Anthology of Spline Based Numerical Techniques for Singularly Perturbed Boundary Value Problems (original) (raw)
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In this study, two polynomial splines are developed and used to obtain the numerical solution of singularly perturbed two-point boundary value problems. The two polynomial splines developed are linear and non-linear polynomial splines. The applications of these splines to singularly perturbed two-point boundary value problems resulted to linear algebraic system of equations which are then solved by Gaussian elimination method to obtain the unknown constants arising from the splines used. Three singularly perturbed boundary value problems are solved.
A parametric spline method for second-order singularly perturbed boundary-value problem
2014
A numerical method based on parametric spline with adaptive parameter is given for the secondorder singularly perturbed two-point boundary value problems of the form 01 () () (); () ; () y p x y q x y r x y a y b The derived method is second-order and fourth-order convergence depending on the choice of the two parameters and . Error analysis of a method is briefly discussed. The method is tested on an example and the results found to be in agreement and support the predicted theory.
A spline method for second-order singularly perturbed boundary-value problems
Journal of Computational and Applied Mathematics, 2002
We consider a di erence scheme based on cubic spline in compression for second-order singularly perturbed boundary value problem of the form y = p(x)y + q(x)y + r(x); y(a) = 0 ; y(b) = 1 : The method is shown to have second-and fourth-order convergence depending on the choice of parameters 1 and 2 involved in the method. The method is tested on an example and the results found to be in agreement with the theory.
Cubic spline method for a class of nonlinear singularly-perturbed boundary-value problems
Journal of Optimization Theory and Applications, 1993
In this paper, we present a numerical method for solving a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. The original second-order problem is reduced to an asymptotically equivalent first-order problem and is solved by a numerical method using a fourth-order cubic spIine in the inner region. The method has been analyzed for convergence and is shown to yield an O(h 4) approximation to the solution. Some test examples have been solved to demonstrate the efficiency of the method.
A spline method for solving fourth order singularly perturbed boundary value problem
2014
In this paper, singularly perturbed boundary value problem of fourth order ordinary differential equation with a small positive parameter multiplying with the highest derivative of the form εu(4)(x) + p(x)u ′′ (x) + q(x)u(x) = r(x), 0 ≤ x ≤ 1, u(0) = γ0, u(1) = γ1, u ′′ (0) = η0, u ′′ (1) = η1, 0 ≤ ε ≤ 1 is considered. We have developed a numerical technique for the above problem using parametric and polynomial septic spline method. The method is shown to have second and fourth order convergent depending on the choice of parameters involved in the method. Truncation error and boundary equations are obtained. The method is tested on an example and the results are found to be in agreement with the theoretical analysis.
Advances in Computational Mathematics, 1996
In this paper, we presented a domain decomposition method via exponential splines for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. The method is distinguished by the following fact: The original singularly perturbed two-point boundary value problem is divided into two problems, namely inner and outer region problems. The terminal boundary condition is obtained from the solution of the reduced problem. Using stretching transformation, a modified inner region problem is constructed. Then, the inner region problem is solved as two-point boundary value problems by employing exponential splines. Several linear and nonlinear problems are solved to demonstrate the applicability of the method.
A combination of spline and spectral approximation for a class of singularly perturbed problems
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1998
We shall consider the selfadjoint boundary layer problem described by the second order differential equation with a small parameter multiplying the highest derivative. The approximate solution will be constructed by the use of spline collocation technique out of the layer and by the use of truncated Chebyshev series inside the layer. The layer subinterval is determined by the use of resemblence function in terms of the chosen degree of the spectral solution. The error at the division point is estimated by the use of the error of the spline function, and it is used to obtain the error estimate inside the layer.Numerical example shows that the combination of these two techniques gives better results than the application of modified spectral methods.
A Spline Collocation Method and a Special Grid of Shishkin Type fora Singularly Perturbed Problem
Communications to Simai Congress, 2007
We consider the two parameter singularly perturbed boundary value problem (0.1) Ly := εy ′′ (x) + µa(x)y ′ (x) − b(x)y(x) = f (x), x ∈ (0, 1), y(0) = p 0 , y(1) = p 1 , where a, b and f are sufficiently smooth functions, 0 < ε ≪ 1, 0 < µ ≪ 1, and a(x) ≥ α > 0, b(x) ≥ β > 0, x ∈ [0, 1].
This paper envisages a fourth-order finite difference method with reference to the solution of a class of singularly perturbed singular boundary value problems especially on a uniform mesh. The non-polynomial spline forms the tool for the solution of the problems. The discretisation equation of the problems are developed using the condition of continuity for the first-order derivatives of the non-polynomial spline at the interior nodes and it is not valid at the singularity. Hence at the singularity, the boundary value problem is modified in order to get a three-term relation. The tridiagonal scheme of the method is processed using discrete invariant imbedding algorithm. The convergence of the method is analysed and maximum absolute errors in the solution are tabulated. Root mean square errors in the solution of the examples are presented in comparison to the methods chosen from the literature to establish the proposed method.