Skip to main content
SpringerLink
Log in

Matrix Laplace transform

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

The article develops operational calculus based on a differential operator with piecewise constant matrix coefficients. The core of a matrix Laplace transformation is the exhibitor of a matrix argument of \(e^{ - Apt }. \) Difference from a Laplace transformation consists that the matrix Laplace transformation affects a vector function; the image is a vector function also. The definition of the Laplace matrix transformation with a division point is given, and its properties and applications to systems of differential equations with piecewise constant coefficients are studied. The Mellin inversion formula for the matrix integral Laplace transform is proved. Significant differences from the scalar case are established; for example, the theorem on the shift of the image argument is valid under the assumption of permutation of matrices. The technique of applying the matrix Laplace transform for solving systems of differential equations and higher order differential equations with piecewise constant coefficients is developed. A method for solving differential equations and systems with piecewise constant coefficients using matrix Laplace transform has been developed. The solution of the vector analog of the problem for the thermal field distribution of a semi-infinite rod is found.

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 statement

Not applicable.

Code Availability

Not applicable.

References

  1. Arman, A.: New trends in Laplace type integral transforms with applications. Boletim da Sociedade Paranaense de Matematica 35(1), 173–193 (2017)

    Article  MathSciNet  Google Scholar 

  2. Ganzha, E.I.: On Laplace and Dini transformations for multidimensional equations with a decomposable principal symbol. Progr. Comput. Softw. 38, 150–155 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fahd, Jarad, Abdeljawad, T.: A modified Laplace transform for certain generalized fractional operators. Results Nonlinear Anal. 1(2), 88–98 (2018)

    Google Scholar 

  4. Wolfram, Koepf, Insuk, Kim, Rathie Arjun, K.: On a new class of laplace-type integrals involving generalized hypergeometric functions. Axioms 8(3), 87 (2019)

    Article  MATH  Google Scholar 

  5. Tsarev, S.P.: Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations, pp. 325–331. Proc. ISSAC, ACM Press (2005)

  6. Zaikina, S.M.: On a new class of laplace-type integrals involving generalized hypergeometric functions. Axioms 8(3), 87 (2019)

    Article  Google Scholar 

  7. Sharma, V.D., Thakare, M.M.: Introduction of generalized Laplace-fractional Mellin transform. Int. J. Eng. Sci. Res. Technol. 5(9), 667–670 (2016)

    Google Scholar 

  8. Pinelas, S., Xavier, G.B.A., Kumar, S.U., Vasantha, M.M.: Laplace–Fibonacci transform by the solution of second order generalized difference equation. Nonauton. Dyn. Syst. 4(1), 22–30 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sharma, V.D., Thakare, M.M.: Generalized laplace–fractional Mellin transform and operators. Int. J. Pure Appl. Sci. Technol. 16(1), 20–25 (2013)

    Google Scholar 

  10. Higham, Nicholas J.: Newton’s method for the matrix square root. Math. Comput. 46(174), 237–549 (1986)

    MathSciNet  MATH  Google Scholar 

  11. Brychkov Yu, A., Prudnikov, A.P., Shishov, V.S.: Itogi Nauki i Tekhn. Ser. Mat. Anal., Operational calculus, Vol 16, VINITI, (1979)

  12. Golub, G.F., van der Vorst, A., Henk, A.D.: Eigenvalue computation in the 20th century. J. Comput. Appl. Math. 123(2), 35–36 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Moler, C., van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. Twenty-five years later. SIAM Rev. 45, 3–49 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carslaw, H., Jaeger, J.: Conduction of Heat in Solids, 2nd edn., p. 510. Oxford University Press, Oxford (1959)

    MATH  Google Scholar 

  15. Yaremko, O.E., Yaremko, N.N.: Generalized double laplace transform and its application for partial differential equations solving. Appl. Math. Phys. 52(4), 239–245 (2020). ([in Russian])

    Google Scholar 

  16. Baeumer, B.: On the inversion of the convolution and Laplace transform. Trans. Am. Math. Soc. 355(3), 1201–1212 (2002)

    Article  MathSciNet  Google Scholar 

  17. Yaremko, O.E., Yaremko, N.N.: Integrals and derivatives of fractional order based on Laplace type integral transformations with applications. Appl. Math. Phys. 53(2), 114–124 (2021). ([in Russian])

    MATH  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Moscow State University of Technology “STANKIN”, Vadkovsky lane, 3a, Moscow, Russia

    O. E. Yaremko & K. R. Zababurin

Authors
  1. O. E. Yaremko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. R. Zababurin
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Not applicable.

Corresponding author

Correspondence to K. R. Zababurin.

Ethics declarations

Conflict of interest

Not applicable.

Ethical approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Financial interests

This work was supported by the Ministry of Science and the Higher Education of the Russian Federation within project 07-2020-0034 non-financial interests.

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 (e.g. a society or other partner) 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

Yaremko, O.E., Zababurin, K.R. Matrix Laplace transform. Bol. Soc. Mat. Mex. 29, 86 (2023). https://doi.org/10.1007/s40590-023-00563-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40590-023-00563-7

Keywords

Mathematics Subject Classification

Search

Navigation