A recursive-operational approach with applications to linear differential systems
-
Gabriel Bengochea
and
Manuel Ortigueira
Abstract
In this paper, an operational method for solving linear and nonlinear systems described by ordinary differential equations is presented. The construction is based on the generalized derivative in the sense of distribution theory. The approach allows the response computation without needing the use of any integral transform as in the Mikusiński operational calculus. A general description of the algorithm is done and some illustrating examples are presented. The algorithm is recursive allowing to add and remove any pole or zero contribution. The extension to nonlinear systems is done by means of the Adomian polynomials.
Funding source: Fundação para a Ciência e a Tecnologia
Award Identifier / Grant number: PEst-UID/EEA/00066/2013
Funding source: Consejo Nacional de Ciencia y Tecnología
Award Identifier / Grant number: SEP-CONACYT 220603
Award Identifier / Grant number: CONACYT 235551
Funding statement: The first author was supported by CONACYT under the project SEP-CONACYT 220603 and also by the Postdoctoral grant CONACYT 235551. The second author was supported by Portuguese National Funds through the FCT Foundation for Science and Technology under the project PEst-UID/EEA/00066/2013.
References
[1] Adomian G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988), no. 2, 501–544. 10.1016/0022-247X(88)90170-9Search in Google Scholar
[2] Bengochea G., Algebraic approach to the Lane–Emden equation, Appl. Math. Comput. 232 (2014), 424–430. 10.1016/j.amc.2014.01.096Search in Google Scholar
[3] Bengochea G., Operational solution of fractional differential equations, Appl. Math. Lett. 32 (2014), 48–52. 10.1016/j.aml.2014.02.011Search in Google Scholar
[4] Bengochea G., An operational approach with application to fractional Bessel equation, Fract. Calc. Appl. Anal. 18 (2015), no. 5, 1201–1211. 10.1515/fca-2015-0069Search in Google Scholar
[5] Bengochea G. and López G., Mikusiński’s operational calculus with algebraic foundations and applications to Bessel functions, Integral Transforms Spec. Funct. 25 (2014), no. 4, 272–282. 10.1080/10652469.2013.838956Search in Google Scholar
[6] Bengochea G. and Verde-Star L., Linear algebraic foundations of the operational calculi, Adv. Appl. Math. 47 (2011), 330–351. 10.1016/j.aam.2010.08.001Search in Google Scholar
[7] Bengochea G. and Verde-Star L., An operational approach to the Emden–Fowler equation, Math. Methods Appl. Sci. 38 (2015), no. 18, 4630–4637. 10.1002/mma.3415Search in Google Scholar
[8] Berg L., General operational calculus, Linear Algebra Appl. 84 (1986), 79–97. 10.1016/0024-3795(86)90308-3Search in Google Scholar
[9] Dimovski L. H., Convolutional Calculus, 2nd ed., Kluwer Academic Publishers, Dordrecht, 1990. 10.1007/978-94-009-0527-6Search in Google Scholar
[10] Dorf R. C. and Bishop R. H., Modern Control Systems, 12th ed., Pentice-Hall, Upper Saddle River, 2011. Search in Google Scholar
[11] Gel’fand I. M. and Shilov G. E., Generalized Functions. Volume I: Properties and Operations, Academic Press, New York, 1964. Search in Google Scholar
[12] Glaeske H. -J., Prudnikov A. P. and Skòrnik K. A., Operational Calculus and Related Topics, Chapman & Hall/CRC, Boca Raton, 2006. 10.1201/9781420011494Search in Google Scholar
[13] Kamen E. and Heck B., Fundamentals of Signals and Systems: Using the Web and Matlab, 2nd ed., Prentice Hall, Upper Saddle River, 2000. Search in Google Scholar
[14] Mikusinski J., Operational Calculus, Pergamon Press, Oxford, 1959. Search in Google Scholar
[15] Ortigueira M. and Coito F., System initial conditions vs derivative initial conditions, Comput. Math. Appl. 59 (2010), no. 5, 1782–1789. 10.1016/j.camwa.2009.08.036Search in Google Scholar
[16] Péraire Y., Heaviside calculus with no Laplace transform, Integral Transforms Spec. Funct. 17 (2006), 221–230. 10.1080/10652460600622365Search in Google Scholar
[17] Roberts M. J., Signals and Systems: Analysis Using Transform Methods & Matlab, McGraw–Hill, New York, 2003. Search in Google Scholar
[18] Rubel L. A., An operational calculus in miniature, Appl. Anal. 6 (1977), 299–304. 10.1080/00036817708839162Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- On the cubic and cubic-quintic optical vortices equations
- Symmetry reductions and exact solutions to the Sharma–Tasso–Olever equation
- Simultaneously proximinal subspaces
- Iteration regularized semigroups of set-valued functions
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- Compactness result and its applications in integral equations
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- Ritz-least squares method for finding a control parameter in a one-dimensional parabolic inverse problem
