Skip to main content
SpringerLink
Log in

Existence of Analytic Solutions for Some Classes of Singular Integral Equations of Non-normal Type with Convolution Kernel

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this work, we deal with the existence and uniqueness of solutions for some classes of singular convolution equations. Firstly, we establish Noethericity theory of solvability, and generalize Sokhotski–Plemelj formula of Fourier transform. In addition, by means of the theory of integral transforms, the analytical methods of boundary value problems, and the principle of analytic continuation, we transform such equations into boundary value problems for analytical functions, and the analytic solutions and conditions of Noether solvability are obtained in the case of non-normal type. Finally, we discuss the asymptotic property of solutions for the equations at nodal points. Compared with Cauchy integral equations, the singular convolution integral equations can more accurately describe the actual physical and engineering problems. Therefore, studying its complex boundary value analytic method is a hot topic, which has important scientific significance and application prospects.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Our manuscript has no associated data.

References

  1. Gahov, F.D., Cherskiy, U.I.: Integral Equations of Convolution Type. NauKa, Moscow (1980)

    Google Scholar 

  2. Duduchava, R.V.: Integral equations of convolution type with discontinuous coefficients. Math. Nachr. 79, 75–78 (1977)

    MathSciNet  Google Scholar 

  3. Muskhelishvilli, N.I.: Singular Integral Equations. NauKa, Moscow (2002)

    Google Scholar 

  4. Litvinchuk, G.S.: Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer Academic, London (2004)

    Google Scholar 

  5. Lu, J.K.: Boundary Value Problems for Analytic Functions. World Sci, Singapore (2004)

    Google Scholar 

  6. Duduchava, R.V.: Wiener-Hopf integral operators. Math. Nachr. 65, 59–82 (1975)

    MathSciNet  MATH  Google Scholar 

  7. Li, P.R.: Generalized convolution-type singular integral equations. Appl. Math. Comput. 311, 314–323 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Li, P.R., Ren, G.B.: Solvability of singular integro-differential equations via Riemann-Hilbert problem. J. Differ. Equ. 265, 5455–5471 (2018)

    Article  MathSciNet  Google Scholar 

  9. Li, P.R.: Non-normal type singular integral-differential equations by Riemann-Hilbert approach. J. Math. Anal. Appl. 483(2), 123643 (2020)

    Article  MathSciNet  Google Scholar 

  10. Li, P.R.: Solvability theory of convolution singular integral equations via Riemann-Hilbert approach. J. Comput. Appl. Math. 370(2), 112601 (2020)

    Article  MathSciNet  Google Scholar 

  11. Li, P.R.: Two classes of linear equations of discrete convolution type with harmonic singular operators. Complex Var. Elliptic Equ. 61(1), 67–73 (2016)

    Article  MathSciNet  Google Scholar 

  12. Li, P.R.: On solvability of singular integral-differential equations with convolution. J. Appl. Anal. Comput. 9(3), 1071–1082 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Li, P.R.: Singular integral equations of convolution type with reflection and translation shifts. Numer. Funct. Anal. Optim. 40(9), 1023–1038 (2019)

    Article  MathSciNet  Google Scholar 

  14. Li, P.R., Ren, G.B.: Some classes of equations of discrete type with harmonic singular operator and convolution. Appl. Math. Comput. 284, 185–194 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Li, P.R.: Generalized boundary value problems for analytic functions with convolutions and its applications. Math. Methods Appl. Sci. 42, 2631–2643 (2019)

    Article  MathSciNet  Google Scholar 

  16. Li, P.R.: Singular integral equations of convolution type with Cauchy kernel in the class of exponentially increasing functions. Appl. Math. Comput. 344–345, 116–127 (2019)

    Article  MathSciNet  Google Scholar 

  17. Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Springer, Berlin (2003). Reprint of the second (1990) edition

    Book  Google Scholar 

  18. Du, H., Shen, J.H.: Reproducing kernel method of solving singular integral equation with cosecant kernel. J. Math. Anal. Appl. 348(1), 308–314 (2008)

    Article  MathSciNet  Google Scholar 

  19. De-Bonis, M.C., Laurita, C.: Numerical solution of systems of Cauchy singular integral equations with constant coefficients. Appl. Math. Comput. 219, 1391–1410 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Begehr, H., Vaitekhovich, T.: Harmonic boundary value problems in half disc and half ring. Funct. Approx. 40(2), 251–282 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Nakazi, T., Yamamoto, T.: Normal singular integral operators with Cauchy kernel. Integral Equ. Oper. Theory 78, 233–248 (2014)

    Article  MathSciNet  Google Scholar 

  22. Chuan, L.H., Mau, N.V., Tuan, N.M.: On a class of singular integral equations with the linear fractional Carleman shift and the degenerate kernel. Complex Var. Elliptic Equ. 53(2), 117–137 (2008)

    Article  MathSciNet  Google Scholar 

  23. Praha, E.K., Valencia, V.M.: Solving singular convolution equations using inverse Fast Fourier Transform. Appl. Math. 57(5), 543–550 (2012)

    Article  MathSciNet  Google Scholar 

  24. Tuan, N.M., Thu-Huyen, N.T.: The solvability and explicit solutions of two integral equations via generalized convolutions. J. Math. Anal. Appl. 369, 712–718 (2010)

    Article  MathSciNet  Google Scholar 

  25. Li, P.R.: The solvability and explicit solutions of singular integral-differential equations of non-normal type via Riemann-Hilbert problem. J. Comput. Appl. Math. 374(2), 112759 (2020)

    Article  MathSciNet  Google Scholar 

  26. Wöjcik, P., Sheshko etc, M.A.: Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel. Differ. Equ. 49(2), 198–209 (2013)

    Article  MathSciNet  Google Scholar 

  27. Gomez, C., Prado, H., Trofimchuk, S.: Separation dichotomy and wavefronts for a nonlinear convolution equation. J. Math. Anal. Appl. 420, 1–19 (2014)

    Article  MathSciNet  Google Scholar 

  28. Gong, Y.F., Leong, L.T., Qiao, T.: Two integral operators in Clifford analysis. J. Math. Anal. Appl. 354, 435–444 (2009)

    Article  MathSciNet  Google Scholar 

  29. Ren, G.B., Kaehler, U., Shi, J.H., Liu, C.W.: Hardy-Littlewood inequalities for fractional derivatives of invariant harmonic functions. Complex Anal. Oper. Theory 6(2), 373–396 (2012)

    Article  MathSciNet  Google Scholar 

  30. Colliander, J., Keel, M., Staffilani etc, G.: Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation. Invent. Math. 181(1), 39–113 (2010)

    Article  MathSciNet  Google Scholar 

  31. Abreu-Blaya, R., Bory-Reyes, J., Brackx, F., De-Schepper, H., Sommen, F.: Cauchy integral formulae in Hermitian Quaternionic Clifford Analysis. Complex Anal. Oper. Theory 6, 971–983 (2012)

    Article  MathSciNet  Google Scholar 

  32. Blocki, Z.: Suita conjecture and Ohsawa-Takegoshi extension theorem. Invent. Math. 193, 149–158 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The authors are very grateful to the anonymous referees for their valuable suggestions and comments, which helped to improve the quality of the paper.

Funding

This work is supported financially by the National Natural Science Foundation of China (11971015).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Pingrun Li

Authors
  1. Pingrun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pingrun Li.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, P. Existence of Analytic Solutions for Some Classes of Singular Integral Equations of Non-normal Type with Convolution Kernel. Acta Appl Math 181, 5 (2022). https://doi.org/10.1007/s10440-022-00522-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-022-00522-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation