Approximation of the solution of the Cauchy problem for nonlinear ODE systems by means of Fourier series in functions orthogonal in the sense of Sobolev

I. I. Sharapudinov

Daghestan Electronic Mathematical Reports, no. 7 (2017), pp. 66-76

Résumé

Consider the systems of functions ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ orthonormal with respect to a Sobolev-type inner product of the form $\langle f,g\rangle= \sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b}f^{(r)}(x)g^{(r)}\rho(x)(x)dx$ generated by a given orthonormal system of functions ${\varphi}_{n}(x)$ $( n=0,1,\ldots)$. It is shown that the Fourier series in the system ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ and their partial sums are a convenient and very effective tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).

Détail

Keywords: the Cauchy problem, Fourier series, Sobolev orthogonal functions.

Bibliographie

[1] Sharapudinov I.I., “Ortogonalnye po Sobolevu sistemy, porozhdennye ortogonalnymi funktsiyami”, Izv. RAN. Ser. Matematicheskaya, 82 (2018) (to appear) | Zbl

[2] Sharapudinov I.I., “Sistemy funktsii, ortogonalnye po Sobolevu, porozhdennye ortogonalnymi funktsiyami”, Materialy 18-i mezhdunarodnoi Saratovskoi zimnei shkoly В«Sovremennye problemy teorii funktsii i ikh prilozheniyaВ», OOO В«Izdatelstvo В«Nauchnaya knigaВ», Saratov, 2016, 329–332

[3] Iserles A., Koch P.E., Norsett S.P. and Sanz-Serna J.M., “On polynomials orthogonal with respect to certain Sobolev inner products”, J. Approx. Theory, 65 (1991), 151–175 | DOI | MR | Zbl

[4] Marcellan F., Alfaro M. and Rezola M.L., “Orthogonal polynomials on Sobolev spaces: old and new directions”, Journal of Computational and Applied Mathematics, 48 (1993), 113–131., North-Holland | DOI | MR | Zbl

[5] Meijer H.G., “Laguerre polynimials generalized to a certain discrete Sobolev inner product space”, J. Approx. Theory, 73 (1993), 1–16 | DOI | MR | Zbl

[6] Kwon K.H. and Littlejohn L.L., “The orthogonality of the Laguerre polynomials $\{L_n^{(-k)}(x)\}$ for positive integers $k$”, Ann. Numer. Anal., 1995, no. 2, 289–-303 | MR | Zbl

[7] Lopez G., Marcellan F., Vanassche W., “Relative Asymptotics for Polynomials Orthogonal with Respect to a Discrete Sobolev Inner-Product”, Constr. Approx., 11:1 (1995), 107–137 | DOI | MR | Zbl

[8] Kwon K.H. and Littlejohn L.L., “Sobolev orthogonal polynomials and second-order differential equations”, Rocky Mountain J. Math., 28 (1998), 547–594 | DOI | MR | Zbl

[9] Marcellan F. and Yuan Xu, “On Sobolev orthogonal polynomials”, Expositiones Mathematicae, 33:3 (2015), 308–352 | DOI | MR | Zbl

[10] Sharapudinov I.I., “Smeshannye ryady po ultrasfericheskim polinomam i ikh approksimativnye svoistva”, Matematicheskii sbornik, 194:3 (2003), 115–148 | DOI | MR | Zbl

[11] Sharapudinov I.I., Smeshannye ryady po ortogonalnym polinomam, Izdatelstvo Dagestanskogo nauchnogo tsentra, Makhachkala, 2004, 176 pp.

[12] Trefethen L.N., Spectral methods in Matlab, SIAM, Fhiladelphia, 2000 | MR | Zbl

[13] Trefethen L.N., Finite difference and spectral methods for ordinary and partial differential equation, Cornell University, 1996

[14] Solodovnikov V.V., Dmitriev A.N., Egupov N.D., Spektralnye metody rascheta i proektirovaniya sistem upravleniya, Mashinostroenie, Moskva, 1986 | MR

[15] Pashkovskii S., Vychislitelnye primeneniya mnogochlenov i ryadov Chebysheva, Nauka, Moskva, 1983 | MR

[16] Arushanyan O.B., Volchenskova N.I., Zaletkin S.F., “Primenenie ryadov Chebysheva dlya integrirovaniya obyknovennykh differentsialnykh uravnenii”, Sib. elektron. matem. izv., 2014, no. 11, 517–531 | MR | Zbl

[17] Lukomskii D.S., Terekhin P.A., “Primenenie sistemy Khaara k chislennomu resheniyu zadachi Koshi dlya lineinogo differentsialnogo uravneniya pervogo poryadka”, Materialy 18-i mezhdunarodnoi Saratovskoi zimnei shkoly В«Sovremennye problemy teorii funktsii i ikh prilozheniyaВ», OOO В«Izdatelstvo В«Nauchnaya knigaВ», Saratov, 2016, 171–173

[18] Magomed-Kasumov M.G., “Priblizhennoe reshenie obyknovennykh differentsialnykh uravnenii s ispolzovaniem smeshannykh ryadov po sisteme Khaara”, Materialy 18-i mezhdunarodnoi Saratovskoi zimnei shkoly В«Sovremennye problemy teorii funktsii i ikh prilozheniyaВ», OOO В«Izdatelstvo В«Nauchnaya knigaВ», Saratov, 2016, 176–178

[19] Sharapudinov I.I., Magomed-Kasumov M.G., “O predstavlenii resheniya zadachi Koshi ryadom Fure po polinomam, ortogonalnym po Sobolevu, porozhdennym mnogochlenami Lagerra”, Differentsialnye uravneniya, 2017 (to appear)

[20] Sharapudinov I.I., Sharapudinov T.I., “Smeshannye ryady po polinomam Yakobi i Chebysheva i ikh diskretizatsiya”, Matematicheskie zametki, 88:1 (2010), 116–147 | DOI | MR | Zbl

[21] Kashin B.S.,Saakyan A.A., Ortogonalnye ryady, AFTs, Moskva, 1999 | MR

[22] Sharapudinov I.I., Muratova G.N., “Nekotorye svoistva r-kratno integrirovannykh ryadov po sisteme Khaara”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:1 (2009), 68–76

[23] Faber G., “Ober die Orthogonalfunktionen des Herrn Haar”, Jahresber. Deutsch. Math. Verein., 19 (1910), 104–112 | Zbl

[24] Sharapudinov I.I., “Asimptoticheskie svoistva polinomov, ortogonalnykh po Sobolevu, porozhdennykh polinomami Yakobi”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 6, 1–24

[25] Sege G., Ortogonalnye mnogochleny, Fizmatgiz, Moskva, 1962