New Iterative Method Research Articles

NUMERICAL STUDY OF OXYGEN DIFFUSION FROM CAPILLARY TO TISSUES DURING HYPOXIA WITH EXTERNAL FORCE EFFECTS

A mathematical model to simulate oxygen delivery through a capillary to tissues under the influence of an external force field is presented. The multi-term general fractional diffusion equation containing force terms and a time dependent absorbent term is taken into account. Fractional calculus is applied to describe the phenomenon of sub-diffusion of oxygen in both transverse and longitudinal directions. A new computational algorithm, i.e., the new iterative method (NIM) is employed to solve the spatio-temporal fractional partial differential equation subject to appropriate physical boundary conditions. Validation of NIM solutions is achieved with a modified adomian decomposition method (MADM). A parametric study is conducted for three loading scenarios on the capillary-radial force alone, axial force alone and the combined case of both forces. The results demonstrate that the force terms markedly influence the oxygen diffusion process. For example, the radial force exerts a more profound effect than axial force on sub-diffusion of oxygen indicating that careful manipulation of these forces on capillary-tissues may assist in the effective reduction of hypoxia or other oxygen depletion phenomena.

An analytical treatment to fractional Fornberg\u2013Whitham equation

In this paper, an analytical technique, namely the new iterative method (NIM), is applied to obtain an approximate analytical solution of the fractional Fornberg–Whitham equation. The obtained approximate solutions are compared with the exact or existing numerical results in the literature to verify the applicability, efficiency, and accuracy of the method.

Approximate solutions of time fractional Kawahara and modified Kawahara equations by Fractional complex transform

In this paper, fractional complex transform with new iterative method (NIM) is used to obtain approximate solutions for the nonlinear time fractional Kawahara and modified Kawahara equations based on He's fractional derivative. Fractional complex transform is proposed to convert time fractional Kawahara and modified Kawahara equations to the nonlinear ordinary differential equations and then NIM is applied to the new obtained equations. The obtained approximate solutions are compared with the exact solutions to verify the applicability, efficiency and accuracy of the method.

New iterative method for solving gas dynamic equation

In this paper, the gas dynamic equation is solved through new iterative method (NIM). The obtained results are compared with those of homotopy perturbation method (HPM), variational iteration method with He's polynomials (VIMHP) and Laplace transform new homotopy perturbation method (LTNHPM). It is noted that the NIM in case of nonhomogeneous problems takes the form of a convergent series with easily computable components. This method is able to solve large class of linear and nonlinear equations effectively, more easily and accurately; and thus the method has been widely applicable to solve any class of equations in sciences and engineering. Keywords : New Iterative Method, Homotopy Perturbation Method, Variational Iteration Method with He's Polynomials, Laplace Transform New Homotopy Perturbation Method, Gas Dynamic Equation.

New Iterative Method for Solving the Fornberg-Whitham Equation and Comparison with Homotopy Perturbation Transform Method

This paper presents an approximate analytical solution of the nonlinear Fornberg-Whitham equation using the new iterative method (NIM). A comparison is made between the NIM results, homotopy perturbation transform method (HPTM) and the Adomian's decomposition method (ADM). The solution procedure reveals that NIM is a reliable, simple and effective. The proposed technique solves nonlinear problems without using Adomian's polynomials and He's polynomials which is a clear advantage of it over the decomposition method. The results reveal that the proposed algorithm is very efficient, simple and can be applied to other nonlinear problems.

An Analytical Treatment to Fractional Gas Dynamics Equation

In this paper, the new iterative method (NIM) is applied to solve nonlinear fractional gas dynamics equation. Further, a coupling of the Sumudu transform and Adomian decomposion (STADM) is used to get an approximate solution of the same problem. The results obtained by the two methods are found to be in agreement. Therefore, the NIM may be considered efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations.

Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method

An extension of the so-called new iterative method (NIM) has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

Numerical solutions of higher order nonlinear boundary value problems by new iterative method

In this work, we present the approximate solutions of higher order nonlinear boundary value problems by an efficient numerical algorithm. The New Iterative Method (NIM) will by used to find such solutions. The solutions thus obtained by NIM, are in the form of rapidly convergent series. The approach developed is tested through examples, which gives the stability and efficiency of the proposed algorithm.

An efficient new iterative method for finding exact solutions of nonlinear time-fractional partial differential equations

In this paper, a new iterative method (NIM) is used to obtain the exact solutions of some nonlinear time-fractional partial differential equations. The fractional derivatives are described in the Caputo sense. The method provides a convergent series with easily computable components in comparison with other existing methods.

A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues

In this article, we propose a new mathematical model, namely a multi-term fractional diffusion equation, for oxygen delivery through a capillary to tissues. Fractional calculus is applied to describe the phenomenon of subdiffusion of oxygen in both transverse and longitudinal directions. A new iterative method (NIM) and a modified Adomian decomposition method (MDM) are used to solve the multi-term fractional diffusion equation for different conditions. The results thus obtained are compared and presented graphically. It is observed that the order of the diffusion equation affects the delivery of oxygen significantly.

New iterative method: Application to partial differential equations

In the present article, we implement the new iterative method (NIM) proposed by Daftardar-Gejji and Jafari [V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753–763] to solve linear/nonlinear partial differential equations of integer and fractional order. The results obtained are compared with those obtained by other iterative methods such as adomian decomposition, homotopy perturbation and variational iteration methods.