Forthcoming articles


International Journal of Applied Nonlinear Science


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International Journal of Applied Nonlinear Science (3 papers in press)


Regular Issues


  • A hybrid numerical treatment of nonlinear reaction-diffusion equations with memory: A prototypical Fisher-Kolmogorov-Petrovskii-Piskunov equation   Order a copy of this article
    by Okey Oseloka Onyejekwe 
    Abstract: In reaction-diffusion systems with non-standard diffusion, the memory of the transport process causes a coupling of reaction and diffusion. A generalization of the Ficks law has been suggested to account for this coupling. Furthermore, the resultant effects of the interplay of transport , memory and reaction lend themselves to some interesting physics which is still not well understood because the governing equation as well as the accompanying memory integral and nonlinear reaction terms are not always amenable to tidy analytic or numerical expressions. Hence the derivation of a suitable governing integro-differential equation as well as the approximate solution demand a non-standard numerical procedure. The main focus of this work can be seen as a contribution towards this objective. In this report, we develop and apply an hybrid boundary integral-finite element finite difference numerical procedure to investigate an integro-differential-FKPP(Fisher-Kolmogorov-Petrovskii-Piskunov) type kinetics. We also focus on scalar evolution for cases where the reaction coefficient takes on relatively large values. Although we are still far from a rigorous mathematical analysis, it has been found that the numerical results obtained compared favorably with existing benchmark solutions. This not only validates the current numerical formulation but also justifies the physics of the resulting front propagation.
    Keywords: Hybrid numerical formulation; boundary integral; finite element; finite differerence; integro-differential partial differential equation; Fisher-Klomogorov-Petrovskii- Piskunov equation; memory term; nonlinear; reaction-diffusion.

  • Pressure curves for compressible flows with slip through asymmetric local constrictions   Order a copy of this article
    by Salahaldeen Rabba, Katrin Rohlf 
    Abstract: A second-order non-linear differential equation is derived for the pressure of a compressible flow with slip at the wall through a constricted cylinder. The ideal gas equation of state is used, and the Karman-Pohlhausen method is utilized to derive the pressure differential equation from the Navier-Stokes equations of motion for a Newtonian viscous fluid. The solution for pressure is determined numerically and assessed in various flow geometries. This work is an extension of existing assessments in that non-linear terms are kept in the differential equation for pressure, as well as second-order derivative terms. Additionally, wall slip and compressibility are incorporated in the equations, as well as geometries that are asymmetric with respect to the location of maximum constriction.
    Keywords: Pressure; Gradient; Compressible; Stenosis; Navier-Stokes; Slip; Karman-Pohlhausen; Asymmetric.

  • Determining an optimal value for the convergence control parameter in the HAM   Order a copy of this article
    by Martin Hermann 
    Abstract: In the applications, the homotopy analysis method (HAM) is an often used method to determine an analytical approximate solution of lower-dimensional nonlinear ordinary differential equations. This approximation consists of an infinite series which depends on an auxiliary real parameter h. This parameter must be adjusted such that the series converges towards the exact solution of the given problem. In this paper we propose an approach, which is based on the residual of the terminated series, to determine an optimal value h_opt or an optimal region for h. Using the numerical computing environment Matlab, we describe several possibilities how this approach can be realized. Finally, by means of three examples (an IVP, a two-point BVP, as well as a BVP on an infinite interval) we show how this mathematically sophisticated strategy can be applied and we present the optimal parameter h_opt for each example.
    Keywords: nonlinear ODEs; homotopy analysis method; HAM; auxiliary parameter.