Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Abstract

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

Formation of Intermetallic Phases in Ion Implantation

“…e implicit-difference scheme has been suggested for the solution of diffusion kinetic problem describing ion implantation by intermetallic phase formation. For further interesting models and methods, we refer the readers to [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. We actually suggest a model of the surface modification of nickel-aluminum ions with the relaxation of mass flows.…”

Fractional residual power series method for the analytical and approximate studies of fractional physical phenomena

“…we consider = k 1, 2, 3, ... in equation (6) and substitute in equation (8). Applying the fractional derivative ( − ) D t k α 1 on both sides for = k 1, 2, 3, ... and finally, we solve…”

“…Fractional calculus is a very important and fruitful tool for describing many physical phenomena. Recently, fractional calculus is used for many purposes in several fields, such as chemistry, physics, dynamics systems, engineering and mathematical biology [1][2][3][4][5][6][7][8][9][10]. Multi-techniques are used to obtain the solutions for differential equations of fractional order such as fractional variational iteration method (VIM), Sumudu transform (ST) method, RBF meshless method, homotopy perturbation method (HPM), exp-function method, homotopy analysis method, quadrature tau method, variational Lyapunov method, adomian decomposition method (ADM), adaptive finite element method, sinc-collocation method, homotopy analysis transform method and adomian decomposition Sumudu transform method (ADSTM).…”

“…The FRPSM is constructed from the generalized Taylor series, which is a prevailing technique for solving nonlinear fractional partial differential equations [2,8]. The advantage of the current method is that it is not timeconsuming and does not require large computer memory and also it is not affected by computational round off errors.…”

Five semi analytical and numerical simulations for the fractional nonlinear space-time telegraph equation

“…Many physicists and mathematicians have focused their attention on formulating some nonlinear phenomena to demonstrate their characterization of undiscovered models [12][13][14]. Deriving accurate analytical, semianalytical, approximate techniques has taken more attention of many researchers; consequently, many distinct techniques have been derived, such as [15][16][17][18][19][20].…”