TY - JOUR
T1 - Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer
AU - Zadorin , Alexander
AU - Tikhovskaya , Svetlana
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 590
EP - 608
PY - 2019
DA - 2019/02
SN - 16
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/13016.html
KW - Function of one variable, exponential boundary layer, formulas of numerical differentiation, an error estimate.
AB - <p style="text-align: justify;">It is known that the solution of a singularly perturbed problem corresponds to the
function with large gradients in a boundary layer. The application of Lagrange polynomial on a
uniform mesh to interpolate such functions leads to large errors. To achieve the error estimates
uniform with respect to a small parameter, we can use either a polynomial interpolation on a
mesh which condenses in a boundary layer or we can use special interpolation formulas which are
exact on a boundary layer component of the interpolating function. In this paper, we construct
and study the formulas of numerical differentiation based on the interpolation formulas which
are exact on a boundary layer component. We obtained the error estimates which are uniform
with respect to a small parameter. Some numerical results validating the theoretical estimates
are discussed.</p>