Geodesic
Application of a Taylor series to approximate a function with large gradients
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1420-1429.

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The method of approximating functions by polynomials based on Taylor series expansion is widely known. However, the residual term of such an approximation can be significant if the function has large gradients. The work assumes that the function has a decomposition in the form of a sum of regular and boundary layer components. The boundary layer component is a function of general form, known up to a factor, and is responsible for large gradients of the given function. This decomposition is valid, in particular, for the solution of a singularly perturbed problem. To approximate the function, a formula is proposed that uses the Taylor series expansion of the function and is exact for the boundary layer component. Under certain restrictions on the boundary layer component, estimates of the error in the approximation of the function are obtained. These estimates do not depend on the boundary layer component. Cases of functions of one and two variables are considered.
Keywords: function of one or two variables with large gradients, boundary layer component, Taylor series approximation
Mots-clés : modification, error estimation. 
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A. I. Zadorin. Application of a Taylor series to approximate a function with large gradients. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1420-1429. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a59/

[1] G.I. Shishkin, Grid approximations of singularly perturbed elliptic and parabolic equations, Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 1992 | Zbl

[2] R.B. Kellogg, A. Tsan, “Analysis of some difference approximations for a singular perturbation problem without turning points”, Math. Comput., 32 (1978), 1025–1039 | DOI | MR | Zbl

[3] H.-G. Roos, M. Stynes, L. Tobiska, Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems, Springer, Berlin, 2008 | MR | Zbl

[4] V.P. Il'in, “Adaptive formulas of numerical differentiation of functions with large gradients”, Phys.: Conf. Ser., 2019, 042003 | DOI

[5] A.M. Il'in, “Differencing scheme for a differential equation with a small parameter affecting the highest derivative”, Math. Notes, 6:2 (1970), 596–602 | DOI | Zbl

[6] A.I. Zadorin, “Non-polynomial interpolation of functions with large gradients and its application”, Comput. Math. Math. Phys., 61:2 (2021), 167–176 | DOI | MR | Zbl

[7] M.K. Kadalbajoo, D. Kumar, “Fitted mesh B-spline collocation method for singularly perturbed differential-equations with small delay”, Appl. Math. Comput., 204:1 (2008), 90–98 | DOI | MR | Zbl

[8] R. Ranjan, H.S. Prasad, “A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts”, J. Appl. Math. Comput., 65:1-2 (2021), 403–427 | DOI | MR | Zbl

[9] N.S. Bakhvalov, “The optimization of methods of solving boundary value problems with a boundary layer”, USSR Comput. Math. Math. Phys., 9:4 (1969), 139–166 | DOI | MR | Zbl

[10] J.J.H. Miller, E. O'Riordan, G.I. Shishkin, Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions, World Scientific, Singapore, 1996 | DOI | MR | Zbl

[11] E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, 1980 | DOI | MR | Zbl