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Construction of nonlocal linear vibration models using a computer algebra system

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Abstract

In the paper, on the examples of a string and beam, resonance vibrations in linear elastic systems that result from nonlocal boundary conditions (even if external forces are lacking) are considered. Formulas of exact solutions of the corresponding boundary-value problems are presented. These formulas are used for evaluation of the solutions and for construction of plots. Computer algebra system Mathematica was used for derivation and analysis of the exact solutions and for development of software packages designed for evaluation of the derived formulas and solution visualization. The authors believe that results of this study can be used to explain the causes for the vibration of the Tacoma and Volgograd bridges in 1940 and 2010, respectively.

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References

  1. Ionkin, N.I., Solution of One Boundary-Value Problem of Heat Conduction Theory with a Nonclassical Boundary Condition, Differ. Uravn., 1977, vol. 13, no. 2, pp. 294–304.

    MATH  MathSciNet  Google Scholar 

  2. Kamynin, L.I., On One Boundary-Value Problem of Heat Conduction Theory with Nonclassical Boundary Conditions, Zh. Vychisl. Mat. Mat. Fiz., 1964, vol. 4, pp. 1006–1024.

    MathSciNet  Google Scholar 

  3. Farlow, S.J., Partial Differential Equations for Scientists and Engineers, New York: Wiley, 1982.

    MATH  Google Scholar 

  4. Beilin, S., Existence of Solutions for One-Dimensional Wave Equation with Nonlocal Conditions, Electronic J. Diff. Equations, 2001, vol. 2001, no. 76, pp. 1–8.

    MathSciNet  Google Scholar 

  5. Cannon, J.R., The Solution of the Heat Equation Subject to the Specification of Energy, Quart. Appl. Math., 1963, vol. 21, pp. 155–160.

    MathSciNet  Google Scholar 

  6. Dimovski, I.H., Convolutional Calculas, Dordrecht: Kluwer, 1990.

    Google Scholar 

  7. Dimovski, I. and Spiridonova, M., Computer Implementation of Solutions of BVP for Finite Vibrating Systems, Mathematica Balkanica, 2004, Fasc. 3–4, pp. 277–285.

  8. Dimovski, I. and Spiridonova, M., Computational Approach to Nonlocal Boundary Value problems by Multivariate Operational Calculus, Math. Sci. Res. J., 2005, vol. 9, no. 12, pp. 315–329.

    MATH  MathSciNet  Google Scholar 

  9. Dimovski, I. and Spiridonova, M., A Feasible Linear Model for Takoma Narrows Bridge Destruction, Abstracts of MASSE Int. Congress on Mathematics MICOM 2009, Ohrid, Macedonia, 2009, pp. 29–30.

  10. Kawohl, B. and Sweers, G., On the Differential Equation for an Anisotropic Stiff Material, SIAM J. Math. Anal., 2006, vol. 37, no. 6, pp. 1828–1853.

    Article  MATH  MathSciNet  Google Scholar 

  11. Saadatmandi Abbas and Dehghan Mehdi, Numerical Solution of the One-Dimensional Wave Equation with an Integral Condition, J. Numerical Methods Partial Diff. Equations, 2007, vol. 23, no. 2, pp. 282–292.

    Article  MATH  Google Scholar 

  12. Spiridonova, M., Operational Methods in the Environment of a Computer Algebra System, Serdica J. Computing, Bulgarian Acad. Sci., Inst. of Math. and Informatics, 2009, no. 3, pp. 381–424.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    I. H. Dimovski & M. N. Spiridonova

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  1. I. H. Dimovski
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  2. M. N. Spiridonova
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Correspondence to I. H. Dimovski.

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Original Russian Text © I.H. Dimovski, M.N. Spiridonova, 2011, published in Programmirovanie, 2011, Vol. 37, No. 2.

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Dimovski, I.H., Spiridonova, M.N. Construction of nonlocal linear vibration models using a computer algebra system. Program Comput Soft 37, 71–77 (2011). https://doi.org/10.1134/S0361768811020046

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  • DOI: https://doi.org/10.1134/S0361768811020046

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