An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline WENO reconstruction with high order flux evaluation. Polyharmonic splines are utilised in the recovery step of the finite volume method yielding a WENO reconstruction that is stable, flexible and optimal in the associated Sobolev (Beppo-Levi) space. The flux evaluation is accomplished by solving generalised Riemann problems across cell interfaces. The mesh adaptation is performed through an a posteriori error indicator, which relies on the polyharmonic spline reconstruction scheme. The performance of the proposed method is illustrated by a series of numerical experiments, including linear advection, Burgers' equation, Smolarkiewicz's deformational flow test, and the five-spot problem.

This paper presents a numerical technique for solving fractional integrals of functions by employing the trapezoidal rule in conjunction with the finite difference scheme. The proposed scheme is only a simple modification of the trapezoidal rule, in which it is treated as an algorithm in a sequence of small intervals for finding accurate approximate solutions to the corresponding problems. This method was applied to solve fractional integral of arbitrary order α > 0 for various values of alpha. The fractional integrals are described in the Riemann-Liouville sense. Figurative comparisons and error analysis between the exact value, two-point and three-point central difference formulae reveal that this modified method is active and convenient.

No abstract

Chicken Pox (also called Varicella) is a disease caused by a virus known as Varicella Zoster Virus (VZV) also known as human herpes virus 3 (HHV -3). Varicella Zoster Virus (VZV) is a DNA virus of the Herpes group, transmitted by direct contact with infective individuals. In this work, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination strategy was solved, using Adomian Decomposition Method (ADM) and Fourth-Fifth Rungekutta Felhberg Method and Approximate solutions were realized. ADM, yields analytical solution in terms of rapidly convergent infinite power series with easily computed terms. This solution was realized by applying Adomian polynomials to the nonlinear terms in the system. Similarly, fourth-fifth-order Runge-Kutta Felberg method with degree four interpolant (RK45F) was used to compute a numerical solution that was used as a reference solution to compare with the semianalytical approximations. The main advantage of the ADM is that it yields an approximate series solution in close form with accelerated convergence. The effect of Varicella was considered in five compartments: The Susceptible, the Vaccinated, the Exposed, the Infective and the Recovered class. The Varicella Zoster virus model which is a nonlinear system can only be solved conveniently using powerful semi-analytic tool such as the ADM. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

No abstract

No abstract

This research work studies the human brain information processing dynamics by transforming the stage model formulated by Atkinson and Shiffrin into two deterministic mathematical models. This makes it more amenable to mathematical analysis. The two models are bottom-up processing mathematical model and top-down processing mathematical model. The bottom-up processing is data driven while the top-down processing is triggered by experience or prior knowledge. Both analytical and numerical methods are used in the analysis of the models. The existence and stability of equilibrium states of the models are investigated, and threshold values of certain parameters of the models arising from the investigation were obtained and interpreted in physical terms. Numerical experiments are also carried out using hypothetical data to further investigate the effect of certain parameters on the human brain information processing process. The results show that attention, repetition and rehearsal play significant roles in learning process. Furthermore, repetition and rehearsal is strongly recommended as an effective way of retaining information. In addition, the instructors should ensure that the students feel physically and psychologically safe in any environment in order to pay adequate attention.