Semisalov B.V.   Kuzmin G.A.  

Modification of Fourier Approximation for Solving Boundary Value Problems Having Singularities of Boundary Layer Type

Reporter: Semisalov B.V.

A method for searching the solutions to the boundary value problems having singularities of boundary-layer type is developed on the basis of mapping of Fourier series domain to the segment [-1,1] that leads to non polynomial basis.High rate of convergence and stability of the proposed method is justified theoretically for four types of coordinate mappings, the dependencies of approximation error on the values of derivatives of approximated function are obtained, the algorithms of expansion of functions into the series with basis consists of Chebyshev polynomials and proposed functions are developed and implemented. It was shown that for functions having high order of smoothness and extremely steep gradients in the vicinity of boundary the accuracy of proposed method cardinally exceeds the accuracy of Chebyshev’s approximation. Moreover for such functions method allows to reach an acceptable accuracy using only N=10 basis elements (relative error does not exceed 1%).