DOI
10.34229/KCA2522-9664.24.2.12
UDC 519.6
THE EQUIVALENCE OF THE FUNDAMENTAL SPLINE AND GREEN’S FUNCTION
IN THE CONSTRUCTION OF THE EXACT FINITE-DIMENSIONAL ANALOGUE
OF THE BOUNDARY-VALUE PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION
OF THE 4TH ORDER
Abstract. The author considers a problem with the main and natural boundary conditions on an interval. A new method for constructing an exact discrete analog of the problem is proposed. The method deals with the projection of the differential equation on local splines, formed by the fundamental system of solutions of the Cauchy problems for the homogeneous equation. A system of linear algebraic equations with a 5-diagonal matrix is obtained for the values of the exact solutions of the original problem at the points of a uniform grid. To implement an exact analog, we recommend using high-order accuracy schemes which are formed by partial sums of series in even powers of the grid step for solving the Cauchy problems.
Keywords: boundary-value problem, ordinary differential equation of the 4th order, Cauchy problem, Wronskian, local spline, superposition of solutions, exact discrete analog, system of linear algebraic equations, 5-diagonal matrix.
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