Narrow Results

The paper considers a class of discrete-time cellular neural networks (DT-CNNs) obtained by applying Euler’s discretization scheme to standard CNNs. Let $T$ be the DT-CNN interconnection matrix which is defined by the feedback cloning template. The paper shows that a DT-CNN is convergent, i.e. each solution tends to an equilibrium point, when $T$ is symmetric and, in the case where $T+E_n$ is not positive-semidefinite, the step size of Euler’s discretization scheme does not exceed a given bound ($E_n$ is the $n\times n$ unit matrix). It is shown that two relevant properties hold as a consequence of the local and space-invariant interconnecting structure of a DT-CNN, namely: (1) the bound on the step size can be easily estimated via the elements of the DT-CNN feedback cloning template only; (2) the bound is independent of the DT-CNN dimension. These two properties make DT-CNNs very effective in view of computer simulations and for the practical applications to high-dimensional processing tasks. The obtained results are proved via Lyapunov approach and LaSalle’s Invariance Principle in combination with some fundamental inequalities enjoyed by the projection operator on a convex set. The results are compared with previous ones in the literature on the convergence of DT-CNNs and also with those obtained for different neural network models as the Brain-State-in-a-Box model. Finally, the results on convergence are illustrated via the application to some relevant 2D and 1D DT-CNNs for image processing tasks.

This paper determines relations between two notions concerning monoids: factorability structure, introduced to simplify the bar complex; and quadratic normalization, introduced to generalize quadratic rewriting systems and normalizations arising from Garside families. Factorable monoids are characterized in the axiomatic setting of quadratic normalizations. Additionally, quadratic normalizations of class $\left(4,3\right)$ are characterized in terms of factorability structures and a condition ensuring the termination of the associated rewriting system.

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with $𝜃$-weighted scheme is utilized to obtain the solution. The method is shown to be unconditionally stable and convergent. Numerical examples indicate that the obtained results compare well with other numerical results available in the literature.

The usage of Lévy processes involving big moves or jumps over a short period of time has proven to be a successful strategy in financial analysis to capture such rare or extreme events of stock price dynamics. Models that follow the Lévy process are FMLS, Kobol, and CGMY models. Such simulations steadily raise the attention of researchers in science because of the certain best options they produce. Thus, the issue of resolving these three separate styles has gained more interest. In the new paper, we introduce the computational method of such models. At first, the left and right tempered fractional derivative with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the third kind (SCPTK). In the second point, by implementing finite difference approximation, we get the semi-discrete structure to solve the tempered fractional B–S model (TFBSM). We show that this system is stable and ${𝒪}(\delta \tau )$ is the convergence order. In practice, the processing time and the calculation time per iteration will be reduced by a quickly stabilized system. Then we use SCPTK to approximate the spatial fractional derivative to get the full design. Finally, two numerical examples are provided to illustrate the established system’s reliability and effectiveness.

In this paper, we are interested in studying the initial value problem for parabolic problem associated with the Caputo–Fabrizio derivative. We deal the problem in two cases: linear inhomogeneous case and nonlinearity source term. For the linear case, we derive the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$ under some various assumptions on the initial datum. For the nonlinear problem, we show the existence and uniqueness of the mild solution using Banach fixed point theory. We also prove the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$.

This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann–Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme.

The purpose of this work is to study the memory effect analysis of Caputo–Fabrizio time fractional diffusion equation by means of cubic B-spline functions. The Caputo–Fabrizio interpretation of fractional derivative involves a non-singular kernel that permits to describe some class of material heterogeneities and the effect of memory more effectively. The proposed numerical technique relies on finite difference approach and cubic B-spline functions for discretization along temporal and spatial grids, respectively. To ensure that the error does not amplify during computational process, stability analysis is performed. The described algorithm is second-order convergent along time and space directions. The computational competence of the scheme is tested through some numerical examples. The results reveal that the current scheme is reasonably efficient and reliable to be used for solving the subject problem.

An efficient high-order computational procedure is going to be created in this paper to determine the solution to the mobile–immobile advection–dispersion model (MIAD) of temporal fractional order $0<\alpha \le 1$, which can be employed to model the solute forwarding in watershed catchments and floods. To do it, the temporal-first derivative of MIAD is discretized by using the finite-difference technique’s first-order precision and the linear interpolation’s temporal-fractional derivative. On either side, the space derivative is simulated using a collocation approach based on the Legendre basis to generate the full-discrete method. The order of MIAD-convergence for the implicit numerical structure is explained. Additionally, a basic conceptual discussion of the temporal-discretized stability mechanism is included in this paper. Finally, two models are provided to show the reliability and excellence of the organized approach.

Deterministic and fractional properties of an epidemic model for the dynamics of the Middle Eastern respiratory syndrome coronavirus (MERS-COV) with various infection stages are proposed in this study whose aim is to show via a mathematical model the transmission of MERS-COV between humans and camels, which are suspected to be the primary source of infection. The mathematical aspects together with biological feasibility of MERS-COV model are provided. The basic reproduction number $R_0$ has been calculated by using the next-generation matrix. With the help of $R_0$, we show the local and global stability analysis of the proposed model. Analysis of sensitivity for the threshold number is performed to understand the most sensitive parameter. Moreover, the concepts of strength number and second-order derivative of the Lyapunov function have been utilized for the waves detection. By using the concept of fixed point approach for the model, concerning if it really exists or not, we prove the existence and uniqueness results for the proposed model. The numerical solutions are obtained with the help of well-known fractional Adams–Bashforth technique. For the approximate solution, with the help of Runge–Kutta technique of order four, we accomplish the numerical simulations to support our analytical outcomes which are believed to have an effective impact on developing preventive measures for MERS-CoV, including disease control as well as prevention of spread and transmission in related populations.

The use of finite element models has gained popularity in the field of foot and footwear biomechanics to predict the stress–strain distribution and the treatment effectiveness of therapeutic insoles for pathological foot conditions. However, a comprehensive evaluation of mesh quality is often ignored, meanwhile no golden standard exists for the mesh density and selection of element size at an acceptable accuracy. Here, we make a convergence test and established anatomically-realistic foot models at different mesh densities. The study compared the discrepancy in output variables to the changes of element type and mesh density under barefoot and footwear conditions with compressive and shear loads, which are commonly encountered in foot and footwear biomechanics simulations. For a range of loading conditions simulated in 125 finite element models, the peak plantar pressure consistently converged with optimal mesh size determined at 2.5mm. The convergence variable of principal strains and stress tensors, however, varies significantly. The max von-Mises stress showed strong sensitive behavior to the changes of the mesh density. The pattern for contact pressure distribution became less accurate when the element sizes increase to 6.0mm; in particular, the locations of the pressure peak do not show remarkable changes, but the size of the area of contact still changes. The current study could offer a general guideline when generating a reasonable accurate finite element models for the analysis of plantar pressure distributions and stress/strain states employed for foot and footwear biomechanics evaluations.

The problems of mathematical biology, as a rule, are formulated in terms of independent and depended (unknown) variables (functions), and operators (mostly differential operators). These variables belong to some spaces and operators acting in these spaces. Both, unknown variables and operators, follow the main rules of these spaces as metrics, topology and etc. In applications a solution of the problem will be constructed by approximation methods based on metrics/topology of the space. In this convergence of the constructed approximations essentially depends on compactness.

Diabetes Mellitus is a metabolic disorder that occurs when the sugar level in blood is not proper. Diabetes is classified into many types and this classification is based upon the etiology of the disease. This disease is classified into three main types: Type 1 Diabetes Mellitus (T1DM), Type 2 Diabetes Mellitus (T2DM) and Gestational Diabetes Mellitus (GDM). In this paper, our aim is to present a model of Diabetes Mellitus Type ${\mathrm{\text{SEII}}}_T$ by considering treatment and genetic factors. We have formulated the model for the epidemic problem. Here, a numerical algorithm based on Fractional Homotopy Analysis Transform Method (FHATM) is constituted to study the fractional form of the model for Diabetes Mellitus Type ${\mathrm{\text{SEII}}}_T$. The suggested technique is obtained by merging the homotopy analysis, Laplace transform and the homotopy polynomials. The convergence and uniqueness of the solution are also discussed and results are also presented.

This study has set to compare the usefulness of the regional integration efforts taken by two important economic blocs, SAARC and ASEAN, within the Asian continent to reduce the regional income inequality. Therefore, the existence of income convergence (or divergence) among the SAARC and ASEAN countries is the aim of this study. To investigate whether (or not) there exists income convergence across the SAARC and ASEAN blocs over the period of 1970-2017, $\beta$-convergence, σ-convergence and club convergence estimation methods have been applied. The results confirm the convergence of income across the ASEAN member countries, which is absent for the SAARC member countries at the intra-regional level. Although we considered all countries of the SAARC and ASEAN blocs together in the panel, an evidence of income convergence over the years has been found. The analysis supports the view of trade liberalization and recommends investing in the human capital to narrow down the regional disparity in future. To achieve a favorable impact of Asian rising growth, regional integration is important, for that a collective policy framework at the regional level is needed for both SAARC and ASEAN.