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DISCRETE APPROXIMATIONS OF DIFFERENTIAL EQUATIONS VIA TRIGONOMETRIC INTERPOLATION OKSANA BIHUN1, AUSTIN BREN2, MICHAEL DYRUD3, AND KRISTIN HEYSSE4 Concordia College, 901 8th Street South, Moorhead, MN 56560, USA Abstract. To approximate solutions of a linear di erential equation, we project, via trigonometric interpolation, its solution space onto a nitedimensional space of trigonometric polynomials and construct a matrix representation of the dif ferential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear di erential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems. PACS 00.02 1. Introduction In this paper, we use trigonometric interpolation to approximate solutions of a di erential equation Au = f, whose di erential operator A with domain D(A) is a formal polynomial of operators 1; x; d dx , and f 2 Range(A). A solution u is projected onto the space T L n of Lperiodic trigonometric polynomials of degree n. The projection T[u], de ned as a trigonometric interpolant of u, is identi ed with a vector ^u of its values at N = 2n + 1 partition points via an isomorphism : T L n ! RN, where n 2 N. The operator A is represented by an N N matrix A de ned implicitly by A^v = T [AT[v]] for all v 2 D(A). The original equation is approximated by a system of linear equations A^u = ^f , where ^f = T[f]. Date: September 14, 2011. 1Corresponding author's email: obihun@cord.edu. 2Email: asbren@cord.edu. 3Email: mdyrud@cord.edu. 4Email: keheysse@cord.edu. 1
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Transcript  DISCRETE APPROXIMATIONS OF DIFFERENTIAL EQUATIONS VIA TRIGONOMETRIC INTERPOLATION OKSANA BIHUN1, AUSTIN BREN2, MICHAEL DYRUD3, AND KRISTIN HEYSSE4 Concordia College, 901 8th Street South, Moorhead, MN 56560, USA Abstract. To approximate solutions of a linear di erential equation, we project, via trigonometric interpolation, its solution space onto a nitedimensional space of trigonometric polynomials and construct a matrix representation of the dif ferential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear di erential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems. PACS 00.02 1. Introduction In this paper, we use trigonometric interpolation to approximate solutions of a di erential equation Au = f, whose di erential operator A with domain D(A) is a formal polynomial of operators 1; x; d dx , and f 2 Range(A). A solution u is projected onto the space T L n of Lperiodic trigonometric polynomials of degree n. The projection T[u], de ned as a trigonometric interpolant of u, is identi ed with a vector ^u of its values at N = 2n + 1 partition points via an isomorphism : T L n ! RN, where n 2 N. The operator A is represented by an N N matrix A de ned implicitly by A^v = T [AT[v]] for all v 2 D(A). The original equation is approximated by a system of linear equations A^u = ^f , where ^f = T[f]. Date: September 14, 2011. 1Corresponding author's email: obihun@cord.edu. 2Email: asbren@cord.edu. 3Email: mdyrud@cord.edu. 4Email: keheysse@cord.edu. 1 