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On the Finding the Other Eigenvalues and Eigen Functions and Ortogonal Basis with a Nonlocal Parity Condition of the Third Kind

Received: 22 September 2015    Accepted: 11 October 2015    Published: 24 October 2015
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Abstract

In the present paper, we find out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition, the completeness and the basis property in the elliptic part of the third kind of a domain in L2(0, π/2). We also consider a new boundaries condition and analyze the orthogonal basis of the eigenfunctions depending on parameters of the problem.

DOI 10.11648/j.pamj.20150406.16
Published in Pure and Applied Mathematics Journal (Volume 4, Issue 6, December 2015)
Page(s) 259-263
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Frankl Problem, Lebesgue Integral, Holder Inequality, Bessel Equation

References
[1] N. Abbasi, Basis property and completeness of the eigenfunctions of the Frankl problem. (Russian) Dokl. Akad.Nauk425 (2009), no. 3, 295-298; translation in Dokl. Math. 79 (2009), no. 2, 193-196.
[2] A. V. Bitsadze, Nekotoryeklassyuravneni v chastnykhproizvodnykh. (Russian) [Some classes of partial differential equations] “Nauka”, Moscow, 1981. 448pp.
[3] F. Frankl, On the problems of Chaplygin for mixed sub- and supersonic flows. (Russian) Bull. Acad. Sci. URSS. Sr. Math. [Izvestia Akad. Nauk SSSR] 9, (1945).121-143.
[4] H.van Haeringen, L.P. Kok, Higher transcendental functions, Vol. II [McGraw-Hill, New York, 1953; MR 15, 419] by A. Erdlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Math. Comp. 41 (1983), no. 164, 778.
[5] E.I. Moiseev, The basis property for systems of sines and cosines. (Russian) Dokl. Akad. Nauk SSSR 275 (1984), no.4, 794–798.
[6] N. Abbasi, E.I. Moiseev, Basis property of eigenfunctins of the Generalized Gasedynamic problem of Frankl with a nonlocal oddness conditions, integral transforms and special Functions.(England) 21 (2010), no. 4,286-294.
[7] E. I. Moiseev, N. Abbasi, The basis property of an eigenfunction of the Frankl problem with a nonlocal parity conditionin the space Sobolev (W1p (0,π)). Integral Transforms Spec. Funct. (England) 22 (2011), no. 6, 415–421.
[8] E. I. Moiseev, On the solution of the Frankl problem in a special domain, Differ. Uravn. 8(4) (1992), pp.721-723.
[9] E. I. Moiseev and N. Abbasi, Basis property of eigen functions of the generalized problem of Frankl with a nonlocal parity condition and with the discontinuity of the gradient of solution , Differ.Uravn.44(10)(2009),pp.155 -160.
[10] K. B. Sabitov, About spectrum of the gasedynamic problem of Frankl for the model equation of mixed type, Differ. Uravn. 39(11) (2003), pp.1568-1570.
[11] K. B. Sabitov and V. V. Tikhomirov, About construction of the gasedynamic problem of Frankl, Differ.Uravn.23 (10) (1990), pp.100-109.
[12] M.M. Smirnov, Uravneniyasmeshannogotipa. (Russian) [Equations of mixed type] Izdat. “Nauka”, Moscow 1970 295pp. New York, 1955.vii+329. pp.
[13] A. Zygmund, Trigonometrical series. Dover Publications, New York, 1955.vii+329. pp.
Author Information
  • Department of Mathematics, Lorestan University, Khoramabad, Iran

  • Department of Mathematics, Islamic Azad University, Ashtian Branch, Ashtian, Iran

  • Department of Mathematics, Lorestan University, Khoramabad, Iran

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  • APA Style

    Naser Abbasi, Hamid Mottaghi Golshan, Mahmood Shakori. (2015). On the Finding the Other Eigenvalues and Eigen Functions and Ortogonal Basis with a Nonlocal Parity Condition of the Third Kind. Pure and Applied Mathematics Journal, 4(6), 259-263. https://doi.org/10.11648/j.pamj.20150406.16

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    ACS Style

    Naser Abbasi; Hamid Mottaghi Golshan; Mahmood Shakori. On the Finding the Other Eigenvalues and Eigen Functions and Ortogonal Basis with a Nonlocal Parity Condition of the Third Kind. Pure Appl. Math. J. 2015, 4(6), 259-263. doi: 10.11648/j.pamj.20150406.16

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    AMA Style

    Naser Abbasi, Hamid Mottaghi Golshan, Mahmood Shakori. On the Finding the Other Eigenvalues and Eigen Functions and Ortogonal Basis with a Nonlocal Parity Condition of the Third Kind. Pure Appl Math J. 2015;4(6):259-263. doi: 10.11648/j.pamj.20150406.16

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  • @article{10.11648/j.pamj.20150406.16,
      author = {Naser Abbasi and Hamid Mottaghi Golshan and Mahmood Shakori},
      title = {On the Finding the Other Eigenvalues and Eigen Functions and Ortogonal Basis with a Nonlocal Parity Condition of the Third Kind},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {6},
      pages = {259-263},
      doi = {10.11648/j.pamj.20150406.16},
      url = {https://doi.org/10.11648/j.pamj.20150406.16},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20150406.16},
      abstract = {In the present paper, we find out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition, the completeness and the basis property in the elliptic part of the third kind of a domain in L2(0, π/2). We also consider a new boundaries condition and analyze the orthogonal basis of the eigenfunctions depending on parameters of the problem.},
     year = {2015}
    }
    

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    AB  - In the present paper, we find out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition, the completeness and the basis property in the elliptic part of the third kind of a domain in L2(0, π/2). We also consider a new boundaries condition and analyze the orthogonal basis of the eigenfunctions depending on parameters of the problem.
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