Optimal Homotopy Asymptotic and Homotopy Perturbation Methods for Linear Mixed Volterra-Fredholm Integral Equations

Abstract

In this paper, we study the mixed Volterra-Fredholm integral equations of the second kind by means of optimal homotopy asymptotic method (OHAM) and Homotopy Perturbation method (HPM).Our approach is independent of time and contains simple computations with quite acceptable approximate solutions in which approximate solutions obtained by these methods are close to

Optimal Homotopy Asymptotic and Multistage Optimal Homotopy Asymptotic Methods for Solving System of Volterra Integral Equations of the Second Kind

“…where and are linear and nonlinear functional operators, respectively. ( ) is a known function, ( ) is an unknown function, and is a boundary operator [20][21][22][23][24].…”

“…Equations 4, (10), (11), (12), and (13) change to (16), 17, (18), (20), and (19), respectively. Also, initial approximation in [ , +1 ), = 0, 1, .…”

“…There are two proofs presented in [19,20] that have some oversight. The following proof covers these shortcoming.…”

“…Substitution of (44) and (45) into (43), regarding (41) and using (18)- (20), determines the residual and the functional…”

Semi-analytical solutions for the hydrodynamic stability based nonlinear fourteenth order differential problem

“…In fact, OHAM gives an easy idea to control the convergence area for strongly nonlinear problems and is successfully applied to such type of problems arising in fluid mechanics and heat transfer as studied in ([29]- [30]). In comparison to OHAM, the HPM is also easy to interpret, can find highly accurate solutions in just small number of iterations [27] with low computational cost as compare to OHAM and eliminated the limitations of traditional perturbation methods. In this direction, Roslen et al [?]…”