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Construction of a global solution for the one dimensional singularly-perturbed boundary value problem
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Abstract
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 -uniform convergence of such gained the approximate solutions, in the maximum norm of the order on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has --uniform convergence, but now of order on [0,1]. In the end a numerical experiment is presented to confirm previously shown theoretical results.
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The paper examines a semilinear singular reaction-diffusion problem. Using the col-
location method with naturally chosen splines of exponential type, a new difference scheme on a mesh of Bakhvalov type is constructed. A difference scheme generates the system of non-linear equations, and the theorem of existence and this system’s solution uniqueness is also provided. At the end, a numerical example, is given as well, which points to the convergence of the numerical solution to the exact one.
In this work we consider the singularly perturbed one-dimensional semi-linear reaction-diffusion problem " y (x) = f (x; y); x 2 (0; 1) ; y(0) = 0; y(1) = 0; where f is a nonlinear function. Here the second-order derivative is multiplied by a small positive parameter and consequently, the solution of the problem has boundary layers. A new difference scheme is constructed on a modified Shishkin mesh with O(N) points for this problem. We prove existence and uniqueness of a discrete solution on such a mesh and show that it is accurate to the order of N^{-2} ln^{2} N in the discrete maximum norm. We present numerical results that verify this rate of convergence.
We consider the singularly perturbed selfadjoint one-dimensional semilinear reaction-diffusion problem () () 2 : , L y y x f x y ε ε ′′ = = , on () 1 , 0 () 0 0 = y ; () 0 1 = y , where f(x,y) is a non-linear function. For this problem, using the spline-method with the natural choice of functions, a new difference scheme is given on a non-uniform mesh. The constructed non-linear difference scheme has uniform convergence in points of uneven division segments. A numerical example is given.
This paper is concerned with discretizing the boundary value problems in ordinary differential equations. We set up a total of 4 schemes for a boundary value problem -x '' =f(t,x) on [0,1], R i x=γ i (i=1,2), with four classes of linear functionals R i on C 1 [0,1] on a nonuniform mesh.
In this paper we are considering a semilinear singular perturbation reaction
-- diffusion boundary value problem, which contains a small perturbation
parameter that acts on the highest order derivative. We construct a difference
scheme on an arbitrary nonequidistant mesh using a collocation method and
Green's function. We show that the constructed difference scheme has a unique
solution and that the scheme is stable. The central result of the paper is
-uniform convergence of almost second order for the discrete
approximate solution on a modified Shishkin mesh. We finally provide two
numerical examples which illustrate the theoretical results on the uniform
accuracy of the discrete problem, as well as the robustness of the method.
We consider a numerical solution of two boundary value problems: (BVP1)-u '' (x)=f(x,u),u(0)=γ 0 ,u(1)=γ 1 , and (BVP2)-u '' (x)-q(x)u(x)=f(x),u(0)=γ 0 ,u(1)=γ 1 , using the authors’ four-point rule of degree 3 for the second derivative. The discretisation meshes depend upon this rule. The discrete problem to (BVP1) has the usual form, but the discrete problem to (BVP2) has an unusual form. Under certain assumptions, our schemes have a third order of convergence.
Collocation with quadratic C
1-splines for a singularly perturbed reaction-diffusion problem in one dimension is studied. A modified Shishkin mesh is used to resolve the layers. The resulting method is shown to be almost second order accurate in the maximum norm, uniformly in the perturbation parameter. Furthermore, a posteriori error bounds are derived for the collocation method on arbitrary meshes. These bounds are used to drive an adaptive mesh moving algorithm. Numerical results are presented.
A singularly perturbed quasi-linear two-point boundary value problem with an exponential boundary layer is discretized on arbitrary nonuniform meshes using first- and second-order difference schemes, including upwind schemes. We give first- and second-order maximum norm a posteriori error estimates that are based on difference derivatives of the numerical solution and hold true uniformly in the small parameter. Numerical experiments support the theoretical results.
An upwind finite-difference scheme for the numerical solution of semilinear convection-diffusion problems with attractive boundary turning points is considered. We show that the maximum nodal error is bounded by a special weighted ℓ1-type norm of the truncation error. This result is used to establish uniform convergence with respect to the perturbation parameter on Shishkin meshes.
The numerical solution of boundary value problems for certain stiff ordinary differential equations is studied. The methods developed use singular perturbation theory to construct approximate numerical solutions which are valid asymptotically; hence, they have the desirable feature of becoming more accurate as the equations become stiffer. Several numerical examples are presented which demonstrate the effectiveness of these methods.
A nonlinear difference scheme is given for solving a semilinear singularly perturbed two-point boundary value problem. Without any restriction on turning points, the solution of the scheme is shown to be first order accurate in the discreteL
1 norm, uniformly in the perturbation parameter. When turning points are excluded, the scheme is first order accurate in the discreteL
norm, uniformly in the perturbation parameter.
We present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders. It is shown that a solution of the scheme converges, uniformly in a perturbation parameter, to that of the continuous problem.
A numerical method, which has an estimate of the rate of convergence which is uniform with respect to a small parameter, is proposed for a non-linear boundary value problem with a small parameter for the highest-order derivative.
Uniform fourth order difference scheme for a singular perturbation problem
, Uniform fourth order difference scheme for a singular perturbation problem, Numer. Math. 56 (1989), no. 7,
675-693.
On numerical solution of semilinear singular perturbation problems by using the hermite scheme
, On numerical solution of semilinear singular perturbation problems by using the hermite scheme, Univ. u Novom
Sadu Zb. Rad, Prirod-Mat. Fak. Ser. Mat 23 (1993), no. 2, 363-379.
An almost sixth-order finite-difference method for semilinear singular perturbation problems
, An almost sixth-order finite-difference method for semilinear singular perturbation problems, Computational methods in applied mathematics 4 (2004), no. 3, 368-383.