nach oben

Erschienen in:

2024 | OriginalPaper | Buchkapitel

On the Exact Solutions of a Sequence of Nonlinear Differential Equations Possessing Polynomial Nonlinearities

verfasst von : Zlatinka I. Dimitrova

Erschienen in: New Trends in the Applications of Differential Equations in Sciences

Verlag: Springer Nature Switzerland

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We apply the Simple Equations Method (SEsM) to a sequence of nonlinear differential equations possessing polynomial nonlinearities and connected to the SEIR model of epidemics spread. Exact solutions are obtained and several of these solutions are discussed from the point of view of application to the model of epidemic waves.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Several Relationships Connected to a Special Function Used in the Simple Equations Method (SEsM)

Nächstes Kapitel Improved Hardy Inequality with Logarithmic Term

Drazin. P.G.: Nonlinear Systems. Cambridge University Press, Cambridge, UK (1992)

Kutner, R., Ausloos, M., Grech, D., Di Matteo, T., Schinckus, C., Stanley, H. E.: Econophysics and sociophysics: Their milestones & challenges. Physica A 516, 240 – 253 (2019). https://doi.org/10.1016/j.physa.2018.10.019

Vitanov, N., Busse, F.: Bounds on the heat transport in a horizontal fluid layer with stress-free boundaries. Z. angew. Math. Phys. 48, 310 – 324 (1997). https://doi.org/10.1007/PL00001478

Vitanov, N. K., Sakai, K., Jordanov, I. P., Managi, S., Demura, K.: Analysis of a Japan government intervention on the domestic agriculture market. Physica A: 382, 330 – 335 (2007). https://doi.org/10.1016/j.physa.2007.02.025

Vitanov N.K.: Science Dynamics and Research Production. Indicators, Indexes, Statistical Laws and Mathematical Models. Springer, Cham (2016)

Lempert, R. J.: A new decision sciences for complex systems. Proceedings of the National Academy of Sciences 99 (suppl 3), 7309 – 7313 (2002). https://doi.org/10.1073/pnas.082081699

Vitanov, N. K.: Upper bound on the heat transport in a horizontal fluid layer of infinite Prandtl number. Physics Letters A 248, 338 – 346 (1998). https://doi.org/10.1016/S0375-9601(98)00674-4

Vitanov, N. K., Chabchoub, A., Hoffmann N.: Deep-Water Waves: on the Nonlinear Schrödinger Equation and its Solutions. Journal of Theoretical and Applied Mechanics 43, 43 – 54 (2013). https://doi.org/10.2478/jtam-2013-0013

Nikolova, E. V., Vitanov, N. K.: On the Possibility of chaos in a ceneralized model of three interacting sectors. Entropy 22, 1388 (2020). https://doi.org/10.3390/e22121388

10.

Vitanov, N. K., Ausloos, M., Rotundo, G.: Discrete model of ideological struggle accounting for migration. Advances in Complex Systems 15, 1250049 (2012). https://doi.org/10.1142/S021952591250049X

11.

Vitanov, N. K., Vitanov, K. N.: Box model of migration channels. Mathematical Social Sciences 80, 108 – 114 (2016). https://doi.org/10.1016/j.mathsocsci.2016.02.001

12.

Vitanov, N. K., Vitanov, K. N.: On the motion of substance in a channel of a network and human migration. Physica A: 490, 1277-1294 (2018). https://doi.org/10.1016/j.physa.2017.08.038

13.

Amaral, L. A. N., Scala, A., Barthelemy, M., Stanley, H. E.: Classes of small-world networks. PNAS USA 97, 11149 – 11152 (2000). https://doi.org/10.1073/pnas.20032719

14.

Vitanov, N. K., Vitanov, K. N., Kantz, H.: On the motion of substance in a channel of a network: Extended model and new classes of probability distributions. Entropy 22, 1240 (2020). https://doi.org/10.3390/e22111240

15.

Vitanov, N. K.: Remarks on the research at the Departemnt of Fluid Mechanics of the Institute of Mechanics of the Bulgarian Academy of Sciences. Journal of theoretical and Applied Mechanics, 53 (2023), https://doi.org/10.55787/jtams.23.53.3.207

16.

Sheard, S. A., Mostashari, A.: Principles of complex systems for systems engineering. Systems Engineering 12, 295 – 311 (2009). https://doi.org/10.1002/sys.20124

17.

Struble R.: Nonlinear Differential Equations. Dover, New York (2018)

18.

Kantz, H., T. Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge, UK (2004)

19.

Boeck, T., Vitanov, N. K.: Low-dimensional chaos in zero-Prandtl-number Benard-Marangoni convection. Physical Review E 65, 037203 (2002). https://doi.org/10.1103/PhysRevE.65.037203

20.

Jordanov, I., Nikolova, E.: On nonlinear waves in the spatio-temporal dynamics of interacting populations. Journal of Theoretical and Applied Mechanics 43, 69 – 76 (2013). arXiv preprint arXiv:1208.5465.

21.

Vitanov, N.K.: Results Connected to Time Series Analysis and Machine Learning. In: Atanassov, K.T. (eds) Research in Computer Science in the Bulgarian Academy of Sciences. Studies in Computational Intelligence, vol 934, pp.363-384. Springer, Cham. (2021). https://doi.org/10.1007/978-3-030-72284-5_17

22.

Jordanov, I. P.: On the nonlinear waves in (2+ 1)-dimensional population systems. Comptes rendus de l’Académie bulgare des Sciences 61, 307-314 (2008).

23.

Kantz, H., Holstein, D., Ragwitz, M., Vitanov, N. K.: Markov chain model for turbulent wind speed data. Physica A 342, 315 – 321 (2004). https://doi.org/10.1016/j.physa.2004.01.070

24.

Vitanov, N. K., Ausloos, M. R.: Knowledge epidemics and population dynamics models for describing idea diffusion. In: Scharnhorst A., Boerner K., van den Besselaar P. (eds.) Models of science dynamics. Understanding Complex Systems. pp. 69-125. Springer, Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642-23068-4_3

25.

Ashenfelter, K. T., Boker, S. M., Waddell, J. R., Vitanov, N.: Spatiotemporal symmetry and multifractal structure of head movements during dyadic conversation. Journal of Experimental Psychology: Human Perception and Performance 35, 1072 – 1091 (2009). https://doi.org/10.1037/a0015017

26.

Brockwell P.J., Davis R.A, Calder M.V.: Introduction to Time Series and Forecasting. Springer, New York (2002)

27.

Vitanov, N. K., Hoffmann, N. P., Wernitz, B.: Nonlinear time series analysis of vibration data from a friction brake: SSA, PCA, and MFDFA. Chaos, Solitons & Fractals 69, 90-99 (2014). https://doi.org/10.1016/j.chaos.2014.09.010

28.

Vitanov, N. K.: Upper bounds on the heat transport in a porous layer. Physica D 136, 322-339 (2000). https://doi.org/10.1016/S0167-2789(99)00165-7

29.

Vitanov, N. K., Dimitrova, Z. I., Kantz, H.: On the trap of extinction and its elimination. Physics Letters A 349, 350-355 (2006). https://doi.org/10.1016/j.physleta.2005.09.043

30.

Borisov, R., Dimitrova, Z. I., Vitanov, N. K.: Statistical characteristics of stationary flow of substance in a network channel containing arbitrary number of arms. Entropy 22, 553 (2020). https://doi.org/10.3390/e22050553

31.

Goldstein, J.: Social psychology and nonlinear dynamical systems theory. Psychological Inquiry 8, 125-128 (1997). https://doi.org/10.1207/s15327965pli0802_6

32.

Vitanov, N. K., Vitanov, K. N.: Statistical distributions connected to motion of substance in a channel of a network. Physica A 527, 121174 (2019). https://doi.org/10.1016/j.physa.2019.121174

33.

Vitanov, N. K., Borisov, R., Vitanov, K. N.: On the motion of substance in a channel and growth of random networks. Physica A 581, 126207 (2021). https://doi.org/10.1016/j.physa.2021.126207

34.

Ablowitz, M. J., Clarkson, P. A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, UK (1991)

35.

Gardner, C. S., Greene, J. M., Kruskal, M. D., Miura, R. M.: Method for solving the Korteweg-deVries equation. Physical review letters, 19(19), 1095 – 1897 (1967). https://doi.org/10.1103/PhysRevLett.19.1095

36.

Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge, UK, (2004)

37.

Vitanov, N. K.: Recent developments of the methodology of the modified method of simplest equation with application. Pliska Studia Mathematica Bulgarica 30, 29 – 42 (2019).

38.

Vitanov, N.K.: Modified method of simplest equation for obtaining exact solutions of nonlinear partia differential equations: history, recent developments of the methodology and studied of classes of equations. Journal of Theoretical and Applied Mechanics 49, 107 – 122 (2019).

39.

Vitanov, N. K.: The simple equations method (SEsM) for obtaining exact solutions of nonlinear PDEs: Opportunities connected to the exponential functions. AIP Conference Proceedings vol. 2159, 030038 (2019). https://doi.org/10.1063/1.5127503

40.

Vitanov, N. K., Dimitrova, Z. I.: Simple equations method (SEsM) and other direct methods for obtaining exact solutions of nonlinear PDEs. AIP Conference Proceedings vol. 2159, 030039 (2019). https://doi.org/10.1063/1.5127504

41.

Vitanov, N.K. Simple Equations Method (SEsM): Areas of possible applications. Springer Proceedings in Mathematics & Statistics, 412, 15 – 24 (2023). https://doi.org/10.1007/978-3-031-21484-4_2

42.

Vitanov, N.K. Special Functions and polynomials connected to the Simple Equations Method (SEsM). Springer Proceedings in Mathematics & Statistics, 412, 73 - 81 (2023). https://doi.org/10.1007/978-3-031-21484-4_7

43.

Martinov, N., Vitanov, N.: On some solutions of the two-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General 25, L419 – L426 (1992). https://doi.org/10.1088/0305-4470/25/8/007

44.

Martinov, N., Vitanov, N.: On the correspondence between the self-consistent 2D Poisson-Boltzmann structures and the sine-Gordon waves. Journal of Physics A: Mathematical and General 25, L51 – L56 (1992). https://doi.org/10.1088/0305-4470/25/2/004

45.

Martinov, N., Vitanov, N.: Running wave solutions of the two-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General 25, 3609 – 3613 (1992). https://doi.org/10.1088/0305-4470/25/12/021

46.

Martinov, N. K., Vitanov, N. K.: On self-consistent thermal equilibrium structures in two-dimensional negative-temperature systems. Canadian Journal of Physics 72, 618 – 624 (1994). https://doi.org/10.1139/p94-079

47.

Martinov, N. K., Vitanov, N. K.: New class of running-wave solutions of the (2+ 1)-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General 27, 4611 – 4618 (1994). https://doi.org/10.1088/0305-4470/27/13/034

48.

Vitanov, N. K.: On travelling waves and double-periodic structures in two-dimensional sine-Gordon systems. Journal of Physics A: Mathematical and General 29, 5195 – 5207 (1996). https://doi.org/10.1088/0305-4470/29/16/036

49.

Vitanov, N. K., Martinov, N. K.: On the solitary waves in the sine-Gordon model of the two-dimensional Josephson junction. Zeitschrift fur Physik B Condensed Matter 100, 129 – 135 (1996). https://doi.org/10.1007/s002570050102

50.

Vitanov, N. K.: Complicated exact solutions to the 2+ 1-dimensional sine-Gordon equation. Zeitschrift für angewandte Mathematik und Mechanik, 78, S787 – S788 (1998).

51.

Vitanov, N. K.: Breather and soliton wave families for the sine-Gordon equation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 2409 – 2423 (1998). https://doi.org/10.1098/rspa.1998.0264

52.

Vitanov, N. K., Jordanov, I. P., Dimitrova, Z. I.: On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations. Communications in Nonlinear Science and Numerical Simulation 14, 2379 – 2388 (2009). https://doi.org/10.1016/j.cnsns.2008.07.015

53.

Vitanov, N. K., Jordanov, I. P., Dimitrova, Z. I.: On nonlinear population waves. Applied Mathematics and Computation 215, 2950 – 2964 (2009). https://doi.org/10.1016/j.amc.2009.09.041

54.

Vitanov, N. K.: Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity. Communications in Nonlinear Science and Numerical Simulation 15, 2050 – 2060 (2010). https://doi.org/10.1016/j.cnsns.2009.08.011

55.

Vitanov, N. K., Dimitrova, Z. I.: Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics. Communications in Nonlinear Science and Numerical Simulation 15, 2836 – 2845 (2010). https://doi.org/10.1016/j.cnsns.2009.11.029

56.

Vitanov, N. K., Dimitrova, Z. I., Kantz, H.: Modified method of simplest equation and its application to nonlinear PDEs. Applied Mathematics and Computation 216, 2587 – 2595 (2010). https://doi.org/10.1016/j.amc.2010.03.102

57.

Vitanov, N. K.: Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs. Communications in Nonlinear Science and Numerical Simulation 16, 1176 – 1185 (2011). https://doi.org/10.1016/j.cnsns.2010.06.011

58.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and b-equation. Communications in Nonlinear Science and Numerical Simulation 16, 3033 – 3044 (2011). https://doi.org/10.1016/j.cnsns.2010.11.013

59.

Vitanov, N. K.: On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation. Communications in Nonlinear Science and Numerical Simulation 16, 4215 – 4231 (2011). https://doi.org/10.1016/j.cnsns.2011.03.035

60.

Vitanov, N. K.: On modified method of simplest equation for obtaining exact solutions of nonlinear PDEs: case of elliptic simplest equation. Pliska Studia Mathematica Bulgarica 21, 257 – 266 (2012).

61.

Vitanov, N. K., Dimitrova, Z. I., Kantz, H.: Application of the method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg-de Vries equation and generalized Camassa-Holm equation. Applied Mathematics and Computation 219, 7480 – 7492 (2013). https://doi.org/10.1016/j.amc.2013.01.035

62.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: Traveling waves and statistical distributions connected to systems of interacting populations. Computers & Mathematics with Applications 66, 1666 – 1684 (2013). https://doi.org/10.1016/j.camwa.2013.04.002

63.

Vitanov, N. K., Dimitrova, Z. I.: Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives. Applied Mathematics and Computation 247, 213 – 217 (2014). https://doi.org/10.1016/j.amc.2014.08.101

64.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications. Applied Mathematics and Computation 269, 363 – 378 (2015). https://doi.org/10.1016/j.amc.2015.07.060

65.

Vitanov, N. K., Dimitrova, Z. I., Ivanova, T. I.: On solitary wave solutions of a class of nonlinear partial differential equations based on the function 1/\(cosh^n\) (\(\alpha \)x+ \(\beta \)t). Applied Mathematics and Computation 315, 372 – 380 (2017). https://doi.org/10.1016/j.amc.2017.07.064

66.

Vitanov, N. K., Dimitrova, Z. I.: On the modified method of simplest equation and the nonlinear Schrödinger equation. Journal of Theoretical and Applied Mechanics 48, 59 – 68 (2018).

67.

Nikolova, E. V., Jordanov, I. P., Dimitrova, Z. I., Vitanov, N. K.: Evolution of nonlinear waves in a blood-filled artery with an aneurysm. AIP Conference Proceedings vol. 1895, 070002 (2017). https://doi.org/10.1063/1.5007391

68.

Jordanov, I.P., Vitanov, N.K.: On the Exact Traveling Wave Solutions of a Hyperbolic Reaction-Diffusion Equation. In: Georgiev, K., Todorov, M., Georgiev, I. (eds.) Advanced Computing in Industrial Mathematics. BGSIAM 2017. Studies in Computational Intelligence, vol. 793, pp. 199 – 210. Springer, Cham. (2019). https://doi.org/10.1007/978-3-319-97277-0_16

69.

Nikolova, E. V., Chilikova-Lubomirova, M., Vitanov, N. K.: Exact solutions of a fifth-order Korteweg-de Vries-type equation modeling nonlinear long waves in several natural phenomena. AIP Conference Proceedings vol. 2321, 030026 (2021). https://doi.org/10.1063/5.0040089

70.

Vitanov, N. K.: Simple equations method (SEsM) and its connection with the inverse scattering transform method. AIP Conference Proceedings vol. 2321, 030035 (2021). https://doi.org/10.1063/5.0040409

71.

Nikolova, E.V., Serbezov, D.Z., Jordanov, I.P., Vitanov, N.K.: Non-linear Waves of Interacting Populations with Density-Dependent Diffusion. In: Georgiev, I., Kostadinov, H., Lilkova, E. (eds.) Advanced Computing in Industrial Mathematics. BGSIAM 2018. Studies in Computational Intelligence, vol. 961, pp. 324 – 332. Springer, Cham. (2021). https://doi.org/10.1007/978-3-030-71616-5_29

72.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: Simple Equations Method (SEsM): Algorithm, connection with Hirota method, Inverse Scattering Transform Method, and several other methods. Entropy 23, 10 (2021). https://doi.org/10.3390/e23010010

73.

Vitanov. N.K.: Schrödinger Equation and Nonlinear Waves. Simpao V., Little H. (eds.). In: Understanding the Schrödinger Equation. pp. 37 - 92. Nova Science Publishers, New York (2020).

74.

Vitanov, N. K., Dimitrova, Z. I.: Simple Equations Method and non-linear differential equations with non-polynomial non-linearity Entropy 23, 1624 (2021). https://doi.org/10.3390/e23121624

75.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: On the use of composite functions in the Simple Equations Method to obtain exact solutions of nonlinear differential equations. Computation 9, 104 (2021). https://doi.org/10.3390/computation9100104

76.

Vitanov, N. K., Dimitrova, Z. I.: Simple equations method (SEsM) and its particular cases: Hirota method. AIP Conference Proceedings vol. 2321, 030036 (2021). https://doi.org/10.1063/5.0040410

77.

Vitanov, N. K.: Simple equations method (SEsM) and nonlinear PDEs with fractional derivatives. AIP Conference Proceedings vol. 2459, 030040 (2022). https://doi.org/10.1063/5.0083566

78.

Dimitrova, Z. I.: Several examples of application of the simple equations method (SEsM) for obtaining exact solutions of nonlinear PDEs. AIP Conference Proceedings vol. 2459, 030005 (2022). https://doi.org/10.1063/5.0083572

79.

Vitanov, N. K.: Simple equations method (SEsM): Review and new results. AIP Conference Proceedings, vol. 2459, 020003 (2022). https://doi.org/10.1063/5.0083565

80.

Dimitrova, Z. I.: On several specific cases of the simple equations method (SEsM): Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method, general projective Riccati equations method, and first intergal method. AIP Conference Proceedings, vol. 2459, 030006 (2022). https://doi.org/10.1063/5.0083573

81.

Vitanov, N. K. (2022). Simple Equations Method (SEsM): An effective algorithm for obtaining exact solutions of nonlinear differential equations. Entropy, 24, 1653 (2022). https://doi.org/10.3390/e24111653

82.

Vitanov, N. K., Vitanov, K. N. Epidemic waves and exact solutions of a sequence of nonlinear differential equations connected to the SIR model of epidemics. Entropy, 25, 438 (2023) https://doi.org/10.3390/e25030438.

83.

Vitanov, N. K., Dimitrova, Z. I. Computation of the exact forms of waves for a set of differential equations associated with the SEIR model of epidemics. Computation, 11, 129 (2023). https://doi.org/10.3390/computation11070129

84.

Bauer, F., van den Driessche, P., Wu, J. Mathematical Epidemiology. Springer: Berlin/Heidelberg (2008).

85.

Hethcote, H.W. Qualitative analyses of communicable disease models. Mathematical Biosciences, 28, 335–356 (1976). https://doi.org/10.1016/0025-5564(76)90132-2.

86.

Keeling, M.J., Rohani, P. Modeling Infectious Diseases in Humans an Animals. Princeton University Press: Princeton, NJ, USA, (2008).

Titel: On the Exact Solutions of a Sequence of Nonlinear Differential Equations Possessing Polynomial Nonlinearities
verfasst von: Zlatinka I. Dimitrova
Verlag: Springer Nature Switzerland
Buch: New Trends in the Applications of Differential Equations in Sciences
Print ISBN: 978-3-031-53211-5

Electronic ISBN: 978-3-031-53212-2

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-53212-2_5

Premium Partner

Bildnachweise

Rittal/© Rittal, NTT Data/© NTT Data, Wildix/© Wildix, Ceyoniq Technology GmbH/© Ceyoniq Technology GmbH, arvato Systems GmbH/© arvato Systems GmbH, Ninox Software GmbH/© Ninox Software GmbH, Everyday Software S.L./© Everyday Software S.L., Redgate/© Redgate, CELONIS Labs GmbH, DC-Datacenter-Group GmbH/© DC-Datacenter-Group GmbH, all for one/© all for one, G Data CyberDefense/© G Data CyberDefense, FAST LTA/© FAST LTA, Vendosoft/© Vendosoft, Kumavision/© Kumavision, Noriis Network AG/© Noriis Network AG, WSW Software GmbH/© WSW Software GmbH, tts GmbH/© tts GmbH

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"