Perturbed Boundary Value Problems Research Articles

A comparative study of numerical methods for singularly perturbed boundary value problems

A comparative study of numerical methods for singularly perturbed boundary value problems

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Exponential B-Spline Method for Second Order Singularly Perturbed Boundary Value Problem with Negative Shift

Exponential B-Spline Method for Second Order Singularly Perturbed Boundary Value Problem with Negative Shift

Semi-analytic PINN methods for boundary layer problems in a rectangular domain

Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.

Fractal quintic spline solutions for singularly perturbed reaction-diffusion boundary-value problems

In this article, with the help of fractal quintic spline, numerical solutions of singularly perturbed boundary value problem of ordinary differential equations are obtained. To check the accuracy of the proposed method, convergence analysis is derived. The developed method has fourth order convergence. To check the correctness of the proposed method, numerical examples are provided.

SIMULATION OF DYNAMIC PROCESSES IN A p - i - n DIODE BY THE METHODS OF PERTURBATION THEORY

In the paper a mathematical model of the dynamic process of the electron-hole plasma formation in the active region ( i - region) of a semiconductor p - i - n - diode in the direct bias mode with a harmonic signal fed on the diode is developed. The basis of the model is a nonlinear nonstationary sin- gularly perturbed boundary value problem for a system of continuity equations of electron-hole currents and Poisson's equation. The algorithm for searching the charge carrier concentration distributions and the potential distribution is based on the asymptotic method of boundary functions and the Fourier method. In the course of the research, a technique for decomposing a nonlinear problem based on the development of perturbation theory me- thods was proposed. The initial nonlinear problem is reduced to a recurrent sequence of linear stationary boundary value problems, solved by classical and partially original analytical-numerical methods. The identification of the boundary functions in the solution of the problem provides, in particular, in contrast to the classical approximation of ambipolar diffusion, an adequate description of the electric field strength behavior in the active region of p - i - n - diodes. The obtained research results shed light on the peculiarities of the formation of the impedance characteristics of p - i - n - structures. The results of the work reveal more deeply the nature of the physical processes in the studied technical system and are aimed at improving the methodology for modeling and designing control semiconductor devices.

Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.

A New Approach for Solving Singularly Perturbed Boundary Value Problems by Using Exponential Spline Method

This article presents the development of families of approaches for numerically solving singularly perturbed two-point boundary-value problems using exponential spline functions. The proposed approaches exhibit second-order and fourth-order accuracy and are suitable for both singular and non-singular problem scenarios. Numerical data are presented to demonstrate the efficacy of our methodologies and are compared with those proposed by various writers.

Fitted mesh methods based on non-polynomial splines for singularly perturbed boundary value problems with mixed shifts

<p>In this paper, numerical schemes based on non-polynomial splines, namely, spline in compression, tension, and adaptive spline, are constructed for singularly perturbed boundary value problems with mixed shifts. A convergence analysis is carried out on the proposed methods. A comparitive study of the results is performed on test problems and presented in the form of tables. Graphs are drawn to illustrate the behavior of the solution to the problems.</p>

Fourth order computational spline method for two-parameter singularly perturbed boundary value problem

The current research work considers a two-parameter singularly perturbed two-point boundary value problem. Here, we suggest a computational scheme derived by using an exponential spline for the numerical solution of the problem on a uniform mesh. The proposed numerical scheme is analyzed for convergence and an accuracy of O(h4) is achieved. Numerical experiments are considered to validate the efficiency of the spline method, and compared comparison with the existing method to prove the superiority of the proposed scheme.

An [formula omitted] finite element method for a two-dimensional singularly perturbed boundary value problem with two small parameters

We consider a singularly perturbed boundary value problem, of reaction–convection–diffusion type with two small parameters, and the approximation of its solution by the hp version of the Finite Element Method on the Spectral Boundary Layer mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples illustrating our theoretical findings are also included.

A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift

A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift

Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up:•The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings.•Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm.•Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature.

The Models of Gradient Mechanics and Singularly Perturbed Boundary Value Problems

The Models of Gradient Mechanics and Singularly Perturbed Boundary Value Problems

(Iq)–Stability and Uniform Convergence of the Solutions of Singularly Perturbed Boundary Value Problems

In this paper, using the notion of (Iq)–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 εy″=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.

Creeping flow of non-Newtonian fluid through membrane of porous cylindrical particles: A particle-in-cell approach

The present study is an attempt to deal with hydrodynamic and thermal aspects of the incompressible Carreau fluid flow past a membrane consisting of uniformly distributed aggregates of porous cylindrical particles enclosing a solid core which aims to provide a comprehensive study of the impact of non-Newtonian nature of Carreau fluid in the filtration process through membranes. The non-Newtonian characteristic of Carreau fluid is adopted to describe the mechanism of the pseudoplastic flow through membranes. The layout of the fluid flow pattern is separated into two distinct areas in which the area adjacent to the solid core of the cylindrical particle is considered as porous. However, the region surrounding the porous cylindrical particle is taken as non-porous (clear fluid region). The Brinkman equation governs the porous region, whereas the non-porous region is regulated by the Stokes equation. The nonlinear governing equations of the Carreau fluid flow in the different regions are solved using an asymptotic series expansion in terms of the small parameters, such as Weissenberg number ( We ≪ 1 ) and a non-dimensional parameter ( S ≪ 1 ), for the higher permeability of the porous material. For large permeability, the expression of velocity is derived, and the same has been used to compute the hydrodynamic permeability, Kozeny constant, and temperature profile. The numerical scheme (NDSolve in Mathematica) is used to solve the singularly perturbed boundary value problems in the case of small permeability of the porous medium [i.e., ( S ≫ 1 )]. The graphical analysis illustrating the outcomes of the effects of varying control parameters such as the power-law index, viscosity ratio parameter, permeability of the porous medium, Weissenberg number, and Nusselt number on the membrane permeability, Kozeny constant and temperature profile are discussed comprehensively and validated with previously published works on the Newtonian fluid in the limiting cases. The notable determination of the present study is that the Carreau fluid parameters, such as the Weissenberg number, power-law index, and viscosity ratio parameter, have a significant impact on the velocity, and hence, the membrane permeability, Kozeny constant, and temperature profile. The results showed a significant increase in the flow velocity and hydrodynamic permeability as the dominance of elastic forces over viscous forces increased in the case of high permeability ( S ≪ 1 ). The velocity gets a slight reduction for lower permeability of the porous material ( S ≫ 1 ); however, the hydrodynamic permeability behaves similar to the higher permeability of the porous material. The findings of the proposed work may be instrumented in analyzing various processes, including wastewater treatment filtration processes, and blood flow through smooth muscle cells. The proposed work, however, requires experimental verification.

On eigenelements of a two-dimensional Steklov-type boundary value problem for the Lamé operator

In this paper, we study a two-dimensional Steklov-type boundary value problem for the Lamé operator in a half-strip, which is the limiting problem for a singularly perturbed boundary-value problem in a half-strip with a small hole. A theorem on the existence of eigenelements of the boundary value problem under study is proved. In particular, we obtain estimates for the eigenvalues expressed in terms of the Lamé constants and a parameter that determines the width of the half-strip, and refine the structure of the corresponding eigenfunctions, which determines their behavior as their argument move away from the base of the half-strip. Moreover, explicit expressions for the eigenvalues of the limiting boundary value problem are found up to the solution of a system of algebraic equations. The results obtained in this paper will make it possible to construct and rigorously justify an asymptotic expansion of the eigenvalue of a singularly perturbed boundary value problem in a half-strip with a small round hole in powers of a small parameter that determines the diameter of the hole.

Asymptotic expansion of the solution for singularly perturbed boundary value problem with boundary jumps

Asymptotic expansion of the solution for singularly perturbed boundary value problem with boundary jumps

A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

We consider two-parameter singularly perturbed problems of reaction-convection-diffusion type in one dimension. The convection coefficient and source term are discontinuous at a point in the domain. The problem is numerically solved using the upwind difference method on an appropriately defined Shishkin-Bakhvalov mesh. At the point of discontinuity, a three-point difference scheme is used. A convergence analysis is given and the method is shown to be first-order uniformly convergent with respect to the perturbation parameters. The numerical results presented in the paper confirm our theoretical results of first-order convergence. Summing up:• The Shishkin-Bakhvalov mesh is graded in the layer region and uniform in the outer region as shown in the graphical abstract.• The method presented here has uniform convergence of order one in the supremum norm.• The numerical orders of convergence obtained in numerical examples with Shishkin- Bakhvalov mesh are better than those for Shishkin mesh.

PARAMETER INDEPENDENT SCHEME FOR SINGULARLY PERTURBED PROBLEMS INCLUDING A BOUNDARY TURNING POINT OF MULTIPLICITY ≥ 1

A numerical scheme is developed for parabolic singularly perturbed boundary value problems, including multiple boundary turning points at the left endpoint of the spatial direction. The highest order derivative of these problems is multiplied by a small parameter $\epsilon\, (0<\epsilon\ll 1)$, and when it is close to zero, the solution exhibits a parabolic type boundary layer near the left lateral surface of the domain of consideration. Thus, large oscillations appear when classical/standard numerical methods are used to solve the problem, and one cannot achieve the expected accuracy. Thus, the Crank-Nicolson scheme on a uniform mesh in the temporal direction and an upwind scheme on a Shishkin-type mesh in the spatial direction is constructed. The theoretical analysis shows that the method converges irrespective of the size of $\epsilon$ with accuracy $\mathcal{O}((\Delta t)^2+N^{-1}\ln N)$. Three test examples are presented to verify that the computational results agree with the theoretical ones.