scholarly journals The New Approximate Analytic Solution for Oxygen Diffusion Problem with Time-Fractional Derivative

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Vildan Gülkaç
Keyword(s):  
Oxygen Diffusion ◽  
Moving Boundary ◽  
Essential Feature ◽  
Power Series Expansion ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Diffusion Problem ◽  
Approximate Analytic Solution ◽  
Stable Case ◽  
Modified Method

Oxygen diffusion into the cells with simultaneous absorption is an important problem and it is of great importance in medical applications. The problem is mathematically formulated in two different stages. At the first stage, the stable case having no oxygen transition in the isolated cell is investigated, whereas at the second stage the moving boundary problem of oxygen absorbed by the tissues in the cell is investigated. In oxygen diffusion problem, a moving boundary is essential feature of the problem. This paper extends a homotopy perturbation method with time-fractional derivatives to obtain solution for oxygen diffusion problem. The method used in dealing with the solution is considered as a power series expansion that rapidly converges to the nonlinear problem. The new approximate analytical process is based on two-iterative levels. The modified method allows approximate solutions in the form of convergent series with simply computable components.

scholarly journals The Adomian Decomposition Method for Solving a Moving Boundary Problem Arising from the Diffusion of Oxygen in Absorbing Tissue

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Lazhar Bougoffa
Keyword(s):  
Oxygen Diffusion ◽  
Decomposition Method ◽  
Moving Boundary ◽  
Approximate Analytical Solution ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Diffusion Problem ◽  
Moving Boundary Problem ◽  
Resolution Method ◽  
Adomian Decomposition

This paper begins by giving the results obtained by the Crank-Gupta method and Gupta-Banik method for the oxygen diffusion problem in absorbing tissue, and then we propose a new resolution method for this problem by the Adomian decomposition method. An approximate analytical solution is obtained, which is demonstrated to be quite accurate by comparison with the numerical and approximate solutions obtained by Crank and Gupta. The study confirms the accuracy and efficiency of the algorithm for analytic approximate solutions of this problem.

Generalized Differential Transformation Method for Solving system of Non linear Volterra integro-differential equations of fractional order

2018 ◽  
Vol 1 (25) ◽  
pp. 523-550
Author(s):  
Basim N.Abood ◽  
Eman A.Hussain ◽  
Mayada T. Wazi
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Modified Method ◽  
Non Linear ◽  
Generalized Differential

       In this paper,  the technique of modified Generalized  Differential Transformation Method (GDTM)  is used to solve a system of Non linear integro-differential equations with initial conditions. Moreover, a particular example has been discussed in three different cases to show reliability and the performance of the modified   method. The fractional derivative is considered in the Caputo sense .The approximate solutions are calculated in the form of a convergent series, numerical results explain that this approach is trouble-free to put into practice and correct when applied to systems integro-differential equations.

scholarly journals An Efficient Method for Systems of Variable Coefficient Coupled Burgers’ Equation with Time-Fractional Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hossein Aminikhah ◽  
Nasrin Malekzadeh
Keyword(s):  
Fractional Derivative ◽  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Burgers Equation ◽  
Power Series Expansion ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Time Fractional Derivative ◽  
Coupled Burgers Equation

A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. Results indicate that the introduced method is promising for solving other types of systems of nonlinear fractional-order partial differential equations.

An improved formulation of the oxygen-diffusion problem and its application to Zircaloy oxidation by steam

1997 ◽  
Vol 47 (5-6) ◽  
pp. 427-444 ◽  
Author(s):  
S. K. Wong ◽  
C. C. Chan ◽  
S. K. Crusher Wong
Keyword(s):  
Oxygen Diffusion ◽  
Diffusion Problem

Application of He’s Homotopy Perturbation Method for Solving Fractional Fokker-Planck Equations

2009 ◽  
Vol 64 (12) ◽  
pp. 788-794 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
External Force Field ◽  
Promising Tool ◽  
Transport Problems ◽  
Fokker Planck

Abstract The fractional Fokker-Planck equation (FFPE) has been used in many physical transport problems which take place under the influence of an external force field and other important applications in various areas of engineering and physics. In this paper, by means of the homotopy perturbation method (HPM), exact and approximate solutions are obtained for two classes of the FFPE initial value problems. The method gives an analytic solution in the form of a convergent series with easily computed components. The obtained results show that the HPM is easy to implement, accurate and reliable for solving FFPEs. The method introduces a promising tool for solving other types of differential equation with fractional order derivatives

scholarly journals Numerical Solution of Higher Order Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Shahid S. Siddiqi ◽  
Muzammal Iftikhar
Keyword(s):  
Boundary Value Problems ◽  
Homotopy Analysis Method ◽  
Present Method ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Higher Order ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

The aim of this paper is to use the homotopy analysis method (HAM), an approximating technique for solving linear and nonlinear higher order boundary value problems. Using HAM, approximate solutions of seventh-, eighth-, and tenth-order boundary value problems are developed. This approach provides the solution in terms of a convergent series. Approximate results are given for several examples to illustrate the implementation and accuracy of the method. The results obtained from this method are compared with the exact solutions and other methods (Akram and Rehman (2013), Farajeyan and Maleki (2012), Geng and Li (2009), Golbabai and Javidi (2007), He (2007), Inc and Evans (2004), Lamnii et al. (2008), Siddiqi and Akram (2007), Siddiqi et al. (2012), Siddiqi et al. (2009), Siddiqi and Iftikhar (2013), Siddiqi and Twizell (1996), Siddiqi and Twizell (1998), Torvattanabun and Koonprasert (2010), and Kasi Viswanadham and Raju (2012)) revealing that the present method is more accurate.

Diffusion of Cations Along the Polymer/Metal Interface Under an Applied Electrical Potential

1993 ◽  
Vol 304 ◽  
Author(s):  
T. Nguyen ◽  
J. M. Pommersheim
Keyword(s):  
Experimental Data ◽  
Moving Boundary ◽  
Electrical Potential ◽  
Potential Gradient ◽  
Polymer Coatings ◽  
Diffusion Problem ◽  
Model Parameters ◽  
Metal Interface ◽  
Problem Model ◽  
Polymer Metal

AbstractDiffusion of cations along the polymer/metal interface controls the rate of blistering of polymer coatings on metals exposed to electrolytes. Cations are driven by both concentration and electrical potential gradients. A theoretical and experimental study was carried out on the diffusion of sodium ion along the polymer coating/steel interface under an applied potential. Mathematical models, consisting of initial and propagation stages, are derived based on a moving boundary diffusion problem. Model variables include ion diffusivity, potential gradient and distance between defects and delamination sites. Models are solved to predict ion fluxes and concentration in the blistering areas. Experimental data are analyzed to extract model parameters. Model predictions agreed well with experimental data and practical observations

Approximate Solution of Time-Fractional Chemical Engineering Equations: A Comparative Study

2010 ◽  
Vol 8 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Asmat Ara ◽  
Amir Mahmood
Keyword(s):  
Approximate Solution ◽  
Fractional Derivatives ◽  
Chemical Engineering ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Chemical Reactor ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
Conventional Numerical Method

In this paper, we present the approximate solutions of the time fractional chemical engineering equations by means of the variational iteration method (VIM) and homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. The solutions of the chemical reactor, reaction, and concentration equations are calculated in the form of convergent series with easily computable components. We compared the HPM against the VIM; an additional comparison will be made against the conventional numerical method. The results show that HPM is more promising, convenient, and efficient than VIM.

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