Abstract: We consider a quadrature method for the numerical solution of hypersingular integral equations on the class of functions that are unbounded at the ends of the integration interval. For a hypersingular integral with a weight function \( p (x) = 1/\sqrt{1-x^2} \), a quadrature formula of the interpolation type is constructed using the zeros of the Chebyshev orthogonal polynomial of the first kind. For a regular integral, the quadrature formula of the highest degree of accuracy is also used with the weight function \(p (x)\). After discretizing the hypersingular integral equation, the singularity parameter is given the values of the roots of the Chebyshev polynomial and, evaluating indeterminate forms when the values of the nodes coincide, a system of linear algebraic equations is obtained. But, as it turned out, the resulting system is incorrect, that is, it does not have a unique solution, there is no convergence. Due to certain additional conditions, the system turns out to be correct. This is proved on numerous test cases, in which the errors of computations are also sufficiently small. On the basis of the considered test problems, we conclude that the constructed computing scheme is convenient for implementation and effective for solving hypersingular integral equations on the class of functions of the integration interval unbound at the ends.

For citation: Khubezhty, Sh. S. On Numerical Solution of Hypersingular Integral Equations of the First Kind, Vladikavkaz Math. J., 2020, vol. 22, no. 1, pp. 85-92 (in Russian). DOI 10.23671/VNC.2020.1.57607

1. Gabdulkhaev, B. G. and Sharipov, R. N. Optimization of Quadrature Formulas for Singular Cauchy and Hadamard Integrals, Konstructivnaya Teoriya Funktsii i Funktsional'nyi Analiz [Constructive Theory of Functions and Functional Analysis], Kazan', Kazan. Gos. Univ., 1987, issue 6, pp. 3-48 (in Russian).

2. Vaynikko, G. M., Lifanov, I. K. and Poltavskiy, I. N.Chislennye Metody v Gipersinguljarnyh Integral'nyh Uravnenijah i ih Prilozhenija [Numerical Methods in Hypersingular Equations and their Applications], Moscow, Yanus-K, 2001, 508 p. (in Russian).

3. Boykov, I. V., Boykov, A. I. and Semov, M. A. Approximate Solution of Nonlinear Hypersingular Integral Equations of First Kind, Izvestiya Vysshih Uchebnyh Zavedenii. Povolzhskii Region. Fiziko-Matematicheskie Nauki. Matematika [University Proceedings. Volga Region. Physical and Mathematical Sciences. Mathematics], 2015, no. 3(35), pp. 11-27 (in Russian).

4. Boykov, I. V. and Boykova, A. I. Approximate Solution of Some Type of Hypersingular Integral Equations, XIII Mezhdunar. Konf. "Differencial'nye Uravneniya i ikh Prilozheniya v Matematicheskom Modelirovanii" (Saransk, 12-16 Ijulja 2017) [XIII International Scientific Conference "Differential Equations and their Applications in Mathematical Modeling" (Saransk, July 12-16, 2017)], Saransk, 2017, pp. 446-461 (in Russian).

5. Khubezhty, Sh. S. and Plieva, L. Yu. O Kvadraturnykh Formulakh dlya Gipersingulyarnykh Integralov na Otrezke Integrirovaniya, Analiticheskie i Chislennye Metody Modelirovaniya Estestvenno-Nauchnykh i Sotsial'nykh Problem: Sbornik Statey IX Mezhdunarodnoy Nauchno-Tekhnicheskoy Konferentsii (28-31 Oktyabrya 2014) [Analytical and Numerical Methods of Modelling of Natural Science and Social Problems: Proceedings of the Ninth International Conference ANM-2014(28-31 October, 2014)], Penza, 2014, pp. 54-59 (in Russian).

6. Krylov, V. I. Priblizhennoe Vychislenie Integralov [Approximate Calculation of Integrals], Moscow, Nauka, 1967, 500 p. (in Russian).

7. Khubezhty, Sh. S. O Chislennom Reshenii Gipersingulyarnykh Integral'nykh Uravneniy I Roda s Yadrom Adamara, Matematicheskoe i Êomp'yuternoe Ìodelirovanie Estestvenno-Nauchnykh i Sotsial'nykh Problem: Materialy X Mezhdunarodnoy Nauchno-Tekhnicheskoy Konferentsii Molodykh Spetsialistov, Aspirantov i Studentov (23-27 Maya 2016) [Mathematical and Computer Modelling of Natural Science and Social Problems: Proceedings of the Tenth International Conference MCM-2016 (23-27 May, 2016)], Penza, PGU, 2016, pp. 83-92 (in Russian).

8. Kantorovich, L .V. and Akilov, G. P. Funkcionalnyj analiz [Functional Analysis], Moscow, Nauka, 1977, 720 p. (in Russian).