Approximation of Solutions of the Forced Duffing Equation with Nonlocal Discontinuous Type Integral Boundary Conditions

Integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow and population dynamics, see for example [16, 17, 24]. In fact, boundary value problems involving integral boundary conditions have received considerable attention, see for instance, [3, 10], [12]–[15], [18, 19, 26] and the references therein. In a recent reference [2], Ahmad, et. al. discussed the existence and uniqueness of the solutions of a boundary value problem with discontinuous type integral boundary conditions. The monotone iterative technique coupled with the method of upper and lower solutions [5, 8, 20, 23, 25] manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions. In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear [11, 21]. To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization (QSL) [9]. This method has been developed for a variety of problems [1, 4, 6, 7, 22]. In view of its diverse applications, this approach is quite an elegant and easier for application algorithms. To the best of our knowledge, the method of quasilinearization has not been developed for Duffing equation with nonlocal discontinuous type integral boundary conditions. In this paper, we apply a quasilinearization technique to obtain the analytic approximation of the solution of the forced Duffing equation with nonlocal discontinuous type integral boundary conditions. In fact, we obtain a sequence of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand. The concept of nonlocal discontinuous integral