in [8, 9].

The construction of high-order methods for the equations is, however, not an easy task because of the singularity in the weakly singular kernel. In fact, in this case, the solution *y* is generally not differentiable at the endpoints of the interval [3], and due to this, to the best of the authors’ knowledge, the best convergence rate ever achieved remains only at polynomial order. For example, if we set uniform meshes with \(n+1\) grid points and apply the spline method of order *m*, then the convergence rate is only \(O(n^{-2P})\) at most (see [4, 12]), and it cannot be improved by increasing *m*. One way of remedying this is to introduce graded meshes [13]. By doing so, the rate is improved to \(O(n^{-m})\), which now depends on m, but still at polynomial order. Rashit Ishik [10] used Bernstein series solution for solving linear integro-differential equations with weakly singular kernels. In [5] and [6], wavelets method was applied for solution of nonlinear fractional integro-differential equations in a large interval and systems of nonlinear singular fractional Volterra integro-differential equations. Authors of [11] applied fractional calculus for solving Abel integral equations. The expansions approach for solving cauchy integral equation of the first kind is discussed in [14].

In this paper, we use the Chebyshev polynomials operational matrices via Galerkin method for solving weakly singular integral equations. Our method consists of reducing the given weakly singular integral equation to a set of algebraic system by expanding the unknown function by Chebyshev polynomials of the first kind. Galerkin method is utilized to solve the obtained system.

The structure of this paper is arranged as follows. The main problem and brief history of some presented methods are expressed in Sect. 1. In Sect. 2, we present some necessary definitions and mathematical preliminaries of the fractional calculus theory in the sense of Riemann–Liouville. Section 3 is devoted to introducing Chebyshev polynomials, properties and some operational matrices of these functions. In Sect. 4, Chebyshev polynomials are applied as testing and weighting functions of Galerkin method for efficient solution of Eq. 1. In Sect. 5, we report our numerical founds and compare with other methods in solving these integral equations, and Sect. 6 contains our conclusion.

## Some preliminaries in fractional calculus

In this section, we briefly present some definitions and results in fractional calculus for our subsequent discussion. The fractional calculus is the name for the theory of integrals and derivatives of arbitrary order, which unifies and generalizes the notions of integer-order differentiation and n-fold integration [7]. There are various definitions of fractional integration and differentiation, such as Grunwald–Letnikov, and Caputo and Riemann–Liouville’s definitions. In this study, fractional calculus in the sense of Riemann–Liouville is considered.

**Definition 1**

*f*be a real function on [

*a*,

*b*] and \(0<\alpha <1\). Then, the left and right Riemann–Liouville fractional integral operators of order \(\alpha\) for the function

*f*are defined, respectively, as

**Definition 2**

In this study, the left Riemann–Liouville fractional integral operator is utilized to transform singular integral equation to some algebraic system. Therefore, for abbreviation, the mentioned operator is denoted by \(I^{\alpha }\).

**Theorem 1**

*The operator*\(I^{\alpha }\) (

*stand for left and right Riemann–Liouville fractional integral operator*)

*satisfies the following properties*:

*Proof*

## Chebyshev polynomials

In this section, a brief summary of orthogonal Chebyshev polynomials is expressed.

**Definition 3**

*n*th degree of Chebyshev polynomials is defined by

### Matrix form

*T*is a \((n+1)\times (n+1)\) matrix defined by

*T*]. To transform the interval \([-1,1]\) to [0,

*T*], we apply the \((n+1)\times (n+1)\) shift matrix

*R*, which is defined by

*WX*, where \(W=T R\).

### Function approximation

*j*. The coefficients \(c_{j}\) are given by

### Operational matrix of fractional Integration

*A*is \((n+1)\times (n+1)\) lower triangular matrix defined by

*g*(

*x*), approximated by shifted Chebyshev functions (\(g(x)=D^{T}WX\)), the fractional integral can be written as

## Numerical implementation

In this section, the shifted Chebyshev polynomials are applied for solving singular integral Eqs. (1) and (2). For this purpose, initially, the singular integral equation is transformed to nonsingular integral equation, utilizing Riemann–Liouville calculus.

*y*(

*x*) is approximated by shifted Chebyshev polynomials as

### The first kind

*i*, which can be written as

*i*th row of the matrix

*W*. Multiplying Eq. (13) by \(\varphi _{i}(x)\), we get

*T*, we have

### The second kind

*T*, we can rewrite the current equation in the following form:

## Illustrative examples

*Example 1*

*y*(

*x*) is a polynomial of degree 2 and the least approximation level for Chebyshev polynomials, in this study, is \(N=4\). Therefore, the approximated solution through the presented method is the same as the exact solution, that is \(y_{4}(x)=x^{2}\).

*Example 2*

*y*(

*x*) is obtained by the method in Sect. 4 for \(N=4, 8\) and 12. The unknown coefficients for \(c_{i}\) are obtained through the method explained in Sect. 4 for \(N=4\):

Exact and approximate solutions of Example 2

| \(N=4\) | \(N=8\) | \(N=12\) | Exact |
---|---|---|---|---|

0.1 | 0.0326 | 0.0316 | 0.0316 | 0.0316 |

0.2 | 0.0896 | 0.0895 | 0.0894 | 0.0894 |

0.3 | 0.1638 | 0.1643 | 0.1643 | 0.1643 |

0.4 | 0.2526 | 0.2529 | 0.2530 | 0.2530 |

0.5 | 0.3536 | 0.3536 | 0.3536 | 0.3536 |

0.6 | 0.4651 | 0.4648 | 0.4648 | 0.4648 |

0.7 | 0.5859 | 0.5856 | 0.5857 | 0.5857 |

0.8 | 0.7154 | 0.7155 | 0.7155 | 0.7155 |

0.9 | 0.8535 | 0.8538 | 0.8538 | 0.8538 |

1 | 1.0006 | 1.0001 | 1.0000 | 1.0000 |

\(c.n.m^{*}\) | 11.23 | 22.32 | 32.85 |

*Example 3*

*y*(

*x*) is obtained by the method in Sect. 4 for \(N=4, 8\), and 12. The unknown coefficients for \(c_{i}\) are obtained through the method explained in Sect. 4 for \(N=4\):

Exact and approximate solutions of Example 3

Approximate | Method of [1] | ||||
---|---|---|---|---|---|

| \(N=4\) | \(N=8\) | \(N=12\) | \(N=20\) | Exact |

0.1 | 0.2870 | 0.2878 | 0.2852 | 0.2848 | 0.2852 |

0.2 | 0.4071 | 0.4017 | 0.4033 | 0.4032 | 0.4033 |

0.3 | 0.4940 | 0.4992 | 0.4936 | 0.4944 | 0.4936 |

0.4 | 0.5706 | 0.5713 | 0.5703 | 0.5704 | 0.5704 |

0.5 | 0.6335 | 0.6366 | 0.6378 | 0.6374 | 0.6377 |

0.6 | 0.6946 | 0.6985 | 0.6986 | 0.6989 | 0.6986 |

0.7 | 0.7556 | 0.7558 | 0.7546 | 0.7547 | 0.7546 |

0.8 | 0.8126 | 0.8055 | 0.8066 | 0.8063 | 0.8067 |

0.9 | 0.8567 | 0.8565 | 0.8556 | 0.8551 | 0.8556 |

1 | 0.8736 | 0.8890 | 0.9018 | 0.8992 | 0.9019 |

\(c.n.m^{*}\) | 13.94 | 26.78 | 43.68 |

*Example 4*

*y*(

*x*) is obtained through the method in Sect. 4 for \(N=4, 8\), and 12. The unknown coefficients for \(c_{i}\) are obtained through the method explained in Sect. 4 for \(N=4\).

Exact and approximate solutions of Example 4

| \(N=4\) | \(N=8\) | \(N=12\) | Exact |
---|---|---|---|---|

0.1 | 0.2732 | 0.2815 | 0.2797 | 0.2799 |

0.2 | 0.4445 | 0.4463 | 0.4433 | 0.4443 |

0.3 | 0.5863 | 0.5823 | 0.5820 | 0.5822 |

0.4 | 0.7063 | 0.7058 | 0.7052 | 0.07052 |

0.5 | 0.8159 | 0.8178 | 0.8185 | 0.8183 |

0.6 | 0.9209 | 0.9239 | 0.9240 | 0.92418 |

0.7 | 1.0237 | 1.0248 | 1.0242 | 1.0241 |

0.8 | 1.1227 | 1.1191 | 1.1194 | 1.1195 |

0.9 | 1.2131 | 1.2110 | 1.2110 | 1.2109 |

1 | 1.2863 | 1.2951 | 1.2995 | 1.2990 |

\(c \cdot n \cdot m^{*}\) | 16.01 | 40.32 | 64.18 |

## Conclusions

In this study, a numerical approach based on Chebyshev polynomials operational matrices was developed to approximate the solution of the weakly singular Volterra integral equations of the first and second kinds. Applying fractional derivative of these polynomials, we have transformed the singular integral equations to some linear algebraic system. The numerical results obtained support the validity and efficiency of the proposed method. It is noteworthy that when the solution of equation is in power series form, the method evaluates the exact solution, such as Example 1. In addition, as can be seen, the operational matrices of first kind integral equations have relatively low condition numbers. Thus, the corresponding matrices are well posed.

### References

- 1.Avazzadeh, Z., Shafiee, B., Loghmani, G.: Fractional calculus for solving Abel’s integral equations using Chebyshev polynomials. Appl. Math. Sci.
**5**(45), 2207–2216 (2011)MathSciNetMATHGoogle Scholar - 2.Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model.
**35**, 5662–5672 (2011)MathSciNetCrossRefMATHGoogle Scholar - 3.Graham, I.G.: Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels. J. Integr. Equ.
**4**, 1–30 (1982)MathSciNetMATHGoogle Scholar - 4.Graham, I.G.: Galerkin methods for second kind integral equations with singularities. Math. Comput.
**39**, 519–533 (1982)MathSciNetCrossRefMATHGoogle Scholar - 5.Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M., Li, M.: Chebyshev wavelets method for solution of nonlinear fractional integro-differential equations in a large interval. Adv. Math. Phys., Article ID 482083 (2013)Google Scholar
- 6.Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F., Cattani, C.: Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun. Nonlinear Sci. Numer. Simul.
**19**(1), 37–48 (2014)MathSciNetCrossRefMATHGoogle Scholar - 7.Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol. 204. Elsevier (2006)Google Scholar
- 8.Nieto, J.J., Okrasinski, W.: Existence, uniqueness, and approximation of solutions to some nonlinear diffusion problems. J. Math. Anal. Appl.
**210**(1), 231–240 (1997)MathSciNetCrossRefMATHGoogle Scholar - 9.Okrasinski, W., Vila, S.: Determination of the interface position for some nonlinear diffusion problems. Appl. Math. Lett.
**11**(4), 85–89 (1998)MathSciNetCrossRefMATHGoogle Scholar - 10.Rasit Isik, O., Sezer, M., Guney, Z.: Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel. Appl. Math. Comput.
**217**(16), 7009–7020 (2011)MathSciNetMATHGoogle Scholar - 11.Saleh, M.H., Amer, S.M., Mohamed, DSh, Mahdy, A.E.: Fractional calculus for solving generalized Abels integral equations using Chebyshev polynomials. Int. J. Comput. Appl.
**100**(8), 19–23 (2014)Google Scholar - 12.Schneider, C.: Product integration for weakly singular integral equations. Math. Comput.
**36**, 207–213 (1981)MathSciNetCrossRefMATHGoogle Scholar - 13.Vainikko, G., Uba, P.: A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel. J. Austral. Math. Soc. Ser. B.
**22**, 431–438 (1981)MathSciNetCrossRefMATHGoogle Scholar - 14.Yaghobifar, M., Nik Long, N.M.A., Eshkuvatov, Z.K.: The expansions approach for solving cauchy integral equation of the first kind. Applied Mathematical Sciences.
**4**(52), 2581–2586 (2010)MathSciNetMATHGoogle Scholar - 15.Zeilon, N.: Sur quelques points de la theorie de l’equation integrale d’Abel [On some points of the theory of integral equation of Abel type]. Arkiv. Mat. Astr. Fysik.
**18**, 1–19 (1924)Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.