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Solving the Cauchy Problem for a Two-Dimensional Difference Equation at a Point Using Computer Algebra Methods

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Abstract

An algorithm for finding the solution to the Cauchy problem for a two-dimensional difference equation with constant coefficients at a point using computer algebra is described. In the one-dimensional case, solving the Cauchy problem is easy; however, already in the two-dimensional case the number of unknowns rapidly increases at each step. To automate the process of computing the solution to the Cauchy problem for a two-dimensional difference equation with constant coefficients at a given point, an algorithm in MATLAB is developed in which the input data are the matrix of coefficients obtained on the basis of the structure of the two-dimensional polynomial difference equation, coordinates of the points that specify the structure and the size of the matrix of initial data, and the matrix of the initial data. The algorithm produces the solution to the Cauchy problem for the given two-dimensional difference equation at the given point.

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REFERENCES

  1. Dudgeon, D.E. and Mersereau, R.M., Dadzhion, D. and Mersero, O., Multidimensional Digital Signal Processing, Englewood Cliffs: Prentice-Hall, 1984.

    Google Scholar 

  2. Isermann, R. Digital Control Systems, Berlin: Springer, 1981.

    Book  Google Scholar 

  3. Ryaben’kii, V.S., Introduction to Computational Mathematics: Textbook, 2nd ed., Moscow: Fizmatlit, 2000.

    MATH  Google Scholar 

  4. Stanley, R., Enumerative Combinatorics, Monterey, Calif. : Wadsworth & Brook, 1986.

    Book  Google Scholar 

  5. Stanley, R., Enumerative Combinatorics: Trees, Generating Functions, and Symmetric Functions, Cambridge: Cambridge University Press, 1999.

    Book  Google Scholar 

  6. Bousquet-Melou, M. and Petkovsek, M., Linear recurrences with constant coefficients: the multivariate case, Discrete Math., 2000, vol. 225, pp. 51–75.

    Article  MathSciNet  Google Scholar 

  7. Abramov, S.A., Gheffar, A., and Khmelnov, D.E., Geffar, A., Rational solutions of linear difference equations: Universal denominators and denominator bounds, Program. Comput. Software, 2011, vol. 37, no. 2, pp. 78–86.

    Article  MathSciNet  Google Scholar 

  8. Abramov, S.A., Search of rational solutions to differential and difference systems by means of formal series, Program. Comput. Software, 2015, vol. 41, no. 2, pp. 65–73.

    Article  MathSciNet  Google Scholar 

  9. Abramov, S.A., Barkatou, M.A., van Hoeij, M., and Petkovsek, M., Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences, J. Symb. Comp., 2011, vol. 46, pp. 1205–1228.

    Article  MathSciNet  Google Scholar 

  10. Abramov, S., Petkovsek, M., and Ryabenko, A., Hypergeometric solutions of first-order linear difference systems with rational-function coefficients, Lect. Notes Comput. Sci. 2015, vol. 9301, pp. 1–14.

    Article  MathSciNet  Google Scholar 

  11. Leinartas, E.K. and Lyapin, A.P., On the rationality of multi-dimentional recusive series, J. Sib. Fed. Univ., Math. & Phys., 2009, vol. 2, no. 2, pp. 449–455.

    Google Scholar 

  12. Kytmanov, A.A., Lyapin, A.P., and Sadykov, T.M., Evaluating the rational generating function for the solution of the Cauchy problem for a two-dimensional difference equation with constant coefficients, Program. Comput. Software, 2017, vol. 43, no. 2, pp. 105–111.

    Article  MathSciNet  Google Scholar 

  13. Rogozina, M.S., On the solvability of the cauchy problem for the polynomial difference operator, Vestn. Novosibirsk. Gos. Univ., 2014, vol. 14, no. 3, pp. 83–94.

    MATH  Google Scholar 

  14. Leinartas, E.K. and Rogozina, M.S., Solvability of the Cauchy Problem for a polynomial difference operator and monomial bases of factors in the ring of polynomials, Sib. Mat. Zh., 2015, vol. 56, no. 1, pp. 111–121.

    Article  MathSciNet  Google Scholar 

  15. Iokhvidov, I.S., Hankel and Toeplitz Matrices, Moscow: Nauka, 1974 [in Russian].

    MATH  Google Scholar 

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Funding

This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2020-1534/1).

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Authors and Affiliations

  1. Kransnoyarsk State Medical University, 660022, Kransnoyarsk, Russia

    M. S. Apanovich & K. V. Shadrin

  2. Siberian Federal University, 660041, Kransnoyarsk, Russia

    M. S. Apanovich & A. P. Lyapin

Authors
  1. M. S. Apanovich
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  2. A. P. Lyapin
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  3. K. V. Shadrin
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Corresponding authors

Correspondence to M. S. Apanovich, A. P. Lyapin or K. V. Shadrin.

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Translated by A. Klimontovich

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Apanovich, M.S., Lyapin, A.P. & Shadrin, K.V. Solving the Cauchy Problem for a Two-Dimensional Difference Equation at a Point Using Computer Algebra Methods. Program Comput Soft 47, 1–5 (2021). https://doi.org/10.1134/S0361768821010023

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  • DOI: https://doi.org/10.1134/S0361768821010023

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