Skip to main content
SpringerLink
Log in

Numerical Solution of the Fredholm Integral Equations of the First Kind by Using Multi-projection Methods

  • Conference paper
  • First Online:
Mathematics and Computing (ICMC 2022)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 415))

Included in the following conference series:

  • 314 Accesses

Abstract

The Fredholm integral equations (fies) of the first kind have been solved by Legendre spectral multi-projection methods by using Tikhonov regularized methods. The theoretical analysis utilizing this method under a priori parameter selection strategy has been explained and the best convergence rates obtained in \(L^2\)-norm. Next, in order to discover an appropriate regularization parameter, Arcangeli’s discrepancy principle has been applied and the order of convergence has been deduced. Numerical example has been furnished which validates our theoretical findings.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994)

    Book  MATH  Google Scholar 

  2. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  3. Chen, Z., Cheng, S., Nelakanti, G., Yang, H.: A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization. Int. J. Comput. Math. 87(3), 565–582 (2010). https://doi.org/10.1080/00207160802155302

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, Z., Xu, Y., Yang, H.: Fast collocation methods for solving ill-posed integral equations of the first kind. Inverse Probl. 24(6), 1–21 (2008). https://doi.org/10.1088/0266-5611/24/6/065007

    Article  MathSciNet  MATH  Google Scholar 

  5. Engl, H.W.: Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates. J. Optim. Theory Appl. 52(2), 209–215 (1987). https://doi.org/10.1007/BF00941281

    Article  MathSciNet  MATH  Google Scholar 

  6. Groetsch, C.W.: Convergence analysis of a regularized degenerate kernel method for Fredholm integral equation of the first kind. Integral Equ. Oper. Theory 13(1), 67–75 (1990). https://doi.org/10.1007/BF01195293

    Article  MathSciNet  MATH  Google Scholar 

  7. Kirsch, A.: Introduction to the Mathematical Theory of Inverse Problems. Springer, New York (1996)

    Book  MATH  Google Scholar 

  8. Maleknejad, K., Mollapourasl, R., Nouri, K., Alizadeh, M.: Convergence of numerical solution of Fredholm integral equation of the first kind with degenerate kernel. Appl. Math. Comput. 181(2), 1000–1007 (2006). https://doi.org/10.1016/j.amc.2006.01.074

    Article  MathSciNet  MATH  Google Scholar 

  9. Nair, M.T.: Linear Operator Equations: Approximation and Regularization. World Scientific, Singapore (2009)

    Book  MATH  Google Scholar 

  10. Neggal, B., Boussetila, N., Rebbani, F.: Projected Tikhonov Regularization method for Fredholm integral equations of the first kind. J. Inequal. Appl. 195, 1–21 (2016). https://doi.org/10.1186/s13660-016-1137-6

    Article  MathSciNet  MATH  Google Scholar 

  11. Patel, S., Panigrahi, B.L., Nelakanti, G.: Legendre spectral projection methods for Fredholm integral equations of first kind. J. Inverse Ill-Posed Probl. 30(5), 677–691 (2022). https://doi.org/10.1515/jiip-2020-0104

    Article  MathSciNet  MATH  Google Scholar 

  12. Patel, S., Panigrahi, B.L., Nelakanti, G.: Legendre spectral multi-projection methods for Fredholm integral equations of the first kind. Adv. Oper. Theory 7(51) (2022). https://doi.org/10.1007/s43036-022-00215-z

  13. Rajan, M.P.: A modified convergence analysis for solving Fredholm integral equations of the first kind. Integral Equ. Oper. Theory 49, 511–516 (2004). https://doi.org/10.1007/s00020-002-1213-9

    Article  MathSciNet  MATH  Google Scholar 

  14. Rostami, Y., Maleknejad, K.: Solving Fredholm integral equations of the first kind by using wavelet bases. Hacet. J. Math. Stat. 48, 1–15 (2019). https://doi.org/10.15672/hujms.553433

    Article  MathSciNet  MATH  Google Scholar 

  15. Tahar, B., Nadjib, B., Faouzia, R.: A variant of projection-regularization method for ill-posed linear operator equations. Int. J. Comput. Methods 18(4) (2021). https://doi.org/10.1142/S0219876221500080

Download references

Acknowledgements

The first author was supported by the INSPIRE Fellowship, Department of Science and Technology, Government of India, New Delhi.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sambalpur University, Burla, 768019, Odisha, India

    Subhashree Patel

  2. Department of Mathematics, Gangadhar Meher University, Sambalpur, 768004, Odisha, India

    Bijaya Laxmi Panigrahi

  3. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Gnaneshwar Nelakanti

Authors
  1. Subhashree Patel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bijaya Laxmi Panigrahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gnaneshwar Nelakanti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Subhashree Patel .

Editor information

Editors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India

    B. Rushi Kumar

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India

    S. Ponnusamy

  3. Department of Information Technology, Maulana Abul Kalam Azad University of Technology, Haringhata, West Bengal, India

    Debasis Giri

  4. Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA

    Bhavani Thuraisingham

  5. Department of Computer Science, Purdue University, West Lafayette, IN, USA

    Christopher W. Clifton

  6. Department of Theoretical and Applied Sciences, University of Insubria, Varese, Italy

    Barbara Carminati

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Patel, S., Panigrahi, B.L., Nelakanti, G. (2022). Numerical Solution of the Fredholm Integral Equations of the First Kind by Using Multi-projection Methods. In: Rushi Kumar, B., Ponnusamy, S., Giri, D., Thuraisingham, B., Clifton, C.W., Carminati, B. (eds) Mathematics and Computing. ICMC 2022. Springer Proceedings in Mathematics & Statistics, vol 415. Springer, Singapore. https://doi.org/10.1007/978-981-19-9307-7_50

Download citation

Publish with us

Policies and ethics

Search

Navigation