Optimum Solutions of Fredholm and Volterra Integro-differential Equations

Abstract

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 integrodifferential 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.

Solutions of system of Volterra integro‐differential equations using optimal homotopy asymptotic method

“…With the approximate solution given in Equation 12 containing auxiliary constants, one can obtain the optimum value of these constants by method of least square discussed in Akbar et al 18…”

Approximate solutions of nonlinear two‐dimensional Volterra integral equations

“…The method is independent of any small or large parameter assumption and acquires less computational work. The technique has applied by different researchers to a number of problems arising in science and engineering, 26–31 which show the efficiency and accuracy of the proposed method.…”