Incomplete trigonometric splines in the problems of constructing approximate solutions of the second order linear differential equations
DOI:
https://doi.org/10.17721/1812-5409.2026/1.15Keywords:
fundamental trigonometric splines, incomplete trigonometric splines, even fundamental splines, odd fundamental splines, boundary value problems, inear differential equationsAbstract
This study introduces a numerical analytical method for deriving approximate solutions to the first boundary value problem associated with second-order linear differential equations. The approach is founded upon the utilization of incomplete fundamental trigonometric splines, both even and odd, whose structural properties closely parallel those of classical polynomial splines and have been extensively investigated in prior works. The proposed methodology employs residual minimization through the established collocation technique, thereby reducing the original problem to the resolution of systems of linear algebraic equations with respect to the spline-defining parameters. A distinctive feature of this method lies in the fact that the values of the sought solution themselves serve as the determining parameters of the fundamental trigonometric splines. Boundary conditions are incorporated by prescribing specific values for the corresponding spline parameters. The theoretical exposition is supplemented with illustrative examples.
It should be noted that incomplete trigonometric splines are functions with a certain evenness. In our opinion, in many cases this property is convenient when setting boundary conditions for the desired solution. The given method assumes generalization. Thus, in particular, the minimization of the residual can be carried out by the method of discrete least squares and its variations. This method can also be applied to linear differential equations of arbitrary orders.
Pages of the article in the issue: 119 - 125
Language of the article: English
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