Modified Method for the Solution of Dual Trigonometric Series Relations
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RESEARCH ARTICLE
Modified Method for the Solution of Dual Trigonometric Series Relations A. Choudhary1 • S. C. Martha1
•
A. Chakrabarti2
Received: 27 February 2015 / Revised: 4 February 2019 / Accepted: 9 February 2019 Ó The National Academy of Sciences, India 2019
Abstract Two basic dual trigonometric series relations involving a countable infinite number of unknowns are considered for the determination of the unknowns. The numerical values of the unknowns are determined with the help of the methods of algebraic least-squares approximation and singular value decomposition. The dual trigonometric series and the corresponding functions are compared with the existing results. The errors are also computed to show the efficiency of these methods. The study indicates that the method of algebraic least squares is more straightforward, simpler and computationally more efficient as compared to the available methods. Keywords Dual trigonometric series Overdetermined system Algebraic least-squares approximation Singular value decomposition method Mathematics Subject Classification 65F20 93E24
& S. C. Martha [email protected] A. Choudhary [email protected] A. Chakrabarti [email protected] 1
Department of Mathematics, Indian Institute of Technology Ropar, Punjab 140001, India
2
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
1 Introduction The dual series relations: 1 X an np sin nx ¼ f1 ðxÞ; 0\x\c
ð1Þ
n¼1
and 1 X
an sin nx ¼ f2 ðxÞ;
c\x\p
ð2Þ
n¼1
occur in a natural way, while handling mixed boundary value problems of mathematical physics [1], and these dual series relations are considered for the solution of the unknown constants involved, where p ¼ 1; 0\c\p, f1 ðxÞ and f2 ðxÞ are two prescribed functions and the coefficients an 0 s; n ¼ 1; 2; . . .; are to be determined. The dual series problem was originally solved by Tranter [2, 3], but the difficulty in his solution lies in the need to find a certain constant by summing a series. Noble [4] solved the problem involving dual series relations by using the method of multiplying factors. Tranter [5] presented an improved method to solve the dual trigonometrical series relations, but there was also a need to find a constant. Srivastav [6] studied the closed-form solution of dual trigonometrical series relations and avoided the difficulty of Tranter [5]. Noble and Whiteman [7] generalized the method of Noble [4]. They used the several properties of Legendre polynomials and obtained the solution of dual trigonometrical series relations in integral form which contains some weakly singular integrals. These singular integrals are further solved in this paper with the help of the Gaussian quadrature rule, and the numerical values of the unknowns are obtained. Kelman and Koper [8] applied the method of analytical least squares. Such analytical method
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A. Choudhary et al.
and the method using Legendre polynomials are not desirable due to the requirement of extra computation in the evaluation of several int
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