Author(s)

Zainidin Eshkuvatov^1,2*, Davron Khayrullaev^1,3, Muzaffar Nurillaev³, Shalela Mohd Mahali¹, Anvar Narzullaev⁴

¹Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu (UMT), Terengganu, Malaysia.
²Faculty of Applied Mathematics and Intellectual Technologies, National University of Uzbekistan, Tashkent, Uzbekistan.
³Faculty of Physics and Mathematics, Tashkent State Pedagogical University (TSPU), Tashkent, Uzbekistan.
⁴Faculty of Science and Technology, University Sains Islam Malaysia (USIM), Bandar Baru Nilai, Malaysia.

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

Integral-Differential Equations, Homotopy Analyses Method, Iterative System, Algebraic Equations, Gauss-Legendre Quadrature Formula

Eshkuvatov, Z. , Khayrullaev, D. , Nurillaev, M. , Mahali, S. and Narzullaev, A. (2023) Application of HAM for Nonlinear Integro-Differential Equations of Order Two. Journal of Applied Mathematics and Physics, 11, 55-68. doi: 10.4236/jamp.2023.111005.