International Journal of Theoretical and Applied Mathematics
Volume 5, Issue 6, December 2019, Pages: 100-112
Received: Apr. 5, 2019;
Accepted: Nov. 29, 2019;
Published: Dec. 6, 2019
Views 579 Downloads 210
Muhammad Akbar, Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
Rashid Nawaz, Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
Sumbal Ahsan, Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed for solving integro-differential equations. In this paper, a powerful semi analytical technique known as Optimal Homotopy Asymptotic Method (OHAM) has been used for finding the approximate solutions of Fredholm type integro-differential equations and Volterra type integro-differential equations. The proposed method does not required discretization like other numerical and approximate method, and it is also free from any small/large parameters. The presented technique provides better accuracy at lower order of approximation, the accuracy of the method can further be increases with higher order of approximation. Moreover, we can easily adjust and control the convergence region. The ability of the method is checked with different problems in literature. The results obtained through OHAM are compared with solutions of Adomian Decomposition Method. It is observed that solutions obtained through the proposed method is more accurate than existing techniques, which proves the validity and stability of the proposed method for solving integro-differential equations. The presented technique is more consistent, effective, suitable and rapidly convergent. The use of Optimal Homotopy Asymptotic Method is simple and straight forward. For the computation of problems, we have used Mathematica 9.0.
Optimum Solutions of Fredholm and Volterra Integro-differential Equations, International Journal of Theoretical and Applied Mathematics.
Vol. 5, No. 6,
2019, pp. 100-112.
Shang, X., & Han, D. (2010). Application of the variational iteration method for solving nth-order integro-differential equations. Journal of Computational and Applied Mathematics, 234 (5), 1442-1447.
Wazwaz, A. M. (2014). The variational iteration method for solving the Volterra integro-differential forms of the Lane–Emden equations of the first and the second kind. Journal of Mathematical Chemistry, 52 (2), 613-626.
Eshkuvatov, Z. K., Zulkarnain, F. S., Muminov, Z., & Long, N. M. A. N. (2017). Convergence of modified homotopy perturbation method for Fredholm-Volterra integro-differential equation of order m. Malaysian Journal of Fundamental and Applied Sciences, 13 (4-1), 340-345.
Elbeleze, A. A., Kılıçman, A., & Taib, B. M. (2016). Modified Homotopy Perturbation Method for Solving Linear Second-Order Fredholm Integro–Differential Equations. Filomat, 30 (7), 1823-1831.
Shah, k., & Singh, T. (2015). • Solution of second kind Volterra Integral and Integro-defferential equation by homotopy analysis method. International Journal of Mathematical Archive EISSN 6 (4), 2229-5046.
Mohamed, M. S., Gepreel, K. A., Alharthi, M. R., & Alotabi, R. A. (2016). Homotopy analysis transform method for integro-differential equations. General Mathematics Notes, 32 (1), 32-48.
Abbas, Z., Vahdati, S., Ismail, F., & Dizicheh, A. K. (2010). Application of homotopy analysis method for linear integro-differential equations. In International Mathematical Forum. 5 (5), 237-249.
Saeidy, M., Matinfar, M., & Vahidi, J. (2010). Analytical solution of BVPs for fourth-order integro-differential equations by using homotopy analysis method. International Journal of Nonlinear Science, 9 (4), 414-421.
Aruchunan, E., & Sulaiman, J. (2011). Half-sweep conjugate gradient method for solving first order linear Fredholm integro-differential equations. Australian Journal of Basic and Applied Sciences, 5 (3), 38-43.
Alao, S., Akinboro, F. S., Akinpelu, F. O., & Oderinu, R. A. (2014). Numerical Solution of Integro-Differential Equation using Adomian Decomposition and Variational Iteration Methods. IOSR Journal of Mathematics, 10 (4), 18-22.
Vahidi, A. R., Babolian, E., Cordshooli, G. A., & Azimzadeh, Z. (2009). Numerical solution of Fredholm integro-differential equation by Adomian’s decomposition method. International Journal of Mathematical Analysis, 3 (36), 1769-1773.
Rahmani, L., Rahimi, B., & Mordad, M. (2011). Numerical Solution of Volterra-Fredholm Integro-Differential Equation by Block Pulse Functions and Operational Matrices. General Mathematics Note, 4 (2), 37-48.
Marinca, V., & Herişanu, N. (2008). Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. International Communications in Heat and Mass Transfer, 35 (6), 710-715.
Herişanu, N., Marinca, V., Dordea, T., & Madescu, G. (2008). A new analytical approach to nonlinear vibration of an electrical machine. Proceedings of the Romanian Academy-Series A, 9 (3), 229-236.
Khan, N., Hashmi, M. S., Iqbal, S., & Mahmood, T. (2014). Optimal homotopy asymptotic method for solving Volterra integral equation of first kind. Alexandria Engineering Journal, 53 (3), 751-755.
Almousa, M., & Ismail, A. (2013). Optimal homotopy asymptotic method for solving the linear Fredholm integral equations of the first kind. Abstract and Applied Analysis, 1-6.
Hashmia, M. S., Khanb, N., Iqbalc, S., & Zahida, M. A. (2016). Exact solution of Fredholmintegro-differential equations using optimal homotopy asymptotic method. Journal of. Applied Environmental and Biological Sciences, 6 (4S), 162-166.
Du Han, Y., & Yun, J. H. (2013). Optimal homotopy asymptotic method for solving integro-differential equations. International Journal of Applied Mathematics, 43 (3).
Karim, M. F., Mohamad, M., Rusiman, M. S., Che-Him, N., Roslan, R., & Khalid, K. (2018) ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations. Journal of Physics, 995 (1), 1-10.