奇摄动非线性边值问题

The singularly perturbed nonlinear boundary value problems are considered. Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained. And then the uniform validity of solution is proved by using the differential inequalities.     

Asymptotic method for solving the time-optimal control problem for a nonlinear singularly perturbed system

The time-optimal control problem for a nonlinear singularly perturbed system with multidimensional controls bounded in the Euclidean norm is considered. An algorithm for constructing asymptotic approximations to its solution is proposed. The main advantage of the algorithm is that the original optimal control problem decomposes into two unperturbed problems of lower dimensions.     

Global convergence method for singularly perturbed boundary value problems

We give uniformly convergent splines difference scheme for singularly perturbed boundary value problems(1)$-\epsilon u^{″}+p(x)u^{\prime }+q(x)u=f(x),u(a)={\alpha }_0,u(b)={\alpha }_1,$by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration. We prove the $\mathrm{O}(\min (h^2,{\epsilon }^2))$ order of uniform convergence with respect to small parameter $\epsilon$ at nodes on uniform mesh and $\mathrm{O}(\min (h,\epsilon ))$ order of uniform global convergence with respect to the approximate solution given by $S(x)={\sum }_{i=1}^N{S}_{{\Delta }_i}(x)H(x_i-x)$ where H is the Heaviside function, which is the approximation for the closed form of the exact solution.     

The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems

A new computational method for solving the second-order nonlinear singularly perturbed boundary value problems (SPBVPs) is provided in this paper. In order to overcome a highly singular behavior very near to the boundary as being not easy to treat by numerical method, we adopt a coordinate transformation from an x-domain to a t-domain via a rescaling technique, which can reduce the singularity within the boundary layer. Then, we construct a Lie-group shooting method (LGSM) to search a missing initial condition through the finding of a suitable value of a parameter r ∈ [0, 1]. Moreover, we can derive a closed-form formula to express the initial condition in terms of r, which can be determined properly by an accurate matching to the right-boundary condition. Numerical examples are examined, showing that the present approach is highly efficient and accurate.     

The Galerkin method for singularly perturbed boundary value problems on adaptive grids

Voronezh. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 5, pp. 138–148, September–October, 1990.     

Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems

We present an exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problem. The convergence analysis is given and the method is shown to have second order uniform convergence. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Piecewise reproducing kernel method for singularly perturbed delay initial value problems

A direct application of the reproducing kernel method presented in the previous works cannot yield accurate approximate solutions for singularly perturbed delay differential equations. In this letter, we construct a new numerical method called piecewise reproducing kernel method for singularly perturbed delay initial value problems. Numerical results show that the present method does not share the drawback of standard reproducing kernel method and is an effective method for the considered singularly perturbed delay initial value problems.     

BOUNDARY VALUE PROBLEMS OF SINGULARLY PERTURBED INTEGRO－DIFFERENTIAL EQUATIONS

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Parameter-robust numerical method for a system of singularly perturbed initial value problems

In this work we study a system of M( ≥ 2) first-order singularly perturbed ordinary differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. A maximum principle does not, in general, hold for this system. It is discretized using backward Euler difference scheme for which a general convergence result is derived that allows to establish nodal convergence of O(N ^− 1ln N) on the Shishkin mesh and O(N ^− 1) on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical experiments are performed to support the theoretical results.     

An algorithm for the asymptotic solution of a singularly perturbed linear time-optimal control problem

An algorithm for the approximate solution (in the asymptotic sense) of a singularly perturbed linear time-optimal control problem is proposed. A computational procedure is outlined, which permits the use of the resulting asymptotic approximation for. the exact solution of the problem with a prescribed value of the small parameter.     

Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.     

A novel method for a class of parameterized singularly perturbed boundary value problems

In this paper a novel approach is presented for solving parameterized singularly perturbed two-point boundary value problems with a boundary layer. By the boundary layer correction technique, the original problem is converted into two non-singularly perturbed problems which can be solved using traditional numerical methods, such as Runge–Kutta methods. Several non-linear problems are solved to demonstrate the applicability of the method. Numerical experiments indicate the high accuracy and the efficiency of the new method.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

Convergence of a Bakhvalov grid adaptation method for singularly perturbed boundary value problems

In this paper, a Galerkin finite element method for non-self-adjoint boundary value problems on Bakhvalov grids is considered. Using the Galerkin projectionmethod, the convergence of a sequence of computational grids with an unknown boundary of the boundary layer is proved. Some numerical examples are presented.     

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods appliedto boundary value problems for the positive definite Helmholtz-typeproblem –U + ²U = 0 in a bounded or unbounded domain,with the parameter real and possibly large. Applications arisein the implementation of space–time boundary integralmethods for the heat equation, where is proportional to 1/(t),and t is the time step. The corresponding layer potentials arisingfrom this problem depend nonlinearly on the parameter and havekernels which become highly peaked as , causing standard discretizationschemes to fail. We propose a new collocation method with arobust convergence rate as . Numerical experiments on a modelproblem verify the theoretical results.     

一类具有无穷边界值的二次奇摄动边值问题

研究一类具有无穷边界值的二次奇摄动Robin边值问题解的存在性与解的渐进行为,重点关注边界值的奇异程度对解的边界层行为的影响;同时将所得的结果与Chang及Howes的结果(带正常边界值)进行比较.研究表明:(1)当边界值大小为O(1/)时,得到的边界层大小为O( ln ),这比Chang及Howes带正常边界值的情形提高了O(ln )量级;(2)增大边界值的奇性至O(1/ r),这里r >1,边界层大小的量级不变,依然为O( ln );(3)若要使得边界层大小为O(1),则边界值的大小需为O(e?1/).最后给出一个算例验证得到的结果.     

A collocation method for singularly perturbed two-point boundary value problems with splines in tension

An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problemay+by+cy=f(·,y),y(0)=y ₀,y(1)=y ₁. Sharp error bounds for the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems (|a|=1), the representation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory.Supported by grant 1-01-254 of the Ministry of Science and Technology, Croatia.     

New initial-value method for singularly perturbed boundary-value problems

An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.This work was supported in part by the Consiglio Nazionale delle Ricerche, Contract No. 86.02108.01, and in part by the Ministero della Pubblica Istruzione.