Ali R. Ansari

Unsteady squeezing flow of a viscous MHD fluid between parallel plates, a solution using the homotopy perturbation method

Abstract The present paper analyses the unsteady 2‐dimensional flow of a viscous MHD fluid between two parallel infinite plates. The two infinite plates are considered to be approaching each other symmetrically, causing the squeezing flow. A similarity transformation is used to reduce the partial differential equations modeling the flow, to a single fourth‐order non‐linear differential equation containing the Reynolds number and the magnetic field strength as parameters. The velocity functions are obtained for a range of values of both parameters by using the homotopy perturbation method. The total resistance to the upper plate is presented.

Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method

Abstract We investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM) in solving time dependent partial differential equations. To this effect we consider the homogeneous, non-homogeneous, linear and nonlinear Klein–Gordon equations with boundary conditions. The results reveal that the method is explicit, effective, and easy to use.

A semi-analytical iterative technique for solving nonlinear problems

We present a semi-analytical iterative method for solving nonlinear differential equations. To demonstrate the working of the method we consider some nonlinear ordinary differential equations with appropriate initial/boundary conditions. In each of the examples we demonstrate the accuracy and convergence of the method to the solution. We demonstrate clearly that the method is accurate, fast and has a high order of convergence.

A new iterative technique for solving nonlinear second order multi-point boundary value problems

Abstract We present a semi-analytical iterative method for solving nonlinear second order multi-point boundary value problems. To demonstrate the working of the method we consider a particular example of this class of problems. In this example, we demonstrate the accuracy and convergence of the method to the solution. We demonstrate clearly that the method is accurate, fast and has a reasonable order of convergence.

Numerical solution of a convection diffusion problem with Robin boundary conditions

We consider a one-dimensional steady-state convection dominated convection-diffusion problem with Robin boundary conditions. We show, both theoretically and with numerical experiments, that numerical solutions obtained using an upwind finite difference scheme on Shishkin meshes are uniformly convergent with respect to the diffusion coefficient.

Use of Optimal Homotopy Asymptotic Method and Galerkin’s Finite Element Formulation in the Study of Heat Transfer Flow of a Third Grade Fluid Between Parallel Plates

We investigate the effectiveness of the optimal homotopy asymptotic method (OHAM) in solving nonlinear systems of differential equations. In particular we consider the heat transfer flow of a third grade fluid between two heated parallel plates separated by a finite distance. The method is successfully applied to study the constant viscosity models, namely plane Couette flow, plane Poiseuille flow, and plane Couette-Poiseuille flow for velocity fields and the temperature distributions. Numerical solutions of the systems are also obtained using a finite element method (FEM). A comparative analysis between the semianalytical solutions of OHAM and numerical solutions by FEM are presented. The semianalytical results are found to be in good agreement with numerical solutions. The results reveal that the OHAM is precise, effective, and easy to use for such systems of nonlinear differential equations.

An evolutionary approach to Wall Shear Stress prediction in a grafted artery

Abstract Restoring the blood supply to a diseased artery is achieved by using a vascular bypass graft. The surgical procedure is a well documented and successful technique. The most commonly cited hemodynamic factor implicated in the disease initiation and proliferation processes at graft/artery junctions is Wall Shear Stress (WSS). WSS distributions are predicted using numerical simulations as they can provide quick and precise results to assess the effects that alternative graft/artery junction geometries have on the WSS distributions in bypass grafts. Validation of the numerical model is required and in vitro studies, using laser Doppler anemometry (LDA), have been employed to achieve this. Numerically, the Wall Shear Stress is predicted using velocity values stored in the computational cell near the wall and assuming zero velocity at the wall. Experimentally obtained velocities require a mathematical model to describe their behavior. This study employs a grammar based evolutionary algorithm termed Chorus for this purpose and demonstrates that Chorus successfully attains this objective. It is shown that even with the lack of domain knowledge, the results produced by this automated system are comparable to the results in the literature.

A note on iterative methods for solving singularly perturbed problems using non-monotone methods on Shishkin meshes

Abstract Non-monotone methods with Shishkin meshes are employed in obtaining finite difference schemes for solving a linear two-dimensional steady state convection–diffusion problem. Preconditioners are used that significantly reduce the number of iterations of the linear solver. Computational results for a Galerkin method are presented which indicate parameter robust, super-linear orders of convergence.

A comparison of the adomian and homotopy perturbation methods in solving the problem of squeezing flow between two circular plates

Abstract The objective of this paper is to compare two methods employed for solving nonlinear problems, namely the Adomian Decomposition Method (ADM) and the Homotopy Perturbation Method (HPM). To this effect we solve the Navier‐Stokes equations for the unsteady flow between two circular plates approaching each other symmetrically. The comparison between HPM and ADM is bench‐marked against a numerical solution. The results show that the ADM is more reliable and efficient than HPM from a computational viewpoint. The ADM requires slightly more computational effort than the HPM, but it yields more accurate results than the HPM.

A note on fitted operator methods for a laminar jet problem

We consider the classical problem of a two-dimensional laminar jet of incompressible fluid flowing into a stationary medium of the same fluid [H. Schlichting, Boundary-Layer Theory, McGraw-Hill, 1979]. The equations of motion are the same as the boundary layer equations for flow over an infinite flat plate, but with different boundary conditions. It has been shown [A.R. Ansari et al., Parameter robust numerical solutions for the laminar freejet, submitted] that using an appropriate piecewise uniform mesh, numerical solutions together with their scaled discrete derivatives are obtained which are parameter (i.e., viscosity v) robust with respect to both the number of mesh nodes and the number of iterations required for convergence. We prove that there do not exist fitted operator schemes which converge v-uniformly if the fitting coefficients are independent of the problem data.