Research Article

Year 2021,
Volume: 18 Issue: 1, 33 - 46, 01.05.2021
### Abstract

### Keywords

### References

This study presents the application of semi-analytical and numerical solution technique to both Volterra and Fredholm integro-differential difference equations by employing Differential Transform Method depending on Taylor series expansion and introducing the new differential transform theorems with their proofs. To illustrate the computational efficiency and the reliability of the method to other common numerical methods in the open literature, some examples are carried out it is found that the results are highly accurate and reliable.

Differential transform method Fredholm and Volterra integrals Integro-differential difference equations Taylor series expansion

- [1] R. Abazari, A. Kilicman, “Application of differential transform method on nonlinear integro-differential equations with proportional delay,” Neural Computing and Applications, vol. 24, pp. 391–397, 2014.
- [2] A. Arikoglu, I. Ozkol, “Solution of boundary value problems for integro-differential equations by using differential transform method,” Applied Mathematics and Computation, vol. 168, pp. 1145-1158, 2005.
- [3] A. Arikoglu, I. Ozkol, “Solution of difference equations by using differential transform method,” Applied Mathematics and Computation, vol. 174, pp. 1216-1228, 2006.
- [4] A. Arikoglu, I. Ozkol, “Solution of differential–difference equations by using differential transform method,” Applied Mathematics and Computation, vol. 181, pp. 153–162, 2006.
- [5] A. Arikoglu, I. Ozkol, “Solution of fractional differential equations by using differential transform method,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1473–1481, 2007.
- [6] A. Arikoglu, I. Ozkol, “Solutions of integral and integro-differential equation systems by using differential transform method,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2411–2417, 2008.
- [7] A. Arikoglu, I. Ozkol, “Solution of fractional integro-differential equations by using fractional differential transform method,” Chaos, Solitons & Fractals, vol. 40, no. 2, pp. 521–529, 2009.
- [8] F. Ayaz, “Solutions of the system of differential equations by differential transform method,” Applied Mathematics and Computation, vol. 147, pp. 547–567, 2004.
- [9] M. M. Bahsi, A. K. Bahsi, M. Cevik, M. Sezer, “Improved Jacobi matrix method for the numerical solution of Fredholm integro-differential-difference equations,” Mathematical Sciences, vol. 10, pp. 83-93, 2016.
- [10] M. Balci, M. Sezer, “Hybrid Euler–Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations,” Applied Mathematics and Computation, vol. 273, pp. 33–41, 2016.
- [11] F. Karakoc, H. Bereketoglu, “Solutions of delay differential equations by using differential transform method,” International Journal of Computer Mathematics, vol. 86, no. 5, pp. 914-923, 2009.
- [12] Z. M. Odibat, “Differential transform method for solving Volterra integral equation with separable kernels,” Mathematical and Computer Modelling, vol. 8, no. 7-8, pp. 1144-1149, 2008.
- [13] S. Y. Reutskiy, “The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type,” Journal of Computational and Applied Mathematics, vol. 296, pp. 724-738, 2016.
- [14] F.A. Rihan, E.H. Doha, M.I. Hassan, N.M. Kamel, “Numerical treatments for Volterra delay integro-differential equations,” Computational Methods in Applied Mathematics, vol. 9, no. 3, pp. 292–308, 2009. [15] A. Saatmandi, M. Dehghan, “Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients,” Computers & Mathematics with Applications, vol. 59, pp. 2996-3004, 2010.
- [16] P. K. Sahu, S. S. Ray, “Legendre spectral collocation method for Fredholm integro-differential difference equation with variable coefficients and mixed conditions,” Applied Mathematics and Computation, vol. 268, pp. 575-580, 2015.
- [17] S. Yuzbasi, “Laguerre approach for solving pantograph-type Volterra integro-differential equations,” Applied Mathematics and Computation, vol. 232, pp. 1183–1199, 2014.
- [18] J.K. Zhou, Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986).

Year 2021,
Volume: 18 Issue: 1, 33 - 46, 01.05.2021
### Abstract

### References

- [1] R. Abazari, A. Kilicman, “Application of differential transform method on nonlinear integro-differential equations with proportional delay,” Neural Computing and Applications, vol. 24, pp. 391–397, 2014.
- [2] A. Arikoglu, I. Ozkol, “Solution of boundary value problems for integro-differential equations by using differential transform method,” Applied Mathematics and Computation, vol. 168, pp. 1145-1158, 2005.
- [3] A. Arikoglu, I. Ozkol, “Solution of difference equations by using differential transform method,” Applied Mathematics and Computation, vol. 174, pp. 1216-1228, 2006.
- [4] A. Arikoglu, I. Ozkol, “Solution of differential–difference equations by using differential transform method,” Applied Mathematics and Computation, vol. 181, pp. 153–162, 2006.
- [5] A. Arikoglu, I. Ozkol, “Solution of fractional differential equations by using differential transform method,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1473–1481, 2007.
- [6] A. Arikoglu, I. Ozkol, “Solutions of integral and integro-differential equation systems by using differential transform method,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2411–2417, 2008.
- [7] A. Arikoglu, I. Ozkol, “Solution of fractional integro-differential equations by using fractional differential transform method,” Chaos, Solitons & Fractals, vol. 40, no. 2, pp. 521–529, 2009.
- [8] F. Ayaz, “Solutions of the system of differential equations by differential transform method,” Applied Mathematics and Computation, vol. 147, pp. 547–567, 2004.
- [9] M. M. Bahsi, A. K. Bahsi, M. Cevik, M. Sezer, “Improved Jacobi matrix method for the numerical solution of Fredholm integro-differential-difference equations,” Mathematical Sciences, vol. 10, pp. 83-93, 2016.
- [10] M. Balci, M. Sezer, “Hybrid Euler–Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations,” Applied Mathematics and Computation, vol. 273, pp. 33–41, 2016.
- [11] F. Karakoc, H. Bereketoglu, “Solutions of delay differential equations by using differential transform method,” International Journal of Computer Mathematics, vol. 86, no. 5, pp. 914-923, 2009.
- [12] Z. M. Odibat, “Differential transform method for solving Volterra integral equation with separable kernels,” Mathematical and Computer Modelling, vol. 8, no. 7-8, pp. 1144-1149, 2008.
- [13] S. Y. Reutskiy, “The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type,” Journal of Computational and Applied Mathematics, vol. 296, pp. 724-738, 2016.
- [14] F.A. Rihan, E.H. Doha, M.I. Hassan, N.M. Kamel, “Numerical treatments for Volterra delay integro-differential equations,” Computational Methods in Applied Mathematics, vol. 9, no. 3, pp. 292–308, 2009. [15] A. Saatmandi, M. Dehghan, “Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients,” Computers & Mathematics with Applications, vol. 59, pp. 2996-3004, 2010.
- [16] P. K. Sahu, S. S. Ray, “Legendre spectral collocation method for Fredholm integro-differential difference equation with variable coefficients and mixed conditions,” Applied Mathematics and Computation, vol. 268, pp. 575-580, 2015.
- [17] S. Yuzbasi, “Laguerre approach for solving pantograph-type Volterra integro-differential equations,” Applied Mathematics and Computation, vol. 232, pp. 1183–1199, 2014.
- [18] J.K. Zhou, Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986).

There are 17 citations in total.

Primary Language | English |
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Subjects | Engineering |

Journal Section | Articles |

Authors | |

Publication Date | May 1, 2021 |

Published in Issue | Year 2021 Volume: 18 Issue: 1 |