# Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Genocchi Polynomials and Their Properties

#### 2.1. Definition of the Genocchi Polynomials

$n$ | 0 | 1 | 2 | 4 | 6 |

${G}_{n}$ | 0 | 1 | −1 | 1 | −3 |

#### 2.2. Approximation of Arbitrary Function by Applying Genocchi Polynomials

#### 2.3. Using the Matrix Approach to Compute the Genocchi Approximation Coefficients

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Implementation of the Genocchi Polynomial Method for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels

## 4. Error Analysis

## 5. Illustrative Examples

**Example**

**1.**

**Example**

**2.**

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Abdou, M.A. On a symptotic methods for Fredholm–Volterra integral equation of the second kind in contact problems. J. Comput. Appl. Math.
**2003**, 154, 431–446. [Google Scholar] [CrossRef] [Green Version] - Datta, K.B.; Mohan, B.M. Orthogonal Functions in Systems and Control; World Scientific: Singapore, 1995. [Google Scholar]
- Smetanin, B.I. On an integral equation for axially-symmetric problems in the case of an elastic body containing an inclusion. J. Appl. Math. Mech.
**1991**, 55, 371–375. [Google Scholar] [CrossRef] - Ramos, J.I.; Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A. Spectral Methods in Fluid Dynamics; Springer: New York, NY, USA, 1988. [Google Scholar]
- Atkinson, K.E. The Numerical Solution of Integral Equations of the Second Kind; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Noeiaghdam, S.; Zarei, E.; Kelishami, H.B. Homotopy analysis transform method for solving Abel’s integral equations of the first kind. Ain Shams Eng. J.
**2016**, 7, 483–495. [Google Scholar] [CrossRef] [Green Version] - Noeiaghdam, S.; Araghi, M.A.; Abbasbandy, S. Finding optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the stochastic arithmetic. Numer. Algorithms
**2019**, 81, 237–267. [Google Scholar] [CrossRef] - Noeiaghdam, S.; Sidorov, D.; Sizikov, V.; Sidorov, N. Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method. Appl. Comput. Math. Int. J.
**2020**, 19, 81–105. [Google Scholar] - Noeiaghdam, S.; Dreglea, A.; He, J.; Avazzadeh, Z.; Suleman, M.; Fariborzi Araghi, M.A.; Sidorov, D.N.; Sidorov, N. Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library. Symmetry
**2020**, 12, 1730. [Google Scholar] [CrossRef] - Allaei, S.S.; Diogo, T.; Rebelo, M. Analytical and computational methods for a class of nonlinear singular integral equations. Appl. Numer. Math.
**2017**, 114, 2–17. [Google Scholar] [CrossRef] - Maleknejad, K.; Hashemizadeh, E.; Basirat, B. Numerical solvability of Hammerstein integral equations based on hybrid Legendre and Block-Pulse functions. In Proceedings of the 2010 International Conference on Parallel and Distributed Processing Techniques and Applications, Las Vegas, NV, USA, 12–15 July 2010; pp. 172–175. [Google Scholar]
- Pourgholi, R.; Tahmasebi, A.; Azimi, R. Tau approximate solution of weakly singular Volterra integral equations with Legendre wavelet basis. Int. J. Comput. Math.
**2017**, 94, 1337–1348. [Google Scholar] [CrossRef] - Eslahchi, M.R.; Dehghan, M.; Parvizi, M. Application of the collocation method for solving nonlinear fractional integro-differential equations. Comput. Appl. Math.
**2014**, 257, 105–128. [Google Scholar] [CrossRef] - Zhu, L.; Fan, Q. Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW. Commun. Nonlinear Sci. Numer. Simul.
**2013**, 18, 1203–1213. [Google Scholar] [CrossRef] - Nemati, S.; Lima, P.M. Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions. Appl. Math. Comput.
**2018**, 327, 79–92. [Google Scholar] [CrossRef] - Singh, S.; Patel, V.K.; Singh, V.K. Operational matrix approach for the solution of partial integro-differential equation. Appl. Math. Comput.
**2016**, 283, 195–207. [Google Scholar] [CrossRef] - Garg, M.; Sharma, A. Solution of space-time fractional telegraph equation by Adomian decomposition method. J. Inequalities Spec. Funct.
**2011**, 2, 1–7. [Google Scholar] - Ray, S.S.; Bera, R.K. An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput.
**2005**, 167, 561–571. [Google Scholar] [CrossRef] - Wu, G.C. A fractional variational iteration method for solving fractional nonlinear differential equations. Comput. Math. Appl.
**2011**, 61, 2186–2190. [Google Scholar] [CrossRef] [Green Version] - Khan, Y.; Faraz, N.; Yildirim, A.; Wu, Q. Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. Comput. Math. Appl.
**2011**, 62, 2273–2278. [Google Scholar] [CrossRef] [Green Version] - Arikoglu, A.; Ozkol, I. Solution of fractional differential equations by using differential transform method. Chaos Solitons Fractals
**2007**, 34, 1473–1481. [Google Scholar] [CrossRef] - Nazari, D.; Shahmorad, S. Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions. J. Comput. Appl. Math.
**2010**, 234, 883–891. [Google Scholar] [CrossRef] [Green Version] - Ebadi, M.A.; Hashemizadeh, E. A new approach based on the Zernike radial polynomials for numerical solution of the fractional diffusion-wave and fractional Klein–Gordon equations. Phys. Scr.
**2018**, 93, 125202. [Google Scholar] [CrossRef] - Shang, Y. On the delayed scaled consensus problems. Appl. Sci.
**2017**, 7, 713. [Google Scholar] [CrossRef] [Green Version] - Isah, A.; Phang, C. On Genocchi operational matrix of fractional integration for solving fractional differential equations. AIP Conf. Proc.
**2017**, 1795, 020015. [Google Scholar] - Loh, J.R.; Phang, C.; Isah, A. New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations. Adv. Math. Phys.
**2017**, 2017, 1–12. [Google Scholar] [CrossRef] - Isah, A.; Phang, C.; Phang, P. Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations. Int. J. Differ. Equ.
**2017**, 2017, 1–10. [Google Scholar] [CrossRef] [Green Version] - Sadeghi Roshan, S.; Jafari, H.; Baleanu, D. Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on Genocchi polynomials. Math. Methods Appl. Sci.
**2018**, 41, 9134–9141. [Google Scholar] [CrossRef] - Diogo, T.; Lima, P.; Rebelo, M. Numerical solution of a nonlinear Abel type Volterra integral equation. Commun. Pure Appl. Anal.
**2006**, 5, 277–288. [Google Scholar] [CrossRef] - Diogo, M.T.; Lima, P.M.; Rebelo, M.S. Comparative study of numerical methods for a nonlinear weakly singular Volterra integral equation. Hermis J.
**2006**, 7, 1–20. [Google Scholar]

**Figure 7.**Plot of approximate solutions by our method (Genocchi polynomials) with different values of $N$ on the interval $[0,\epsilon ]$ with $\epsilon =0.002$ for Example 1.

**Table 1.**Approximate and exact values of nonlinear Volterra integral equations with $N=5,10,15,20$ for Example 1.

$\mathit{N}=5$ | $\mathit{N}=10$ | $\mathit{N}=15$ | $\mathit{N}=20$ | ${\mathit{y}}_{\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}$ | |
---|---|---|---|---|---|

0.0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

0.2 | 0.585076 | 0.584768 | 0.584793 | 0.584797 | 0.584804 |

0.4 | 0.736620 | 0.736795 | 0.736802 | 0.736804 | 0.736806 |

0.6 | 0.843508 | 0.843427 | 0.843431 | 0.843434 | 0.843433 |

0.8 | 0.928164 | 0.928313 | 0.928317 | 0.928319 | 0.928318 |

1.0 | 1.00041 | 0.99996 | 1.000001 | 1.000001 | 1.000000 |

**Table 2.**Numerical results of ${\zeta}_{N}$ for different values $N$ on the interval $[0,1]$ for Example 1.

$\mathit{N}$ | ${\mathit{\zeta}}_{\mathit{N}}$ | Computing Time (s) |
---|---|---|

5 | 5.98532 × 10^{−4} | 0.321 |

10 | 1.14944 × 10^{−4} | 0.357 |

15 | 4.48214 × 10^{−5} | 0.420 |

20 | 2.85973 × 10^{−5} | 0.451 |

$\mathit{t}$ | ${\mathit{e}}_{5}(\mathit{t})$ | ${\mathit{e}}_{10}(\mathit{t})$ | ${\mathit{e}}_{15}(\mathit{t})$ | ${\mathit{e}}_{20}(\mathit{t})$ |
---|---|---|---|---|

0.0 | 0.000000000 | 0.0000000000 | 0.00000000 | 0.00000000 |

0.2 | 0.000272294 | 0.0000359627 | 0.00001089 | 6.67632 × 10^{−6} |

0.4 | 0.000185829 | 0.0000111075 | 3.8548 × 10^{−6} | 2.25757 × 10^{−6} |

0.6 | 0.000075540 | 5.5081 × 10^{−6} | 1.83763 × 10^{−6} | 9.72768 × 10^{−6} |

0.8 | 0.000153622 | 4.62581 × 10^{−6} | 1.11576 × 10^{−6} | 2.18163 × 10^{−6} |

1.0 | 0.000406512 | 0.0000397521 | 8.73408× 10^{−6} | 9.0017 × 10^{−6} |

**Table 4.**Comparison of maximum absolute errors between a new approach approximate solution and Euler’s method on $[0,\epsilon ]$ for Example 1.

Euler’s Method [10] | Our Method (Genocchi Polynomials) | ||||||||
---|---|---|---|---|---|---|---|---|---|

N | $\mathit{\epsilon}=0$ | $\mathit{\epsilon}=0.01$ | $\mathit{\epsilon}=0.02$ | $\mathit{\epsilon}=0.03$ | N | $\mathit{\epsilon}=0$ | $\mathit{\epsilon}=0.01$ | $\mathit{\epsilon}=0.02$ | $\mathit{\epsilon}=0.03$ |

${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ||

80 | 0.67 × 10^{−2} | 6.60 × 10^{−3} | 6.50 × 10^{−3} | 6.30 × 10^{−3} | 5 | 0.000 | 1.851 × 10^{−3} | 2.343 × 10^{−3} | 2.427 × 10^{−3} |

160 | 3.21 × 10^{−3} | 3.10 × 10^{−3} | 3,10 × 10^{−3} | 3.03 × 10^{−3} | 10 | 0.000 | 6.595 × 10^{−4} | 6.704 × 10^{−4} | 6.704 × 10^{−4} |

320 | 1.55 × 10^{−3} | 1.50 × 10^{−3} | 1.50 × 10^{−3} | 1.50 × 10^{−3} | 15 | 0.000 | 3.193 × 10^{−4} | 3.193 × 10^{−4} | 3.178 × 10^{−4} |

640 | 753 × 10^{−4} | 7.40 × 10^{−4} | 7.20 × 10^{−4} | 7.20 × 10^{−4} | 20 | 0.000 | 2.305 × 10^{−4} | 2.305 × 10^{−4} | 2.305 × 10^{−4} |

$\mathit{N}$ | ${\mathit{\zeta}}_{\mathit{N}}$ | Computing Time (s) |
---|---|---|

5 | 1.912914 × 10^{−4} | 0.351 |

10 | 1.087754 × 10^{−4} | 0.402 |

15 | 9.106063 × 10^{−5} | 0.457 |

20 | 7.200394 × 10^{−5} | 0.530 |

${\Vert {\mathit{y}}_{\mathit{N}}-{\mathit{y}}_{\mathit{M}}\Vert}_{\infty}$ | ||||
---|---|---|---|---|

$\mathit{t}$ | $\mathit{N}=5;\mathrm{M}=7$ | $\mathit{N}=7;\mathrm{M}=10$ | $\mathit{N}=10;\mathrm{M}=12$ | $\mathit{N}=12;\mathrm{M}=13$ |

0.0000 | 1.11022 × 10^{−16} | 1.12022 × 10^{−16} | 0.000000000 | 0.000000000 |

0.0004 | 0.000507344 | 0.000477524 | 0.000214749 | 0.0000875408 |

0.0008 | 0.000800212 | 0.000750437 | 0.000336102 | 0.0001366621 |

0.0012 | 0.001041861 | 0.000973492 | 0.000434214 | 0.0001761062 |

0.0016 | 0.001254042 | 0.001167461 | 0.000518585 | 0.0002097873 |

0.002 | 0.001445845 | 0.001341082 | 0.000593241 | 0.000239374 |

**Table 7.**Comparison of maximum absolute errors ${e}_{\infty}(N)={\Vert {y}_{2}-{y}_{N}\Vert}_{\infty}$ between our method (Genocchi polynomials) and Euler’s method: the Picard iterate ${y}_{2}$ was used on $[0,\epsilon ]$ for Example 2.

Euler’s Method [10] | Our Method (Genocchi Polynomials) | ||||||
---|---|---|---|---|---|---|---|

N | $\mathit{\epsilon}=0.002$ | $\mathit{\epsilon}=0.003$ | $\mathit{\epsilon}=0.008$ | N | $\mathit{\epsilon}=0.002$ | $\mathit{\epsilon}=0.003$ | $\mathit{\epsilon}=0.008$ |

${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ${\mathit{e}}_{\infty}(\mathit{N})$ | ||

40 | 3.60 × 10^{−2} | 3.00 × 10^{−2} | 1.70 × 10^{−2} | 5 | 6.165 × 10^{−3} | 7.441 × 10^{−3} | 9.878 × 10^{−3} |

80 | 2.10 × 10^{−2} | 1.7 × 10^{−2} | 9.10 × 10^{−3} | 10 | 3.378 × 10^{−3} | 3.386 × 10^{−3} | 4.162 × 10^{−3} |

160 | 1.01 × 10^{−2} | 8.4 × 10^{−3} | 4.00 × 10^{−3} | 12 | 2.785 × 10^{−3} | 3.113 × 10^{−3} | 3.222 × 10^{−3} |

320 | 4.00 × 10^{−3} | 3.00 × 10^{−3} | 1.30 × 10^{−3} | 15 | 2.151 × 10^{−3} | 2.311 × 10^{−3} | 2.232 × 10^{−3} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hashemizadeh, E.; Ebadi, M.A.; Noeiaghdam, S.
Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels. *Symmetry* **2020**, *12*, 2105.
https://doi.org/10.3390/sym12122105

**AMA Style**

Hashemizadeh E, Ebadi MA, Noeiaghdam S.
Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels. *Symmetry*. 2020; 12(12):2105.
https://doi.org/10.3390/sym12122105

**Chicago/Turabian Style**

Hashemizadeh, Elham, Mohammad Ali Ebadi, and Samad Noeiaghdam.
2020. "Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels" *Symmetry* 12, no. 12: 2105.
https://doi.org/10.3390/sym12122105