A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation
DOI:
https://doi.org/10.71310/pcam.2_64.2025.07Keywords:
Chebyshev polynomials, discrete variant, pre-integration, high accuracy, small parameterAbstract
The article presents a numerical modeling approach for solving a non-homogeneous singularly perturbed fourth-order equation using by a discrete variant of the pre integration method. In the proposed method, the basis functions are Chebyshev poly nomials of the first kind. The highest derivative of the differential equation is expanded into a finite series in terms of these polynomials, with unknown expansion coefficients. All lower derivatives and the solution of the differential equation are represented in the form of the chosen expansion. The resulting series are substituted into the differential equation, leading to a system of discrete equations. For Chebyshev polynomials, there is a discrete integration formula that reduces the order of the derivative. By applying this formula, the discrete system is integrated four times, resulting in a system of algebraic equations where the number of equations is less than the number of unknown coefficients. The missing equations are obtained from the boundary conditions of the problem. The integration operation improves the smoothness of the approximating polynomials. For example, a zeroth-order polynomial, after four integrations, transforms into a fourth order polynomial. Numerical calculations demonstrate the high accuracy and efficiency of the proposed method, especially for very small values of the parameter e.
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