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First-order linear differential equations whose data are complex random variables: Probabilistic solution and stability analysis via densities

  • Received: 20 July 2021 Accepted: 28 September 2021 Published: 27 October 2021
  • MSC : 34M03, 34F05

  • Random initial value problems to non-homogeneous first-order linear differential equations with complex coefficients are probabilistically solved by computing the first probability density of the solution. For the sake of generality, coefficients and initial condition are assumed to be absolutely continuous complex random variables with an arbitrary joint probability density function. The probability of stability, as well as the density of the equilibrium point, are explicitly determined. The Random Variable Transformation technique is extensively utilized to conduct the overall analysis. Several examples are included to illustrate all the theoretical findings.

    Citation: J.-C. Cortés, A. Navarro-Quiles, J.-V. Romero, M.-D. Roselló. First-order linear differential equations whose data are complex random variables: Probabilistic solution and stability analysis via densities[J]. AIMS Mathematics, 2022, 7(1): 1486-1506. doi: 10.3934/math.2022088

    Related Papers:

  • Random initial value problems to non-homogeneous first-order linear differential equations with complex coefficients are probabilistically solved by computing the first probability density of the solution. For the sake of generality, coefficients and initial condition are assumed to be absolutely continuous complex random variables with an arbitrary joint probability density function. The probability of stability, as well as the density of the equilibrium point, are explicitly determined. The Random Variable Transformation technique is extensively utilized to conduct the overall analysis. Several examples are included to illustrate all the theoretical findings.



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    [1] Y. Sibuya, Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, Rhode Island: American Mathematical Society, vol. 82, 1990.
    [2] E. Hille, Ordinary Differential Equations in the Complex Domain, New York: John While & Sons, 1976.
    [3] A. Savin, B. Sternin, Introduction to Complex Theory of Differential Equations, Switzerland: Birkhauser Basel, 2017. doi: 10.1007/978-3-319-51744-5.
    [4] A. Joohy, Ordinary Differential Equations in the Complex Domain with Applications: In Physics and Engineering, Latvia: Scholars' Press, 2018.
    [5] I. Laine, Nevanlinna Theory and Complex Differential Equations, Berlin: Walter de Gruyter, 1993. doi: 10.1515/9783110863147.
    [6] H. Davis, Introduction to Nonlinear Differential and Integral Equations, Eastford, USA: Martino Fine Books, 2014.
    [7] G. Filipuk, A. Lastra, S. Michalik, Y. Takei, H. Zoladeka, Complex Differential and Difference Equations: Proceedings of the School and Conference Held at Bedlewo, Poland, September 2-15, 2018, Walter de Gruyter GmbH and Co KG, 2019.
    [8] R. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, ser. Computational Science and Engineering, Philadelphia: SIAM, 2014.
    [9] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Berlin: Springer, 2007. doi: 10.1007/978-3-662-03185-8.
    [10] X. Mao, Stochastic Differential Equations and Applications, 2nd ed, New York: Woodhead Publishing, 2007.
    [11] T. T. Soong, Random Differential Equations in Science and Engineering, New York: Academic Press, 1973.
    [12] T. Neckel, F. Rupp, Random Differential Equations in Scientific Computing, London: Versita, 2013.
    [13] C. Braumann, Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance, Wiley, 2019. doi: 10.1002/9781119166092.
    [14] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, ser. Computational Science and Engineering, New Jersey: Princeton University Press, 2010. doi: 10.1515/9781400835348.
    [15] J. Eriksson, E. Ollila, V. Koivunen, Statistics for complex random variables revisited, Proceedings of the 34th IEEE International Conference on Acoustics, Speech and Signal Processing, (2009), 3565–3568. doi: 10.1109/ICASSP.2009.4960396.
    [16] J. Eriksson, E. Ollila, V. Koivunen, Essential statistics and tools for complex random variables, IEEE T. Signal Proces., 58 (2010), 5400–5408. doi: 10.1109/TSP.2010.2054085. doi: 10.1109/TSP.2010.2054085
    [17] F. D. Neeser, J. L. Massey, Proper complex random processes with applications to information theory, IEEE T. Inform. Theory, 29 (1993), 1293–1302. doi: 10.1109/18.243446. doi: 10.1109/18.243446
    [18] A. Khurshid, Z. A. Al-Hemyari, S. Kamal, On complex random variables, Pak. J. Stat. Oper. Res., 8 (2012), 645–654. doi: 10.18187/pjsor.v8i3.534.
    [19] A. Lapidoth, A Foundation in Digital Communication, Cambridge University Press, 2009.
    [20] M. C. Casabán, J. C. Cortés, J. V. Romero, M. D. Roselló, Determining the first probability density function of linear random initial value problems by the Random Variable Transformation (RVT) technique: A comprehensive study, Abstr. Appl. Anal., 2014 (2014), 1–25. doi: 10.1155/2013/248512. doi: 10.1155/2013/248512
    [21] J. C. Cortés, A. Navarro-Quiles, J. V. Romero, M. D. Roselló, Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function, Appl. Math. Comput., 331 (2018), 33–45. doi: 10.1016/j.amc.2018.02.051. doi: 10.1016/j.amc.2018.02.051
    [22] G. Falsone, D. Settineri, Explicit solutions for the response probability density function of linear systems subjected to random static loads, Probabilist. Eng. Mech., 33 (2013), 86–94. doi: 10.1016/j.probengmech.2013.03.001. doi: 10.1016/j.probengmech.2013.03.001
    [23] G. Falsone, D. Settineri, On the application of the probability transformation method for the analysis of discretized structures with uncertain properties, Probabilist. Eng. Mech., 35 (2014), 44–51. doi: 10.1016/j.probengmech.2013.10.001. doi: 10.1016/j.probengmech.2013.10.001
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