A seventh order numerical method for singular perturbation problems

Chakravarthy, P. Pramod; Phaneendra, K.; Reddy, Y. N.

doi:10.1016/j.amc.2006.08.022

Cited by 12 publications

(8 citation statements)

References 9 publications

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“…In Table 1 and Table 2; the actual absolute errors, the estimated absolute errors and the improved absolute errors are compared for (N,M)= (12,13) and 3 10    , 5 10    respectively. In Table 3, a comparison is given between the exact solution and the results of other methods such as seventh order numerical method (SNM) [10], the Chebyshev method (CM) [4], the Bessel collocation method [6] and the present method for () yx.On the other hand in Figs. 1-2; the exact solutions and the improved approximate solutions are drawn for () yx for different values of truncation limits.…”

Section: mentioning

confidence: 99%

A Muntz-Legendre Approach to Obtain Solutions of Singular Perturbed Problems

2020

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Singularly perturbed differential equations are encountered in mathematical modelling of processes in physics and engineering. Aim of this study is to give a collocation approach for solutions of singularly perturbed two-point boundary value problems. The method provides obtaining the approximate solutions in the form of Müntz-Legendre polynomials by using collocation points and matrix relations. Singularly perturbed problem is transformed into a system of linear algebraic equations. By solving this system, the approximate solution is computed. Also, an error estimation is done using the residual function and the approximate solutions are improved by means of the estimated error function. Two numerical examples are given to show the applicability of the method.

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Section: mentioning

confidence: 99%

A Muntz-Legendre Approach to Obtain Solutions of Singular Perturbed Problems

2020

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“…We derived high-order schemes, higher than the seventh-order scheme given in [4]. The obtained results may be directly applied to schemes of higher order.…”

Section: 2mentioning

confidence: 99%

High-order exponentially fitted difference schemes for singularly perturbed two-point boundary value problems

Marušić¹

2018

etna

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We introduce a family of exponentially fitted difference schemes of arbitrary order as numerical approximations to the solution of a singularly perturbed two-point boundary value problem: εy + by + cy = f. The difference schemes are derived from interpolation formulae for exponential sums. The so-defined k-point differentiation formulae are exact for functions that are a linear combination of 1, x,. .. , x k−2 , exp (−ρx). The parameter ρ is chosen from the asymptotic behavior of the solution in the boundary layer. This approach allows a construction of the method with arbitrary order of consistency. Using an estimate for the interpolation error, we prove consistency of all the schemes from the family. The truncation error is bounded by Ch k−2 , where C is a constant independent of ε and h. Therefore, the order of consistency for the k-point scheme is k − 2 (k ≥ 3) in case of a small perturbation parameter ε. There is no general proof of stability for the proposed schemes. Each scheme has to be considered separately. In the paper, stability, and therefore convergence, is proved for three-point schemes in the case when c < 0 and b = 0.

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“…For ( N , M ) = (14,15) and ε = 2 −6 , Figure (c) denotes a comparison of actual error function

(||, e_{N}^{ε} (x))

and improved actual error functions

(||, E_{N, M}^{ε} (x))

. Example Let us consider the singularly perturbed differential equation

ε y^{′′} (x) - y^{'} (x) - (1 + ε) y (x) = 0, 0 \leq x \leq 1,

with the boundary conditions

y (0) = 1 + e^{- \frac{1 + ε}{ε}} and y (1) = 1 + \frac{1}{e},

and the exact solution

y (x) = e^{- x} + e^{\frac{(1 + ε) (x - 1)}{ε}} .

The improved approximate solution for ( N , M ) = (12,13) and ε = 10 −3 by our method is obtained. We compare the numerical results of the exact solution, the Chebyshev method , the seventh order numerical method , and the present method in Table and Figure . Example We consider the singularly perturbed differential equation

ε y^{′′} (x) + (1 - x / 2) y^{'} (x) - \frac{1}{2} y (x) = 0, 0 \leq x \leq 1,

with the boundary conditions

…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The improved approximate solution for .N, M/ D .12, 13/ and " D 10 3 by our method is obtained. We compare the numerical results of the exact solution, the Chebyshev method [7], the seventh order numerical method [2], and the present method in Table III and Figure 4.…”

Section: Examplementioning

confidence: 99%

“…In the recent years, for solving the mentioned problems, some authors have published the papers including various methods such as the seventh order numerical method , the finite difference methods , the B‐spline collocation method , the B‐splines with artificial viscosity , the second order spline finite difference method , the Chebysev method , and a reliable algorithm of finite difference method .…”

Section: Introductionmentioning

confidence: 99%

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A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

Yüzbaşı

2014

Math Methods in App Sciences

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In this paper, a collocation method is given to solve singularly perturbated two-point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results.

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A seventh order numerical method for singular perturbation problems

Cited by 12 publications

References 9 publications

A Muntz-Legendre Approach to Obtain Solutions of Singular Perturbed Problems

A Muntz-Legendre Approach to Obtain Solutions of Singular Perturbed Problems

High-order exponentially fitted difference schemes for singularly perturbed two-point boundary value problems

A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

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