Fathi M. Allan

Derivation of the Adomian decomposition method using the homotopy analysis method

Abstract Adomian decomposition method has been used intensively to solve nonlinear boundary and initial value problems. It has been proved to be very efficient in generating series solutions of the problem under consideration under the assumption that such series solution exits. However, very little has been done to address the mathematical foundation of the method and its error analysis. In this article the mathematical derivation of the method using the homotopy analysis method is presented. In addition, an error analysis is addressed as well as the convergence criteria.

Comparison of nonideal sorption formulations in modeling the transport of phthalate esters through packed soil columns

Sorption of dimethyl phthalate (DMP), diethyl phthalate (DEP) and dipropyl phthalate (DPP) to two soil materials that vary in organic matter content was investigated using miscible displacement experiments under saturated flow conditions. Generated breakthrough curves (BTCs) were inversely simulated using linear, equilibrium sorption (LE), nonlinear, equilibrium sorption (NL), linear, first-order nonequilibrium sorption (LFO), linear, radial diffusion (LRD), and nonlinear, first-order nonequilibrium sorption (NFO) models. The Akaike information criterion was utilized to determine the preferred model. The LE model could not adequately describe phthalate ester (PE) BTCs in higher organic matter soil or for more hydrophobic PEs. The LFO and LRD models adequately described the BTCs but a slight improvement in curve-fitting was gained in some cases when the NFO model was used. However, none of the models could properly describe the desorptive tail of DPP for the high organic matter soil. Transport of DPP through this soil was adequately predicted when degradation or sorption hysteresis was considered. Using the optimized parameter values along with values reported by others it was shown that the organic carbon distribution coefficient (K(oc)) of PEs correlates well with the octanol/water partition coefficient (K(ow)). Also, a strong relationship was found between the first-order sorption rate coefficient normalized to injection pulse size and compound residence time. A similar trend of timescale dependence was found for the rate parameter in the radial diffusion model. Results also revealed that the fraction of instantaneous sorption sites is dependent on K(ow) and appears to decrease with the increase in the sorption rate parameter.

Decomposition method for solving nonlinear integro-differential equations

This paper outlines a reliable strategy for solving nonlinear Volterra-Fredholm integro-differential equations. The modified form of Adomian decomposition method is found to be fast and accurate. Numerical examples are presented to illustrate the accuracy of the method.

Construction of solutions for the shallow water equations by the decomposition method

This paper deals with the implementation of Adomians decomposition method for the variable-depth shallow water equations with source term. Using this method, the solutions were calculated in the form of a convergent power series with easily computable components. The convergence of the method is illustrated numerically.

An efficient algorithm for solving higher-order fractional Sturm–Liouville eigenvalue problems

Abstract In this paper, we present a simple and efficient computational algorithm for solving eigenvalue problems of high fractional-order differential equations with variable coefficients. The method of solution is based on utilizing the series solution to convert the governing fractional differential equation into a linear system of algebraic equations. Then, the eigenvalues can be calculated by finding the roots of the corresponding characteristic polynomial. Notice that this class of eigenvalue problems is very promising to the solution of linear fractional partial differential equations (FPDE). The numerical results demonstrate reliability and efficiency of the proposed algorithm. Based on our simulations some theoretical conjectures are reported.

Fluid-particle model of flow through porous media: The case of uniform particle distribution and parallel velocity fields

Under the assumption of a uniform distribution of dust particles, we develop a mathematical model to describe the flow of a dusty fluid through isotropic consolidated and granular porous materials. Intrinsic volume averaging is employed to develop continuity and momentum equations when the fluid- and dust-phase velocities are everywhere parallel.

Infiltration of oil into porous sediments

The continuum flow of a dilute system through a porous sediment is considered. The fluid system is composed of a carrier fluid-phase and an oil-phase. Model equations governing the time-dependent flow of the incompressible two-phase fluid are developed based on volume averaging.

On the similarity solution of nano-fluid flow over a moving flat plate using the homotopy analysis method

The present article discusses the characteristics of Newtonian water-base-Copper nano-fluid flowing over an infinite flat plate moving with a constant velocity in the direction of the flow. The non-classical similarity transformation is employed to transform the Navier-Stokes equation into a nonlinear ordinary differential equation with specific boundary conditions. The Homptopy analysis method (HAM) is employed to solve the resulting nonlinear differential equation to study the effects of the nanoparticle volume fraction and wall velocity on the flow velocity profile, the boundary layer thickness and the local skin friction coefficient. The existence and non-uniqueness of the solution as a function of the wall velocity will be also discussed.

Solving nonlinear boundary value problems using the homotopy analysis method

Homotopy analysis method (HAM) has been employed recently by many authors to solve nonlinear problems, in particular nonlinear initial and boundary values problems. Such nonlinear problems are usually derived from physical problems such as fluid mechanics; heat transfer, boundary layer equations and many others. In the suggested work we will extend the use of the HAM to solve a certain class of boundary value problems. Focus will be on multi-layer boundary problems. Examples of these kind of problems include fluid flow through multi-layer porous media.

Multi-layer parallel shooting method for multi-layer boundary value problems

Multi-layer boundary value problems have received a great deal of attention in the past few years. This is due to the fact that they model many engineering applications. Examples of applications include fluid flow though multi-layers porous media such as ground water and oil reservoirs. In this work, we present a new method for solving multilayer boundary value problems. The method is based on an efficient adaption of the classical shooting method, where boundary value problems are solved for by means of solving a sequence of initial value problems. Illustration of the method is presented on application to fluid flow through multi-layer porous media. The examples presented suggested that the method is reliable and accurate.