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We study positive stationary solutions for the following Neumann problem in one-dimension space arising from population genetics: { u t = D u x x + g (x) u 2 (1 − u) in (0 , 1) × (0 , ∞) , u x (0 , t) = u x (1 , t) = 0 in (0 , ∞) , where g changes sign once in (0 , 1) and D is a positive parameter. This equation has a stationary positive solution u , where u − 1 has n zeros in (0 , 1). We denote this solution by an M (n) -solution (n = 1 , 2 , ⋯). We show that the M (n) -solution branch bifurcates from the trivial solution u = 1 and the M (n) -solution branch does not meet other bifurcating points and is extended globally to D → 0. When D is sufficiently small, the M (n) -solution has a very characteristic shape. Especially when n = 1 , we will show all possible shapes of M (1) -solutions. [ABSTRACT FROM AUTHOR]

Abstract.This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of solutions to a class of right-hand discontinuous 2nd-order semilinear singularly perturbed boundary value problems with turning points was studied. Firstly, the original problem was divided into left and right problems at the discontinuity, the accuracy of the asymptotic solution to the left problem was improved through modification of the regularization equation for the left problem degradation problem, and the existence of the smooth solution to the left problem was proved by means of the Nagumo theorem. Secondly, the solution to the right problem was proved to have a spatial contrast structure, and the asymptotic solution to the original problem was obtained through smooth joints at the discontinuity points. Finally, the correctness of the results was verified by an example. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points. [ABSTRACT FROM AUTHOR]

We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings. [ABSTRACT FROM AUTHOR]

This article deals with the numerical analysis of a class of singularly perturbed delay Volterra integro-differential equations exhibiting multiple boundary layers. The discretization of the considered problem is done using an implicit difference scheme for the differential term and a composite numerical integration rule for the integral term. The analysis of the discrete scheme consists of two parts. First, we establish an a priori error estimate that is used to prove robust convergence of the discrete scheme on Shishkin and Bakhvalov type meshes. Next, we establish the maximum norm a posteriori error estimate that involves difference derivatives of the approximate solution. The derived a posteriori error estimate gives the computable and guaranteed upper bound on the error. Numerical experiments confirm the theory. [ABSTRACT FROM AUTHOR]

Design of feedback control, for a system with slow and fast eigenvalue modes, using single sampling rate results in either information loss (for a larger sampling period) or increased computations (for a smaller sampling period). This paper contributes to designing multirate state feedback controllers for such systems. In this, it is shown that the feedback control for a linear time-invariant system, having slow and fast eigenvalue modes, can be successfully designed by multirate sampling of states in two stages. Multirate sampling refers to sampling slow- and fast-varying states at different rates, that is sampling slow states at a lower rate than the fast states. Here, two approaches for multirate sampling of states are presented, depending on the sampling sequence. In the first approach, fast subsystem states are sampled initially, and then, slow subsystem states are sampled, whereas in the second approach, slow subsystem states are sampled before sampling the fast subsystem states. As far as the two-stage design is concerned, the first stage of the design of feedback control is initiated just after sampling the first subsystem. Then, the left subsystem is sampled, and the second stage of the design of feedback control is accomplished. It is proved that the feedback controls derived with the multirate sampling of states stabilize the full-order system in both approaches. The design and implementation aspects of both approaches are compared. Finally, the applicability of the proposed control is demonstrated by simulating two examples. Simulations are also compared with other methods proposed in the literature. [ABSTRACT FROM AUTHOR]

In the study of singularly perturbed boundary problems with turning points, the solution undergoes sharp changes near these points and exhibits various interior phenomena. We employ the matching asymptotic expansion method to analyze and solve a singularly perturbed boundary and interior layers problem with multiple turning points, resulting in a composite expansion that fits well with the numerical solution. The solution demonstrates a strong association with special functions, which is verified by the theory of differential inequalities. [ABSTRACT FROM AUTHOR]

A three‐species food chain model with Allee and fear effects on both prey and predator is investigated in this work. The model consists of ODEs describing the evolution and interaction of prey, predator, and top predator populations in a biological system. We assume that the intrinsic growth rate of the prey is much smaller than a threshold value determined by other biological parameters of the model. Consequently, the model is transformed into a singularly perturbed system that acts on two different time scales, ranging from slow for both prey and predator to fast for the top predator. Positivity and boundedness of the solutions of the model along with the stability for its equilibria are investigated. The model possesses the exact invariant manifold with changing stability (black swan) and exhibits the canard phenomenon. Some numerical simulations are performed to validate our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and the linear stability of spiky stationary solutions for the Gierer-Meinhardt model with heterogeneity on the Y-shaped compact metric graph. The existence is shown by the Liapunov-Schmidt reduction method, and the stability is shown by investigating the associated linearized eigenvalue problem. In particular, it is revealed that the location, amplitude, and stability of spikes are decided by the interaction of the heterogeneity functions with the geometry of the graph, represented by the associated Green's function. Moreover, by applying our abstract theorem to several concrete examples, we study the detailed effects of the geometry of the graph and the heterogeneity function on spiky solutions. [ABSTRACT FROM AUTHOR]

Some popular stabilization techniques, such as nonsymmetric interior penalty Galerkin (NIPG) method, have important application value in computational fluid dynamics. In this paper, we analyze a NIPG method on Shishkin mesh for a singularly perturbed convection diffusion problem, which is a typical simplified fluid model. According to the characteristics of the solution, the mesh and the numerical scheme, a new interpolation is designed for convergence analysis. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. On the basis of that, by selecting special penalty parameters at different mesh points, we establish supercloseness of almost k + 1 order in an energy norm. Here k ≥ 1 is the degree of piecewise polynomials. Then, a simple post-processing operator is constructed, and it is proved that the corresponding post-processing can make the numerical solution achieve higher accuracy. In this process, a new analysis is proposed for the stability analysis of this operator. Finally, superconvergence is derived under a discrete energy norm. These conclusions can be verified numerically. Furthermore, numerical experiments show that the increase of polynomial degree k and mesh parameter N, the decrease of perturbation parameter ε or the use of over-penalty technology may increase the condition number of linear system. Therefore, we need to cautiously consider the application of high-order algorithm. [ABSTRACT FROM AUTHOR]

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of 'Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions' (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

Localized solutions are known to arise in a variety of singularly perturbed reaction–diffusion systems. The Gierer–Meinhardt (GM) system is one such example and has been the focus of numerous rigorous and formal studies. A more recent focus has been the study of localized solutions in systems exhibiting anomalous diffusion, particularly with Lévy flights. In this paper, we investigate localized solutions to a one-dimensional fractional GM system for which the inhibitor’s fractional order is supercritical. Specifically, we assume the fractional orders of the activator and inhibitor are, respectively, in the ranges s 1 ∈ (1 / 4 , 1) and s 2 ∈ (0 , 1 / 2) . Using the method of matched asymptotic expansions, we reduce the construction of multi-spike solutions to solving a nonlinear algebraic system. The linear stability of the resulting multi-spike solutions is then addressed by studying a globally coupled eigenvalue problem. In addition to these formal results, we also rigorously establish the existence and stability of ground state solutions when the inhibitor’s fractional order is nearly critical. The fractional Green’s function, for which we present a rapidly converging series expansion, is prominently featured throughout both the formal and rigorous analysis in this paper. Moreover, we emphasize that the striking similarities between the one-dimensional supercritical GM system and the classical three-dimensional GM system can be attributed to the leading-order singular behaviour of the fractional Green’s function. [ABSTRACT FROM AUTHOR]

In this paper, we address a class of boundary layer originated singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions having large time delay; for the higher order numerical analysis. The boundary layer originated nonuniform mesh, is generated by adaptive moving mesh approach based on equidistribution principle. The mesh is obtained by solving a nonlinear system; involving the discrete problem and the discrete nonlinear mesh generating functions. We provide the theoretical mesh structure. The difference scheme for Robin type boundary conditions having normalized flux; uses a combination of space and time discretizations so that the order of convergence in space is higher than the usual linear accurate upwind discretization in space. The present theoretical convergence analysis requires less regularity of the solution and proves to be exactly second order parameter uniform accurate in space. Numerical experiments highly validate the theoretical findings and also show that the present approach is better than the well-known upwind discretizations. [ABSTRACT FROM AUTHOR]

We employ the Craig-Wayne-Bourgain (CWB) method to construct quasi-periodic solutions for the nonlinear delayed perturbation equations. The linearized equation at each step of the iterations is a differential-difference equation on the torus. We shall combine the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, which generalizes the applicability of the CWB method. As an application, we study the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay. [ABSTRACT FROM AUTHOR]

The small-amplitude damped oscillations of a gas-filled spherical bubble surrounded by a viscous liquid is studied analytically using a symbolically computable version of the two-time scale expansion ("two-timing") technique. The developed uniformly valid asymptotic expansions have closed analytical forms under arbitrary specifications for the ideal gas polytropic exponent and the damping coefficient. Heretofore, a comparable level of analytical tractability has been afforded only by a linearized mass-spring-damper model; the corresponding regular perturbation expansion is, however, nonuniform for weak damping. Predictions from the two-timing and the mass-spring-damper models are compared with the full-scale numerical solution of the Rayleigh-Plesset equation. The two-scale analysis resolves several characteristics of the numerical data, such as the disparately large/small acceleration of the bubble wall near the most compressed/expanded states of the bubble, which the linearized model does not. The predictions of the two-scale model are shown to perform favorably against experimental data from the literature. New asymptotic predictions, which are very accurate for overdamped settings, are also developed by extending the mass-spring-damper model. • Novel computer extension of singular perturbation analysis on bubble dynamics. • Analytical closed form in time, initial data, bubble expansion and damping parameters. • Numerical validation of asymptotic results. • Comparison to experimental data from literature on cavitating bubbles. • Simultaneous improvement of asymptotic accuracy and long-time-limit validity. [ABSTRACT FROM AUTHOR]

In this paper, we study the two-parameter spectral element method based on weighted shifted orthogonal polynomials for solving singularly perturbed diffusion equation on an interval [0, 1] which are modeled with singular parameters. We continue our study to estimate the lower bound of the weighted orthogonal polynomial coefficient and the upper bound of a posteriori error estimates of the method through different weighted norms to minimize the computational cost. Numerical examples are implemented to study the applicability and efficiency of the technique. The obtained error bounds for the coefficient of orthogonal polynomials and the posteriori estimates fall within the bounds derived in the theoretical section. It is also observed that the two weighted norms decreases when the values of N1 and N2 increases for the three choices of 휖 and for different values of x and y. The quality and accuracy of the solution can be realized through figures and tables. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel semi-analytic physics informed neural network (PINN) method for solving singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that shows great promise for finding approximate solutions to partial differential equations. PINNs have demonstrated impressive performance in solving a variety of differential equations, including time-dependent and multi-dimensional equations involving complex domain geometries. However, when it comes to stiff differential equations, neural networks in general struggle to capture the sharp transition of solutions, due to the spectral bias. To address this limitation, we develop a semi-analytic PINN approach, which is enriched by incorporating the so-called corrector functions obtained from boundary layer analysis. Our enriched PINN approach provides accurate predictions of solutions to singular perturbation problems. Our numerical experiments cover a wide range of singularly perturbed linear and nonlinear differential equations. Overall, our approach shows great potential for solving challenging problems in the field of partial differential equations and machine learning. [ABSTRACT FROM AUTHOR]

A generalized finite element method (FEM) is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter\varepsilon, based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size ƍ\ast. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard finite element (FE) discretizations. Exponential decay rates for local approximation errors with respect to ƍvarepsilon and ƍ(at the discrete level with h denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to\varepsilon in the standard H 1 norm, and that if the oversampling size is relatively large with respect to\varepsilon and h (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

As a popular stabilization technique, the nonsymmetric interior penalty Galerkin (NIPG) method has significant application value in computational fluid dynamics. In this paper, we study the NIPG method for a typical two-dimensional singularly perturbed convection diffusion problem on a Shishkin mesh. According to the characteristics of the solution, the mesh and numerical scheme, a new composite interpolant is introduced. In fact, this interpolant is composed of a vertices-edges-element interpolant within the layer and a local L 2 -projection outside the layer. On the basis of that, by selecting penalty parameters on different types of interelement edges, we further obtain supercloseness of almost k + 1 2 order in an energy norm. Here k is the degree of piecewise polynomials. Numerical tests support our theoretical conclusion. [ABSTRACT FROM AUTHOR]

This paper purposes to present a new discrete scheme for the singularly perturbed semilinear Volterra–Fredholm integro-differential equation including two integral boundary conditions. Initially, some analytical properties of the solution are given. Then, using the composite numerical integration formulas and implicit difference rules, the finite difference scheme is established on a uniform mesh. Error approximations for the approximate solution and stability bounds are investigated in the discrete maximum norm. Finally, a numerical example is solved to show -uniform convergence of the suggested difference scheme. [ABSTRACT FROM AUTHOR]

In this study, we consider singularly perturbed large negative shift parabolic reaction-diffusion with integral boundary condition. The continuous solution's properties are discussed. On a non-uniform Shishkin mesh, the spatial derivative is discretized using the tension spline method, and the temporal derivative is discretized using the Crank-Nicolson method. In order to handle the integral boundary condition, Simpson's rule is used. After conducting an error analysis, it was determined that the method was uniformly convergent. To support the theoretical findings, numerical examples are taken into account and solved for different values of the perturbation parameter and mesh sizes. It is proved to be uniformly convergent to almost second order. [ABSTRACT FROM AUTHOR]

In this study, Couette-Poiseuille flow inside a channel partially filled with a porous medium is treated. Considering the applicability to the practical scenarios, we assume that the porous medium is anisotropic in nature. More precisely, the model under steady-state assumption is described by the coupling of Brinkman-Forchheimer and Stokes equations. We develop the existence and uniqueness results corresponding to this two-dimensional Stokes-Brinkman-Forchheimer coupled system using the Browder-Minty theorem. For this we restrict the permeability to a diagonal matrix. The coupled system involves a non-linear term due to the Brinkman-Forchheimer equation. We propose an algorithm based on shooting and searching techniques to develop a numerical solution for this system, however, for an arbitrary anisotropic permeability matrix. The robustness of the presented numerical solution is established by comparing it with asymptotic solutions corresponding to small and large Darcy numbers. For the case of a large Darcy number, the problem of interest is a regular perturbation while the same is a singular perturbation problem for the case of a small Darcy number, which is dealt with using Prandtl's matching principle. With the velocity distribution determined, we have presented a comparative analysis of the shear stress distribution at the liquid-porous interface and at the bottom plate, which is of interest to analyze the stress distribution on the arterial wall. The asymptotic results are found to be in good agreement with those obtained numerically. [ABSTRACT FROM AUTHOR]

In this paper, the initial value problem for singularly perturbed nonlinear holomorphic differential equations with a small parameter is considered in a complex domain, and the asymptotic behavior of solutions with respect to the parameter is studied. Under suitable conditions, a formal power series solution is constructed, and then it is proved that this formal solution is summable in a suitable direction. This guarantees the existence of a holomorphic solution that admits the formal power series solution as an asymptotic expansion of Gevrey type. It is then proved that this solution also has a Gevrey type asymptotic expansion with respect to the parameter. [ABSTRACT FROM AUTHOR]

To reduce the influence of system dynamic coupling and external disturbance on the performance of flight control system of fixed-wing UAV thus improve its flight control accuracy, this paper proposes a singular perturbation model for fixed-wing UAV and then designs a sliding mode control approach based on disturbance observer. The velocities and attitudes of the fixed-wing UAV are modeled based on the dynamics of action. Then the dynamic model is transformed into a singular perturbation one and then decomposed to complete decoupling. Two reduced-order uncoupled subsystems are obtained, which is a fast subsystem with angular velocity as fast variable and a slow subsystem with linear velocity and attitude as slow variables. Then anti-disturbance sliding mode controllers are designed for angular velocity loop, and angle ＆ attitude loop. Finally, the feasibility and effectiveness of the sliding mode control approach based on fast and slow decomposition are verified by Simulink simulation. [ABSTRACT FROM AUTHOR]

The position-force tracking issue of an object being moved by collaborative Multiple Flexible-Joint arms, facing dynamic uncertainties, model nonlinearities, and unknown perturbations is studied in this paper. Toward this end, an adaptive control scheme applying the function approximation technique is suggested, enabling the object to track the reference trajectory. This capability arises from the universal approximation property of the FAT-based approaches. Herein, the q-Bernstein-Schurer operators are exploited to disturbances/uncertain dynamic approximations. Since the parameters of the system are not exactly recognized, adaptive rules are suggested for tuning the coefficients of the operator. The Lyapunov stability analysis guarantees uniformly ultimately bounded stability of all the error signals. Two Flexible-Joint manipulators transporting a rigid object are employed to validate the theoretical achievements. The state-of-the-art Chebyshev Neural Network approximator is also used to compare the suggested methodology. The outcomes exhibit the usefulness of the presented approach, handling the system even in the incidence of uncertainties and disturbances needless to the system's state feedback for function approximation with a low computational burden. [ABSTRACT FROM AUTHOR]

In this article, a closed-form iterative analytic approximation to a class of nonlinear singularly perturbed parabolic partial differential equation is developed and analysed for convergence. We have considered both parabolic reaction diffusion and parabolic convection diffusion type of problems in this paper. The solution of this class of problem is polluted by a small dissipative parameter, due to which solution often shows boundary and interior layers. A sequence of approximate analytic solution for the above class of problems is constructed using Lagrange multiplier approach. Within a general frame work, the Lagrange multiplier is optimally obtained using variational theory. The sequence of approximate analytical solutions so obtained is proved to converge the exact solution of the problem. To demonstrate the proposed method's efficiency and accuracy, linear and nonlinear test problems have been taken into account. From numerical experiments, it is observed that the proposed method is highly accurate, straightforward. [ABSTRACT FROM AUTHOR]

In this paper, the initial value problem for the second order singularly perturbed Volterra integro-differential equation is considered. To solve this problem, a finite difference scheme is constructed, which based on the method of integral identities using interpolating quadrature rules with remainder terms in integral form. As a result of the error analysis, it is proved that the method is first-order convergent uniformly with respect to the perturbation parameter in the discrete maximum norm. Numerical experiments supporting the theoretical results are also presented. [ABSTRACT FROM AUTHOR]

We investigate the singularly perturbed monotone systems with respect to cones of rank 2 and obtain the so called Generic Poincaré-Bendixson theorem for such perturbed systems, that is, for a bounded positively invariant set, there exists an open and dense subset \mathcal {P} such that for each z\in \mathcal {P}, the \omega-limit set \omega (z) that contains no equilibrium points is a closed orbit. [ABSTRACT FROM AUTHOR]

With the continuous development of science and technology, singularly perturbed problems have attracted more and more attentions from computational fluid scholars. In this paper, we discuss a singularly perturbed convection–diffusion problem with a discontinuous convection. Generally speaking, due to this discontinuity, an interior layer appears in the solution. A streamline diffusion finite element method on Shishkin mesh is used to solve this problem, and the optimal order of convergence in a modified streamline diffusion norm is derived. Numerical results are presented to support the theoretical conclusion. [ABSTRACT FROM AUTHOR]

In this paper, N -cluster games with coupling and private constraints are studied, where each player's cost function is nonsmooth and depends on the actions of all players. In order to seek the generalized Nash equilibrium (GNE) of the nonsmooth N -cluster games, a distributed seeking neurodynamic approach with two-time-scale structure is proposed. An adaptive leader-following consensus technique is adapted to dynamically adjust parameters according to the degree of consensus violation, so as to quickly obtain accurate estimation information of other players' actions which facilitates the evaluation of its own cost. Benefitting from the unique structure of the approach based on primal dual and adaptive penalty methods, the players' actions enter the constraints while completing the seeking for GNE. As a result, the neurodynamic approach is completely distributed, and prior estimation of penalty parameters is avoided. Finally, two engineering examples of power system game and company capacity allocation verify the effectiveness and feasibility of the neurodynamic approach. [ABSTRACT FROM AUTHOR]

We consider a singularly perturbed problem for a nonlinear first-order Fredholm integro-differential equation. Our aim is to build and analyze a numerical approach with uniform convergence in the ε -parameter. The numerical solution of problem is discretized utilizing interpolation quadrature formulas for the differential part and the composite rectangular rule for the integral part. Error estimations for the approximate solution are established. In support of the idea, numerical examples are provided. [ABSTRACT FROM AUTHOR]

Understanding the interplay of different traits in a co-infection system with multiple strains has many applications in ecology and epidemiology. Because of high dimensionality and complex feedback between traits manifested in infection and co-infection, the study of such systems remains a challenge. In the case where strains are similar (quasi-neutrality assumption), we can model trait variation as perturbations in parameters, which simplifies analysis. Here, we apply singular perturbation theory to many strain parameters simultaneously and advance analytically to obtain their explicit collective dynamics. We consider and study such a quasi-neutral model of susceptible–infected–susceptible (SIS) dynamics among N strains, which vary in 5 fitness dimensions: transmissibility, clearance rate of single- and co-infection, transmission probability from mixed coinfection, and co-colonization vulnerability factors encompassing cooperation and competition. This quasi-neutral system is analyzed with a singular perturbation method through an appropriate slow–fast decomposition. The fast dynamics correspond to the embedded neutral system, while the slow dynamics are governed by an N-dimensional replicator equation, describing the time evolution of strain frequencies. The coefficients of this replicator system are pairwise invasion fitnesses between strains, which, in our model, are an explicit weighted sum of pairwise asymmetries along all trait dimensions. Remarkably these weights depend only on the parameters of the neutral system. Such model reduction highlights the centrality of the neutral system for dynamics at the edge of neutrality and exposes critical features for the maintenance of diversity. [ABSTRACT FROM AUTHOR]

This paper is to analyze a finite element method of any order on a Bakhvalov-type mesh in the case of 2D. By introducing a new interpolation according to the characteristics of layers, we show that the finite element method has uniform convergence of the optimal order with respect to the singular perturbation parameter. The result partially resolves an open problem introduced by Roos and Stynes (Comput. Methods Appl. Math. 15(4):531–550, 2015). [ABSTRACT FROM AUTHOR]

We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection–diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the convective derivative of the error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error, which functions as a maximum-norm counterpart to the dual norm used in the energy norm setting. The total error is then defined as the sum of this seminorm, the maximum norm of the error and data oscillation. The natural maximum norm residual error estimator is shown to be equivalent to this total error notion, with constant independent of singular perturbation parameters. These estimates are proved under the assumption that certain natural estimates hold for the Green's function for the problem at hand. Numerical experiments confirm that our estimators effectively capture the maximum-norm error behavior for singularly perturbed problems, and can effectively drive adaptive refinement in order to capture layer phenomena. [ABSTRACT FROM AUTHOR]

This paper aims to establish the finite difference scheme on Bakhvalov-type mesh for the singularly perturbed pseudo-parabolic problems with time-delay. Using the energy estimates, the bounds of the solution are investigated. An error analysis is derived in the discrete energy norm and the almost second order convergence is obtained. The reliability of the suggested method is displayed by means of a numerical example. [ABSTRACT FROM AUTHOR]

This article is concerned with a two-dimensional singularly perturbed convection-reaction-diffusion interface problem. In the considered problem, the convection and reaction coefficients and the source term have discontinuities along interface lines, which are parallel to the x - and y -axes. The coefficient of the highest-order term is a small positive parameter denoted by ε. Due to discontinuities in the coefficients and the source term, interior and boundary layers appear in the solution when ε approaches zero. A Local Discontinuous Galerkin method is constructed on an appropriate Shishkin mesh. The test functions in the Local Discontinuous Galerkin method are piecewise polynomials that lie in the space Q r of piecewise polynomials of degree at most r in each variable, where r is a positive integer. We established that the error in the computed solution converges at the rate of O ((N − 1 ln ⁡ N) r + 1 2 ) in an energy-norm. Numerical results are given to support the theoretical results. • A singularly perturbed 2D convection-reaction-diffusion interface problem is considered. • The problem has discontinuities in the coefficients and source term along the interfaces. • For discretization an LDG method is used on a Shishkin mesh. • Uniform convergence of order almost r + 1 2 is obtained. • Some numerical experiments are conducted to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

We examine a nonlinear initial value problem both singularly perturbed in a complex parameter and singular in complex time at the origin. The study undertaken in this paper is the continuation of a joined work with Lastra published in 2015. A change of balance between the leading and a critical subdominant term of the problem considered in our previous work is performed. It leads to a phenomenon of coalescing singularities to the origin in the Borel plane with respect to time for a finite set of holomorphic solutions constructed as Fourier series in space on horizontal complex strips. In comparison to our former study, an enlargement of the Gevrey order of the asymptotic expansion for these solutions relatively to the complex parameter is induced. [ABSTRACT FROM AUTHOR]

We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient. We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale. Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models. [ABSTRACT FROM AUTHOR]

This paper studies the time-scale characteristics, model reduction, and oscillation stability of the grid-connected variable speed pumped storage unit (VSPSU). Firstly, a method based on the modal analysis and time-scale separation ratio indicator is established to analyze the time-scale characteristics of VSPSU. The results indicate that the state variables of VSPSU can be approximately separated on four-time scales. On this basis, the singular perturbation method (SPM) is applied to obtain the reduced-order model (ROM) of VSPSU under different conditions. The results show that the accuracy of ROMs can be affected by changes in operating conditions. To balance the complexity and accuracy of ROM, a model reduction strategy combining SPM and reduction margin is proposed. The results indicate that the proposed method enables the order minimization of ROM while the selected oscillation mode reaches a given accuracy condition. Finally, the feasibility of ROM for oscillation stability analysis is verified. The ROMs obtained with the proposed strategy are more accurate than those obtained with SPM for oscillation stability analysis of VSPSU. The influence of grid strength, turbine speed, and torque conditions on system oscillation stability is also revealed. This study provides new insights for model order reduction and stability analysis of VSPSU. [ABSTRACT FROM AUTHOR]

In this article, we examine a higher order convergent approximation for a class of singularly perturbed two-dimensional (2-D) convection-diffusion-reaction elliptic problems with discontinuous convection and source terms. The coefficient of the highest-order derivative of this problem, is considered to be arbitrarily small. The solutions of these problems exhibit interior layers due to the discontinuity in the convection and source terms. In this article, a numerical approach is proposed by using a hybrid-difference technique (a suitable combination of the midpoint upwind technique in the smooth regions and the central difference technique in the layer regions) on a propitious layer-adaptive piecewise-uniform Shishkin mesh which leads to an almost second-order estimate. We present numerical experiments with several combinations of possible discontinuities to verify that our proposed results are sharp. [ABSTRACT FROM AUTHOR]

The objective of this work is to provide an efficient and higher-order numerical scheme for the solution of a singularly perturbed 2D time-delayed parabolic convection-diffusion problem. Each time step is split into two partial time steps by using the Peaceman-Rachford splitting algorithm. The splitting is done over the idea of the Crank-Nicholson scheme. The hybrid scheme is a combination of the central difference scheme in the layer region and the mid-point upwind scheme in the outer region defined on a Shishkin mesh for the discretization in the spatial direction. This makes the order of convergence to two (up to a logarithmic factor) in space. Again, the presence of the logarithmic effect is rectified by the use of the Bakhvalov-Shishkin mesh. The proposed method is proved to be parameter uniform and also it attains second-order spatial accuracy in the discrete supremum norm. Numerical tests validate the efficacy of the proposed scheme. [ABSTRACT FROM AUTHOR]

A two-class Processor-Sharing queue with one impatient class is studied. Local exponential decay rates for its stationary distribution (N (∞) , M (∞)) are established in the heavy traffic regime where the arrival rate of impatient customers grows proportionally to a large factor A. This regime is characterized by two time-scales, so that no general Large Deviations result is applicable. In the framework of singular perturbation methods, we instead assume that an asymptotic expansion of the solution of associated Kolmogorov equations exists for large A and derive it in the form P (N (∞) = A x , M (∞) = A y) ∼ g (x , y) 2 π A · e − A H (x , y) , x > 0 , y > 0 , with explicit functions g and H. This result is then applied to the model of mobile networks proposed in [Olivier and Simonian] and accounting for the spatial movement of users. We give further evidence of a unusual growth behavior in heavy traffic in that the stationary mean queue length E (N mob (∞)) and E M mob (∞)) of each customer-class increases proportionally to E (N mob (∞)) ∝ E (M mob (∞)) ∝ log (1 1 − ϱ tot ) with system load ϱ tot tending to 1, instead of the usual 1 / (1 − ϱ tot) growth behavior. [ABSTRACT FROM AUTHOR]

The symmetric interior penalty discontinuous Galerkin method and its version with weighted averages are considered on shape-regular nonconforming meshes with an arbitrarily large number of mesh faces contained in any element face. For this method, residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. The error constants are independent of the diameters of mesh elements and of the small perturbation parameter. The theoretical findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, using the notion of ( I q )–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [ a , b ] have been established for four different types of boundary conditions for a singularly perturbed differential equation ε y ″ = f (x , y , y ′) ,   a ≤ x ≤ b . In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f (x , y , y ′) = 0 . [ABSTRACT FROM AUTHOR]

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction–diffusion equations where m of the equations (with m < n ) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable, together with classical finite difference approximations. Some analytical properties and error analyses are derived. Furthermore, a bound of the error is provided. Under certain assumptions, it is proved that the proposed scheme has almost second-order convergence in the space direction and almost first-order convergence in the time variable. Errors do not increase when the perturbation parameter ε → 0 , proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results. [ABSTRACT FROM AUTHOR]