Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters

We study a model linear convection-diffusion-reaction problem where both the diffusion term and the convection term are multiplied by small parameters ε_d and ε_c, respectively. Depending on the size of the parameters the solution of the problem may exhibit exponential layers at both end points of the domain. Sharp bounds for the derivatives of the solution are derived using a barrier-function technique. These bounds are applied in the analysis of a simple upwind-difference scheme on Shishkin meshes. This method is established to be almost first-order convergent, independently of the parameters ε_d and ε_c.     

Geometric mesh FDM for self-adjoint singular perturbation boundary value problems

A numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors max_i|y(x_i)-y_i|, for different values of parameter , number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.     

Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem

In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.     

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.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

A CLASS OF SINGULAR PERTURBATIONS FOR SECOND ORDER QUASI－LINEAR BOUNDARY VALUE PROBLEMS ON INFINITE INTERVAL

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一类双参数奇摄动边值问题的研究

研究了一类双参数高阶非线性奇摄动方程边值问题,利用合成展开法构造出形式渐近解,再利用微分不等式理论证明了解的存在一致有效性.     

The quasilinearization method for boundary value problems on time scales

In this paper, we apply the method of quasilinearization to a family of boundary value problems for second order dynamic equations −y^Δ∇+q(t)y=H(t,y) on time scales. The results include a variety of possible cases when H is either convex or a splitting of convex and concave parts and whether lower and upper solutions are of natural form or of natural coupled form.     

Fourth-order problems with fully nonlinear boundary conditions

Differential inequality method, bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth-order differential equations under nonlinear boundary conditions, and many existing results are complemented.     

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.     

Variable step size initial value algorithm for singular perturbation problems using locally exact integration

We consider quasilinear singular perturbation problems of the form εy^″+p(x)y^′+q(x,y)=h(x),x[0,1];y(0)=,y(1)=β with a boundary layer at one end point. The original problem is reduced to an asymptotically equivalent linear first order initial-value problem (IVP). Then, a variable step size initial value algorithm is applied to solve this (IVP). The algorithm is based on the locally exact integration of quadratic linearized problem coefficients on a non-uniform mesh. Two term-recurrence relation with controlled step size is obtained. Several problems are solved to demonstrate the applicability and efficiency of the algorithm. It is observed that the present method approximates the exact solution very well.     

An exponentially fitted special second-order finite difference method for solving singular perturbation problems

An exponentially fitted special second-order finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) 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 several linear and non-linear problems. From the results, it is observed that the present method approximates the exact solution very well.     

Fredholm alternative and boundary value problems

This note presents a simple proof of A. Lasota's application of the nonlinear Fredholm alternative to the existence proofs of the boundary value problems involving ordinary differential equations. It then uses Lasota's result to get a stronger version of the theorem of Herzog and Lemmert on the Dirichlet boundary value problem for the second-order systems of ordinary differential equations.

     

Application of orthogonal collocation on finite elements for solving non-linear boundary value problems

A simple, convenient and easy approach to solve non-linear boundary value problems (BVP) using orthogonal collocation on finite elements (OCFE) is presented. The algorithm is the conjunction of finite element method (FEM) and orthogonal collocation method (OCM). The stability of the numerical results is checked by a novel algorithm which not only justifies the stability of the results but also checks the convergence of the method. The method is applied to the non-symmetric boundary value problems having Dirichlet’s and mixed Robbin’s boundary conditions.     

Positive solutions for nonlinear singular boundary value problems

Some sufficient conditions for the existence of positive solutions to Dirichlet boundary value problems of a class of nonlinear second order differential equations are given.     

Existence and uniqueness of solutions for third-order nonlinear boundary value problems

In this paper, we present some general results of existence and uniqueness of solutions of nonlinear two-point boundary value problems for third-order nonlinear differential equations by using the Shooting method. As applications we give certain concrete sufficient conditions for the existence and uniqueness.