Search

The study of biofilm formation is becoming increasingly important. Microbes that produce biofilms have complicated impact on medical implants. In this paper, we construct an unconditionally positive non-standard finite difference scheme for a mathematical model of biofilm formation on a medical implant. The unknowns in many applications reflect values that cannot be negative, such as chemical component concentrations or population numbers. The model employed here uses the bistable Allen-Cahn partial differential equation, which is a generalization of Fisher’s equation. We study consistency and convergence of the scheme constructed. We compare the performance of our scheme with a classical finite difference scheme using four numerical experiments. The technique used in the construction of unconditionally positive method in this study can be applied to other areas of mathematical biology and sciences. The results here elaborate the benefits of the non-standard approximations over the classical approximations in practical applications.

In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

Two new explicit finite difference schemes for the solution of the one-dimensional Korteweg-de-Vries equation are proposed. This equation describes the character of a wave generated by an incompressible fluid. We analyse the spectral properties of our schemes against two existing schemes proposed by Zabusky and Kruskal (1965) and Wang et al. (2008). An optimisation technique based on minimisation of the dispersion error is implemented to compute the optimal value of the spatial step size at a given value of the temporal step size and this is validated by some numerical experiments. The performance of the four methods are compared in regard to dispersive and dissipative errors and their ability to conserve mass, momentum and energy by using two numerical experiments which involve solitons. [ABSTRACT FROM AUTHOR]

In this paper, we use three existing schemes namely, upwind forward Euler, non-standard finite difference (NSFD) and unconditionally positive finite difference (UPFD) schemes to solve two numerical experiments described by a linear and a nonlinear advection-diffusion-reaction equation with constant coefficients. These equations model exponential travelling waves and biofilm growth on a medical implant respectively. We study the exact and numerical dissipative and dispersive properties of the three schemes for both problems. Moreover, L1 error, dispersion and dissipation errors, at some values of temporal and spatial step sizes have been computed for the three schemes for both problems. [ABSTRACT FROM AUTHOR]

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted by h and k, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to the L1 norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advectiondiffusion equation at some values of k and h. Two optimisation techniques are then implemented to find the optimal values of k when h = 0.02 for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments. [ABSTRACT FROM AUTHOR]

We describe briefly howa third-order Weighted Essentially Nonoscillatory (WENO) scheme is derived by coupling aWENO spatial discretization scheme with a temporal integration scheme. The scheme is termed WENO3. We perform a spectral analysis of its dispersive and dissipative properties when used to approximate the 1D linear advection equation and use a technique of optimisation to find the optimal cfl number of the scheme. We carry out some numerical experiments dealing with wave propagation based on the 1D linear advection and 1D Burger's equation at some different cfl numbers and show that the optimal cfl does indeed cause less dispersion, less dissipation, and lower L1 errors. Lastly, we test numerically the order of convergence of the WENO3 scheme. [ABSTRACT FROM AUTHOR]

We perform a spectral analysis of the dispersive and dissipative properties of two time-splitting procedures, namely, locally onedimensional (LOD) Lax-Wendroff and LOD (1, 5) [9] for the numerical solution of the 2D advection-diffusion equation. We solve a 2D numerical experiment described by an advection-diffusion partial differential equation with specified initial and boundary conditions for which the exact solution is known. Some errors are computed, namely, the error rate with respect to the L1 norm, dispersion and dissipation errors. Lastly, an optimization technique is implemented to find the optimal value of temporal step size that minimizes the dispersion error for both schemes when the spatial step is chosen as 0.025, and this is validated by numerical experiments. [ABSTRACT FROM AUTHOR]

The numerical simulation of aeroacoustic phenomena requires high-order accurate numerical schemes with low dispersion and low dissipation errors. A technique has recently been devised in a Computational Fluid Dynamics framework which enables optimal parameters to be chosen so as to better control the grade and balance of dispersion and dissipation in numerical schemes (Appadu and Dauhoo, 2011; Appadu, 2012a; Appadu, 2012b; Appadu, 2012c). This technique has been baptised as the Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation (MIEELDLD) and has successfully been applied to numerical schemes discretising the 1-D, 2-D, and 3-D advection equations. In this paper, we extend the technique of MIEELDLD to the field of computational aeroacoustics and have been able to construct high-order methodswith Low Dispersion and Low Dissipation properties which approximate the 1-D linear advection equation. Modifications to the spatial discretization schemes designed by Tam and Webb (1993), Lockard et al. (1995), Zingg et al. (1996), Zhuang and Chen (2002), and Bogey and Bailly (2004) have been obtained, and also a modification to the temporal scheme developed by Tam et al. (1993) has been obtained. These novel methods obtained using MIEELDLD have in general better dispersive properties as compared to the existing optimised methods. [ABSTRACT FROM AUTHOR]

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher's information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced. [ABSTRACT FROM AUTHOR]

Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight | x | α exp (− c x 6) , c > 0 , α > − 1 . Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat's quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara's symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials. [ABSTRACT FROM AUTHOR]