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This paper develops a second-order explicit predictor-corrector numerical approach for solving a mathematical model on the dynamic of cytokine expressions and human immune cell activation in response to the bacterium staphylococcus aureus (S. aureus). The proposed algorithm is at least zero-stable and second-order accurate. Mathematical modeling works that analyze the human body in response to some antigens have predicted concentrations of a broad range of cells and cytokines. This study deals with a coupled cellular-cytokine model which predicts cytokine expressions in response to gram-positive bacteria S. aureus. Tumor necrosis factor alpha, interleukin 6, interleukin 8 and interleukin 10 are included to assess the relationship between cytokine release from macrophages and the concentration of the S. aureus antigen. Ordinary differential equations are used to model cytokine levels while the cellular responses are modeled by partial differential equations. Interactions between both components provide a more robust and complete systems of immune activation. In the numerical simulations, a low concentration of S. aureus is used to measure cellular activation and cytokine expressions. Numerical experiments indicate how the human immune system responds to infections from different pathogens. Furthermore, numerical examples suggest that the new technique is faster and more efficient than a large class of statistical and numerical schemes discussed in the literature for systems of nonlinear equations and can serve as a robust tool for the integration of general systems of initial-boundary value problems., Comment: 20 pages, 15 figures, 4 tables

This paper develops a robust three-level time split high-order Leapfrog/Crank-Nicolson technique for solving the two-dimensional unsteady sobolev and regularized long wave equations arising in fluid mechanics. A deep analysis of the stability and error estimates of the proposed approach is considered using the $L^{\infty}(0,T;H^{2})$-norm. Under a suitable time step requirement, the theoretical studies indicate that the constructed numerical scheme is strongly stable (in the sense of $L^{\infty}(0,T;H^{2})$-norm), temporal second-order accurate and convergence of order $O(h^{\frac{8}{3}})$ in space, where $h$ denotes the grid step. This result suggests that the proposed algorithm is less time consuming, fast and more efficient than a broad range of numerical methods widely discussed in the literature for the considered problem. Numerical experiments confirm the theory and demonstrate the efficiency and utility of the three-level time split high-order formulation., Comment: 20 pages, 3 tables, 12 figures

This paper constructs a third-step second-order numerical approach for solving a mathematical model on the dynamic of corruption and poverty. The stability and error estimates of the proposed technique are analyzed using the $L^{2}$-norm. The developed algorithm is at less zero-stable and second-order accurate. Furthermore, the new method is explicit, fast and more efficient than a large class of numerical schemes applied to nonlinear systems of ordinary differential equations and can serve as a robust tool for integrating general systems of initial-value problems. Some numerical examples confirm the theory and also consider the corruption and poverty in Cameroon., Comment: 21 pages, 5 tables, 5 figures

This paper considers a two-step fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving the time-variable fractional mobile-immobile advection-dispersion model subjects to suitable initial and boundary conditions. Both stability and error estimates of the new approach are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theoretical studies show that the proposed technique is unconditionally stable with convergence of order $O(k+h^{4})$, where $h$ and $k$ are space step and time step, respectively. This result indicate that the two-step fourth-order formulation is more efficient than a broad range of numerical schemes widely studied in the literature for the considered problem. Numerical experiments are performed to verify the unconditional stability and convergence rate of the developed algorithm., Comment: 28 pages, 16 figures, 4 tables

This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theory indicates that the method is unconditionally stable with convergence of order $O(k^{2-\frac{\lambda}{2}}+h^{4})$, where $k$ and $h$ are time step and mesh size, respectively, and $\lambda\in(0,1)$. This result suggests that the two-level fourth-order technique is more efficient than a large class of numerical techniques widely studied in the literature for the considered problem. Some numerical evidences are provided to verify the unconditional stability and convergence rate of the proposed algorithm., Comment: 37 pages, 4 tables, 16 figures

This paper considers an explicit numerical scheme for solving the mathematical model of the propagation of Covid-19 epidemic with undetected infectious cases. We analyze the stability and convergence rate of the new approach in $L^{\infty}$-norm. The proposed method is less time consuming. Furthermore, the method is stable, at least second-order convergent and can serve as a robust tool for the integration of general systems of ordinary differential equations. A wide set of numerical evidences which consider the case of Cameroon are presented and discussed., Comment: 26 pages, 11 figures, 11 tables

In this paper, we analyze the three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions. The differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second order accurate in time and fourth convergent in space, while it is second order convergent in both time and space for high Reynolds numbers problems. This shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature for solving the two-dimensional time dependent nonlinear coupled Burgers' equations. Numerical examples which confirm the theoretical results are presented and discussed., Comment: 17 pages, 4 tables, 11 figures

Starting from the definition of an $n\times n$ $g$-Toeplitz matrix, $T_{n,g}(u)=\left[\widehat{u}_{r-gs}\right]_{r,s=0}^{n-1},$ where $g$ is a given nonnegative parameter, $\{\widehat{u}_{k}\}$ is the sequence of Fourier coefficients of the Lebesgue integrable function $u$ defined over the domain $\mathbb{T}=(-\pi,\pi]$, we consider the product of $g$-Toeplitz sequences of matrices, $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\},$ which extends the product of Toeplitz structures, $\{T_{n}(f_{1})T_{n}(f_{2})\},$ in the case where the symbols $f_{1},f_{2}\in L^{\infty}(\mathbb{T}).$ Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of $g$-Toeplitz structures. Specifically, for $g\geq2$ our result shows that the sequences $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\}$ are clustered to zero. This extends the well-known result, which concerns the classical case (that is, $g=1$) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed., Comment: 28 pages, 16 tables

A three-level explicit time-split MacCormack scheme is proposed for solving the two-dimensional nonlinear reaction-diffusion equations. The computational cost is reduced thank to the splitting and the explicit MacCormack scheme. Under the well known condition of Courant-Friedrich-Lewy (CFL) for stability of explicit numerical schemes applied to linear parabolic partial differential equations, we prove the stability and convergence of the method in $L^{\infty}(0,T;L^{2})$-norm. A wide set of numerical evidences which provide the convergence rate of the new algorithm are presented and critically discussed., Comment: 25 pages; 12 figures; 6 tables

In the last decades, more or less complex physically-based hydrological models, have been developed to solve the shallow water equations or their approximations using various numerical methods. The MacCormack method was developed for simulating overland flow with spatially variable infiltration and microtopography using the hydrodynamic flow equations. The basic MacCormack scheme is enhanced when it uses the method of fractional steps to treat the friction slope or a stiff source term and to upwind the convection term in order to control the numerical oscillations and stability. In this paper we describe, the MacCormack scheme for 1D complete shallow water equations with source terms, analyze the stability condition of the method and we provide the convergence rate of the algorithm. This work improves some well known results deeply studied in the literature which concern the Saint-Venant problem and it represents an extension of the time dependent shallow water equations without source terms. The numerical evidences consider the rate of convergence of the method and compares the numerical solution respect to the analytical one., Comment: 27 pages; 16 figures; 1 table

The error estimates and convergence rate of a two-level MacCormack rapid solver method for solving a two-dimensional incompressible Navier-Stokes equations are analyzed. This represents a continuation of the work on the stability analysis of the method. The theoretical result suggests that the rapid solver method is both convergent and second order accurate with respect to time step $\Delta t.$ A wide set of numerical evidences confirm this theoretical analysis., Comment: 26 pages, 16 figures, 2 tables

For a given nonnegative integer alpha, a matrix A_{n} of size n is called alpha-Toeplitz if its entries obey the rule A_{n}=[a_{r-alpha*s}]_{r,s=0}^{n-1}. Analogously, a matrix A_{n} again of size n is called alpha-circulant if A_{n}= [a_{(r-alpha*s)mod n}]_{r,s=0}^{n-1}. Such kind of matrices arises in wavelet analysis, subdivision algorithms and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of alpha-circulants and we provide an asymptotic analysis of the distribution results for the singular values of alpha-Toeplitz sequences in the case where {a_{k}} can be interpreted as the sequence of Fourier coeffcients of an integrable function f over the domain (-pi;pi). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered., Comment: 38 pages, submitted to SIMAX