Singularly Perturbed Boundary Value Problems Research Articles

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.

Modified asymptotic solutions for second-order nonlinear singularly perturbed boundary value problems

We introduce a coordinate transformation of independent variable, such that the second-order nonlinear singularly perturbed boundary value problem (SPBVP) in the transformed coordinate is less stiff within the boundary layer. An initial value problem for a new dependent variable can be derived easily through the variable transformation. While the zero initial values are given, an unknown terminal value of the new variable at the right end is determined iteratively. We propose the modifications of the asymptotic solution and the uniform approximate solution of the SPBVP; hence, the modified analytic solutions can exactly satisfy both the boundary conditions at two ends. Some examples confirm that the novel methods can achieve better analytic and numerical solutions of the nonlinear SPBVP.

A neural network training algorithm for singular perturbation boundary value problems

A neural network training algorithm for singular perturbation boundary value problems

A class of fourth order nonlinear boundary value problem with singular perturbation

A class of singularly perturbed fourth order nonlinear differential equation is considered in this article. The existence of solution to boundary value problem (BVP) without singularly perturbed is obtained via nonlinear analysis method and upper and lower solution theory firstly. In addition, based upon some differential inequality techniques, as well as appropriate upper–lower solutions, the existence, uniqueness and corresponding asymptotic estimation of the solution to singularly perturbed boundary value problem (SPBVP) is considered.

Trigonometric quintic B-spline collocation method for singularly perturbed turning point boundary value problems

A trigonometric quintic B-spline method is proposed for the solution of a class of turning point singularly perturbed boundary value problems (SP-BVPs) whose solution exhibits either twin boundary layers near both endpoints of the interval of consideration or an interior layer near the turning point. To resolve the boundary/interior layer(s) trigonometric quintic B-spline basis functions are used with a piecewise-uniform mesh generated with the help of a transition parameter that separates the layer and regular regions. The proposed method reduces the problem into a system of algebraic equations which can be written in matrix form with the penta-diagonal coefficient matrix. The well-known fast penta-diagonal system solver algorithm is used to solve the system. The method is shown almost fourth-order convergent irrespective of the size of the diffusion parameter ϵ. The theoretical error bounds are verified by taking some relevant test examples computationally.

A Sixth-Order Numerical Method Based on Shishkin Mesh for Singularly Perturbed Boundary Value Problems

Recently, Lodhi and Mishra (J Comput Appl Math 319:170–187, 2017) presented the standard B-spline method based on quintic B-spline basis functions to solve a type of singularly perturbed boundary value problems (SPBVP). We note that their method provides only fourth-order convergence approximation to the solution of such problem. In this paper, we present a novel optimal B-spline technique, based on same quintic B-spline basis function as used in Lodhi and Mishra (2017), for solving linear and nonlinear SPBVP. The advantage of the suggested method over the method in Lodhi and Mishra (2017) is that our method has sixth-order rate of convergence. To obtain higher order of convergence, a high-order perturbation of the SPBVP is generated. The method is tested for its efficiency by applying it on five test problems.

Boundary shape functions methods for solving the nonlinear singularly perturbed problems with Robin boundary conditions

Abstract For a second-order nonlinear singularly perturbed boundary value problem (SPBVP), we develop two novel algorithms to find the solution, which automatically satisfies the Robin boundary conditions. For the highly singular nonlinear SPBVP the Robin boundary functions are hard to be fulfilled exactly. In the paper we first introduce the new idea of boundary shape function (BSF), whose existence is proven and it can automatically satisfy the Robin boundary conditions. In the BSF, there exists a free function, which leaves us a chance to develop new algorithms by adopting two different roles of the free function. In the first type algorithm we let the free functions be the exponential type bases endowed with fractional powers, which not only satisfy the Robin boundary conditions automatically, but also can capture the singular behavior to find accurate numerical solution by a simple collocation technique. In the second type algorithm we let the BSF be solution and the free function be another variable, such that we can transform the boundary value problem to an initial value problem (IVP) for the new variable, which can quickly find accurate solution for the nonlinear SPBVP through a few iterations.

A subdivision-based approach for singularly perturbed boundary value problem

A numerical approach for solving second order singularly perturbed boundary value problems (SPBVPs) is introduced in this paper. This approach is based on the basis function of a 6-point interpolatory subdivision scheme. The numerical results along with the convergence, comparison and error estimation of the proposed approach are also presented.

Gaussian Quadrature for Two-Point Singularly Perturbed Boundary Value Problems with Exponential Fitting

In this paper, the Gaussian quadrature method with exponential fitting is proposed for the solution of two-point singularly perturbed boundary value problems with layer at one endpoint, dual boundary layers and internal boundary layers. The given boundary value problem is reduced into an equivalent first order differential equation with the perturbation parameter as deviating argument. Then, Gaussian two-point quadrature technique with exponential fitting is implemented to solve the first order equation with deviating parameter. The analysis of the convergence of the method is discussed. Several numerical examples are illustrated with a layer at one end, a layer at two ends and internal layers. Comparison of maximum errors in the solution of the examples with other methods is shown to justify the method.

Discrete approximation for a two-parameter singularly perturbed boundary value problem having discontinuity in convection coefficient and source term

In this study we present a method for approximating the solution of a Singularly Perturbed Boundary Value Problem (SPBVP) containing two parameters ( ε 1 , ε 2 ), which multiply the diffusion coefficient and the convection term, respectively. Moreover, we consider that the convection coefficient and the source term present a discontinuity at an intermediate point. Theoretical bounds for the solution and its derivatives are derived for two complementary cases. A parameter uniform numerical scheme is constructed, which involves an upwind finite difference method with an appropriate piecewise uniform mesh. The error estimation and convergence analysis are presented, which show that the scheme provides a parameter uniform convergence of almost first order. Some numerical examples are discussed to illustrate the performance of the present method.

The Existence and Asymptotic Estimates of Solutions for a Third-Order Nonlinear Singularly Perturbed Boundary Value Problem

In this paper, we consider a class of third-order nonlinear differential equation with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a boundary layer at one endpoint. By using the Schauder fixed point theorem, Green’s function and the method of upper–lower solutions, we first establish an existence result of corresponding boundary value problem without perturbation. Furthermore, by constructing an appropriate lower solution-upper solution pair, as well as analysis technique, the existence and asymptotic estimates of the solutions for the singularly perturbed boundary value problems are obtained.

Simulation of second order singularly-perturbed boundary-value problems using improved meshless method

In this study, numerical integration based on Block-Pulse functions and Chebyshev wavelets are employed for Element Free Galerkin approximation. Moving Least Squares (MLS) approach is used to construct shape functions with optimized weight functions and basis. The proposed techniques are implemented on singularly-perturbed boundary-value problems with two-point boundary conditions. Numerical results for two examples are presented in this article to show the pertinent features for the proposed technique. Comparison with existing techniques shows that our proposed method based on integration technique provides better approximation at reduced computational cost. Moreover the effect of perturbation parameter on solution of test problems has been studied.

Quintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems

In this communication, we have studied an ecient numerical ap- proach based on uniform mesh for the numerical solutions of fourth order singular perturbation boundary value problems. Such type of problems arises in various elds of science and engineering, such as electrical network and vi- bration problems with large Peclet numbers, Navier-Stokes ows with large Reynolds numbers in the theory of hydrodynamics stability, reaction-diusion process, quantum mechanics and optimal control theory etc. In the present study, a quintic B-spline method has been discussed for the approximate solu- tion of the fourth order singular perturbation boundary value problems. The convergence analysis is also carried out and the method is shown to have con- vergence of second order. The performance of present method is shown through some numerical tests. The numerical results are compared with other existing method available in the literature.

Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

We consider an approximate solution for the one--dimensional semilinear singularly--perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.

NUMERICAL METHOD FOR SINGULARLY PERTURBED THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of reaction-diffusion type of third order Ordinary Differential Equations (ODEs). The SPBVP is reduced into a weakly coupled system of one first order and one second order ODEs, one without the parameter and the other with the parameter <TEX>${\varepsilon}$</TEX> multiplying the highest derivative subject to suitable initial and boundary conditions, respectively. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. The weakly coupled system is decoupled by replacing one of the unknowns by its zero-order asymptotic expansion. Finally the present numerical method is applied to the decoupled system. In order to get a numerical solution for the derivative of the solution, the domain is divided into three regions namely two inner regions and one outer region. The Shooting method is applied to two inner regions whereas for the outer region, standard finite difference (FD) scheme is applied. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing. The main advantage of this method is that due to decoupling the system, the computation time is very much reduced.

A Spectral Collocation Method for Systems of Singularly Perturbed Boundary Value Problems

Abstract We present a spectrally accurate method for solving coupled singularly perturbed second order two-point boundary value problems (BVPs). The method combines analytical coordinate transformations with a standard Chebyshev spectral collocation method; it is applicable to linear and to nonlinear problems. The method performs well in resolving very thin boundary layers. Compared to other methods which had been proposed for systems of BVPs this method is competitive in terms of accuracy, allows for different perturbation parameters in each of the equations, and does not require special properties of the coefficient functions.

Computational Approach for Fourth-Order Self-Adjoint Singularly Perturbed Boundary Value Problems via Non-polynomial Quintic Spline

This paper proposes a non-polynomial quintic spline method for approximate solutions of fourth-order self-adjoint singular perturbation boundary value problems, which comprises the polynomial part of degree three and two non-polynomial terms, i.e., sine and cosine. It is explicitly shown that the free parameter \(\tau\) of the non-polynomial part can be used to raise the order of accuracy of the scheme. The method has been proved for the second and fourth-order convergence. The proposed scheme is tested on two examples. The experimental results are compared with the existing method.

SHOOTING METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE

A class of singularly perturbed boundary value problems (SPBVPs) for fourth-order ordinary differential equations (ODEs) is considered. The SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. In order to solve them numerically, a method is suggested in which the given interval is divided into two inner regions (boundary layer regions) and one outer region. Two initial-value problems associated with inner regions and one boundary value problem corresponding to the outer region are derived from the given SPBVP. In each of the two inner regions, an initial value problem is solved by using fitted mesh finite difference (FMFD) scheme on Shishkin mesh and the boundary value problem corresponding to the outer region is solved by using classical finite difference (CFD) scheme on Shishkin mesh. A combination of the solution so obtained yields a numerical solution of the boundary value problem on the whole interval. First, in this method, we find the zeroth-order asymptotic expansion approximation of the solution of the weakly coupled system. Error estimates are derived. Examples are presented to illustrate the numerical method. This method is suitable for parallel computing.

AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH A SMALL NEGATIVE SHIFT

Abstract. In this paper, we present an initial value technique for solvingsingularly perturbed differential difference equations with a boundary layerat one end point. Taylor’s series is used to tackle the terms containing shiftprovided the shift is of small order of singular perturbation parameter andobtained a singularly perturbed boundary value problem. This singularlyperturbed boundary value problem is replaced by a pair of initial valueproblems. Classical fourth order Runge–Kutta method is used to solvethese initial value problems. The effect of small shift on the boundarylayer solution in both the cases, i.e., the boundary layer on the left sideas well as the right side is discussed by considering numerical experiments.Several numerical examples are solved to demonstate the applicability ofthe method.AMS Mathematics Subject Classification : 65L11.Key words and phrases : Initial value technique, singular perturbation,differential–difference equation, boundary layer. 1. IntroductionThe numerical treatment of singular perturbations is far from the trivial be-cause of the boundary layer behavior. In recent decades this is a field of increas-ing interest to applied mathematicians and numerical analysts in view of thechallenges the problems there in pose to the researchers. For a detailed theoryand analytical discussion on singular perturbation problems one may refer tothe books and high level monographs: O’ Malley [13], Nayfeh [12], Kevorkianand Cole [11], Bender and Orszag [10]. A Singularly perturbed delay differentialequation is an ordinary differential equation in which the highest derivative ismultiplied by a small parameter and involving atleast one delay term. In 1979,the Japanese physicist Kensuke Ikeda considered a nonlinear absorbing medium

Fourth Order Fitted Scheme for Second Order Singular Perturbation Boundary Value Problems

Fitted fourth order central difference scheme is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end point. A fitting factor is introduced in a tri-diagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system and its stability is investigated. To demonstrate the applicability of the method, we have solved linear and nonlinear problems. From the results, it is observed that the present method approximates the exact solution very well.