In this paper, modified variational iteration algorithm-II is investigated for finding approximate solutions of nonlinear Parabolic equations. Comparisons of the MVIA-II with trigonometric B-spline collocation method, variational iteration method, homotopy perturbation transform method, Adomian decomposition method, and modified variational iteration method are carried out, which show that the proposed algorithm (MVIA-II) is robust one. Some nonlinear Parabolic equations are given to demonstrate the implementation and accuracy of the MVIA-II.

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In this work, the variational iteration algorithm-I with an auxiliary parameter is used for the analytical treatment of the wave equations and wave-like vibration equations. The technique has the capability of reducing the size of computational work and easily overcomes the difficulty of the perturbation method or Adomian polynomials. Comparison with the classic variational iteration algorithm-I (VIA-1) is carried out, showing that the modification is more efficient and reliable. Keywords Variational iteration algorithm-I, wave equation, wave-like vibration equation, VIA-I with an auxiliary parameter where g(u) is a linear function. The vibrational behavior of beams and shafts can be expressed in terms of waves. 1 The motion of the system of vibrating beams, rods, and springs reduces to that of the wave equation. The wave equation is probably the most used partial differential equation (PDE) in practical applications. Waves exist in different media like as in geophysics there are surface and internal waves in the ocean, but there are also more complex nonlinear models comparing to PDE. There are wave processes in the Earth (seismic waves), acoustic waves in the air and water, electromagnetic waves in different media. Initial equations differ for each case, but the wave equation can be derived in some approximation in most of the cases.

The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.

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