Singular Perturbation Problems Research Articles

Numerical solution of time‐fractional singularly perturbed convection–diffusion problems with a delay in time

This work is concerned with the development of a stable finite difference method (SFDM) for time‐fractional singularly perturbed convection–diffusion problems with a delay in time. The fractional derivative is considered in the Caputo sense. The SFDM is constructed based on the stability of the analytical solution. Unlike the other classical numerical methods for singular perturbation problems, this method works nicely on a uniform mesh. The method is easily adaptable on Shishkin mesh and also on any graded mesh in space. Error estimates are presented to show the convergence of the numerical scheme. To support the theory, numerical results are presented in tables for different values of the fractional derivative parameter and perturbation parameter.

Singular perturbations of the Holling I predator-prey system with a focus

We study the occurrence of limit cyles in a model for two-species predator-prey interaction (referred to as Holling I). This model is based on the phenomenological fitting of the so-called functional response function to observed data in nature by a continuous, piecewise differentiable function. Of the original models presented by Holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet.We extend the work of Liu who showed that for certain values of parameters of the Holling I system at least two nested limit cycles will occur surrounding a focus. His case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities, i.e. a singular perturbation problem with slow-fast solutions.We prove that exactly two hyperbolic limit cycles occur after perturbation.

On the Solutions of Linear Fractional Differential Equations of Order \(2q\), Including Small Delay Where \( 0< q

The main goal of this study is to find the solutions of linear fractional differential equations of order \(2q\), including small delay, where \(0<q<1\) which has various applications. The fractional derivatives are taken in the sense of Caputo which is more suitable than Riemann-Liouville sense. We assume that the order \(q\) satisfy the condition \(nq=1\) for some natural number \(n\) which determines the number of the linearly independent solutions. Since the delay term is small, the linear fractional differential equation is expanded in powers series of which reduce the problem to regular or singular perturbation problem for which it is easier to find the solution. The solution is obtained in the form of a series expansion of \(E\). To demonstrate the accuracy and the effectiveness of the proposed approach, some illustrative examples are presented.

Two-stroke relaxation oscillators

Two-stroke relaxation oscillations consist of two distinct phases per cycle—one slow and one fast—which distinguishes them from the well-known van der Pol-type ‘four-stroke’ relaxation oscillations. This type of oscillation can be found in singular perturbation problems in non-standard form, where the slow-fast timescale splitting is not necessarily reflected in a slow-fast variable splitting. The existing literature on such non-standard problems has developed primarily through applications—we complement this by illustrating the suitability of a more general framework for geometric singular perturbation theory to prove existence and uniqueness for a general class of two-stroke relaxation oscillators. While this result can be derived from a more general result in de Maesschalck et al (2011 Indagationes Math. 22 165–206), our methods emphasise the scope, simplicity and applicability of this non-standard approach. We apply this non-standard geometric singular perturbation toolbox to a collection of examples arising in the dynamics of nonlinear transistors and models for mechanical oscillators with friction.

Non-standard fitted operator scheme for singularly perturbed boundary value problem

A non-standard fitted operator finite difference technique is designed for the solution of singularly perturbed two-point boundary value problems. Using the solution of reduced problem of the singular perturbation problem, it is transformed to an equivalent equation of first order. This first order singularly perturbed problem is solved by using an exact non-standard fitted operator method with an appropriate condition. To demonstrate the method computationally, it is implemented on several examples. From the numerical results, it is noticed that the proposed scheme is in respectable agreement with the exact solution with minimal computational effort.

Study of the NIPG method for two–parameter singular perturbation problems on several layer adapted grids

In this paper, we apply the non-symmetric interior penalty Galerkin (NIPG) method to obtain the numerical solution of two-parameter singularly perturbed convection-diffusion-reaction boundary-value problems. In order to discretize the domain, here, we use the layer-adapted piecewise-uniform Shishkin mesh, the Bakhvalov mesh and the exponentially-graded mesh. We establish a superconvergence result of the NIPG method, that is, the proposed method is parameter-uniformly convergent with the order almost $$(k+1)$$ on the Shishkin mesh and $$(k+1)$$ on the Bakhvalov mesh and on the exponentially graded mesh in the energy norm, where k is the order of the polynomials. Numerical results comparing the three different types of meshes are presented at the end of the article supporting the theoretical error estimates.

The Super-Diffusive Singular Perturbation Problem

In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] , 1 &lt; α &lt; 2 , ϵ &gt; 0 , for the parabolic equation ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem ( P ϵ ) has a unique solution u ϵ ( t ) for each small ϵ &gt; 0 . Moreover u ϵ ( t ) converges to u 0 ( t ) as ϵ → 0 + , the unique solution of equation ( P ) .

A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport

We present a new discontinuous ordinary differential equation (ODE) model of the glacial cycles. Model trajectories flip from a glacial to an interglacial state, and vice versa, via a switching mechanism motivated by ice sheet mass balance principles. Filippov’s theory of differential inclusions is used to analyze the system, which can be viewed as a nonsmooth geometric singular perturbation problem. We prove the existence of a unique limit cycle, corresponding to the Earth’s glacial cycles. The diffusive heat transport component of the model is ideally suited for investigating the competing temperature gradient and transport efficiency feedbacks, each associated with ice-albedo feedback. It is the interplay of these feedbacks that determines the maximal extent of the ice sheet. In the nonautonomous setting, model glacial cycles persist when subjected to external forcing brought on by changes in Earth’s orbital parameters over geologic time. The system also exhibits various bifurcation scenarios as key parameters vary.

The nonconforming virtual element method for fourth-order singular perturbation problem

We present the nonconforming virtual element method for the fourth-order singular perturbation problem. The virtual element proposed in this paper is a variant of the C0-continuous nonconforming virtual element presented in our previous work and allows to compute two different projection operators that are used for the construction of the discrete scheme. We show the optimal convergence in the energy norm for the nonconforming virtual element method. Further, the lowest order nonconforming method is proved to be uniformly convergent with respect to the perturbation parameter. Finally, we verify the convergence for the nonconforming virtual element method by some numerical tests.

Capillary surfaces arising in singular perturbation problems

In this paper we prove Bernstein type theorems for a class of stationary points of the Alt-Caffarelli functional in $\mathbb R^2$ and $\mathbb R^3$.

Computational results of differential difference equations with mixed shifts having layer structure using cubic non-polynomial spline

A computational scheme using non-polynomial spline is suggested for solving differential - difference equations having delay and advance terms, solutions with layer structure at the left-end of the interval. First, the small shifts are tackled with Taylor's expansion and accordingly the problem is transformed to a second order singular perturbation problem. The domain is decomposed into inner and outer regions using a terminal boundary point and the problem is treated as inner region and outer region problems. Terminal boundary condition has been determined by using the reduced problem of the singular perturbation problem. In order to solve the inner and exterior region problems, a fourth order method was suggested using cubic non-polynomial spline. The method is repeated for numerous terminal point choices, until the solution profiles do not differ greatly from iteration to iteration. To illustrates the process, numerical examples were solved for specific values of the parameters of perturbation, delay and advance. The results of the computations are tabulated and compared to exact solutions. Convergence of the scheme was also studied.

СИНГУЛЯРНО ЗБУРЕНІ ЗАДАЧІ ТЕРМОМЕХАНІКИ ШАРУВАТИХ ДОРОЖНІХ ПОКРИТТІВ

The problem of singular perturbation of the equations of elastic bending of layered pavements and thermal conductivity is discussed. It is shown that the complexity of solving these equations significantly exceeds the complexity of singularly perturbed equations known in the scientific literature. First, this is due to the fact that in the problems of road construction, the mechanics of elastic bending are described by the partial differential equations, and the integration domain reaches large dimensions. The problem of thermal deformation of the pavement and distribution in its array of temperature, displacement, and stress fields is also considered. It is proved that thesolution of the problemof thermal conductivity has the appearance of a boundary effect, localized in the elements of the coating adjacent to its free surface.Also is noted that high-gradient temperature distribution deep in the soil leads to large values of normal and tangential stresses that provoke cracking and delamination of the upper layers of the coating. They are confirmed by the results of computer simulation. KEYWORDS: SINGULARLY PERTURBED PROBLEM, LAYERED ROAD COVERING, TEMPERATURE FIELD, TRANSPORT LOAD, STRESS FIELD, THERMAL STRAIN STATE

Piecewise reproducing kernel-based symmetric collocation approach for linear stationary singularly perturbed problems

The aim of this paper is to develop an accurate symmetric collocation scheme for a class of linear stationary singular perturbation problems with two boundary layers. To adapt to the character of solutions, piecewise reproducing kernels is constructed. In the boundary layers intervals, inverse multiquadrics kernel function is employed. In the regular interval, exponential kernel function is used. On the basis of the piecewise reproducing kernels, a new symmetric collocation technique is presented for the considered linear stationary singular perturbation problems. Results of numerical tests illustrate that our method is easy to implement and is uniformly effective for any small e.

Solving a System of Singular Perturbation Problems via Neural Networks

In this work, we will introduce a new procedure to solve a system of singular perturbation problems (SSPPs) via artificial neural networks. The neural networks use the code of back propagation with altered training algorithms such as quasi-Newton, Levenberg-Marquardt, and Bayesian regularization. In our research, we provide examples of two different types of systems, showing the accuracy, speed, resolution, and convergence of the new technology, the effectiveness of using the network techniques for solving this type of equations. The convergence properties of the technique and accuracy of the interpolation technique are considered.

Fitted difference approach for differential equations with delay and advanced parameters

A difference scheme involving acceptable fitting parameters is suggested for differential equations with delay and advanced terms, the solutions of which show boundary layer behaviour. First, the original problem is reshaped into asymptotically comparable second order singular perturbation problem using Taylor series approximation for the retarded terms. In order to obtain precise solution, fitting parameters are introduced in difference scheme using modified upwind differences for the first order derivatives. Thomas procedure is used to solve the resulting tri-diagonal difference system. The method is tested on numerical examples for various values of the perturbation, delay and advance parameters. Computed maximum absolute errors are tabulated. Numerical experiments are shown in graphs and the effects of small shifts have been studied on the boundary layer region. Also, convergence has been established of the proposed method.

Numerical Approach for Differential-Difference Equations with Layer Behaviour

A numerical scheme is proposed using a non polynomial spline to solve the differential-difference equations having layer behaviour, with delay as well advanced terms. The retarded terms are handled by using Taylor’s series, subsequently the given problem is substituted by an equivalent second order singular perturbation problem. A finite difference scheme using non polynomial spline is derived and it is applied to the singular perturbation problem using non standard differences of the first derivatives. Tridiagonal algorithm is used to solve the resulting system. The method is exemplified on numerical examples with various values of perturbation, delay and advance parameters. Maximum absolute errors are computed and tabulated to support the method. Numerical solutions are pictured in graphs and the effects of small shifts on the boundary layer region has been investigated. Also, the convergence of the proposed method has also been established.

Velocity-defect laws, log law and logarithmic friction law in the convective atmospheric boundary layer

The mean velocity profile in the convective atmospheric boundary layer (CBL) is derived analytically. The shear-stress budget equations and the mean momentum equations are employed in the derivation. The multi-point Monin–Obukhov similarity (MMO) recently proposed and analytically derived by Tong & Nguyen (J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) and Tong & Ding (J. Fluid Mech., vol. 864, 2019, pp. 640–669) provides the scaling properties of the statistics in the shear-stress budget equations. Our previous and present studies have shown that the CBL is mathematically a singular perturbation problem. Therefore, we obtain the mean velocity profile using the method of matched asymptotic expansions. Three scaling layers are identified: the outer layer, which includes the mixed layer, the inner-outer layer and the inner-inner layer, which includes the roughness layer. There are two overlapping layers, the local-free-convection layer and the log layer, respectively. Two new velocity-defect laws are discovered: the mixed-layer velocity-defect law and the surface-layer velocity-defect law. The local-free-convection mean profile is obtained by asymptotically matching the expansions in the first two layers. The log law is obtained by matching the expansions in the last two layers. The von Karman constant is obtained using velocity and length scales, and therefore has a physical interpretation. A new friction law, the convective logarithmic friction law, is obtained. The present work provides an analytical derivation of the mean velocity profile hypothesized in the Monin–Obukhov similarity theory, and is part of a comprehensive derivation of the MMO scaling from first principles.

Computational Solution of Two- Point Boundary Value Problem by Quadrature method in terms of Liouville-Green Transformation

In this research paper we are selected a two-pointedge value problem ofsingular perturbation with Dirichlet type of margin conditions. The selected differential equation is transformedinto to the required form by using a Liouville –Green transformation. Then the computational process has been implemented for solving thetwo-pointborder line value problem ofsingular perturbation with either right or left end frontier layer in the specific interval [0,1]. The transformation reduces the mathematical complexity with some assumptions and applied numerical integration method to get the computations for different choices of the perturbation parameter,which is very near to zero. In the present research problem, we have observed the uniform convergence in the computational solution in the regular region and some chaotic behavior near the periphery layer region. We are implemented this method for several linear differential equations and observed that the numerically obtained resultsare validated with literature

Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem

The main aim of this paper is to present a two-grid method for the fourth order nonlinear Bi-wave singular perturbation problem with low order nonconforming finite element based on the Ciarlet–Raviart scheme. The existence and uniqueness of the approximation solution are demonstrated through the Brouwer fixed point theorem and the uniform superconvergent estimates in the broken H1− norm and L2− norm are obtained, which are independent of the perturbation parameter δ. Some numerical results indicate that the proposed method is indeed an efficient algorithm.

Variable mesh non polynomial spline method for singular perturbation problems exhibiting twin layers

In this paper, we descend a variable mesh finite difference scheme based on non polynomial spline approximation for the solution of singular perturbation problems with twin boundary layers. We develop the discretization equation for the problem using the condition of continuity for the first order derivatives of the variable mesh non polynomial spline at the interior nodes. The discrete invariant imbedding algorithm is utilized to solve the tridiagonal system obtained by the method. Endeavor examples are illustrated and maximum absolute errors in comparison to the other methods in the literature are shown to vindicate the method.