Editorial Perturbation Methods and Formal Modeling for...

EditorialPerturbation Methods and Formal Modeling forDynamic Systems

Saeed Islam,1 Sher Afzal Khan,2 Gul Zaman,3 and Il Hyo Jung4

1Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan2Faculty of Computing and Information Technology in Rabigh, King Abdulaziz University, Jeddah, Saudi Arabia3Department of Mathematics, University of Malakand, Lower Dir, Pakistan4Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

Correspondence should be addressed to Saeed Islam; saeedislam@awkum.edu.pk

Received 1 March 2015; Accepted 1 March 2015

Copyright © 2015 Saeed Islam et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The nonlinear differential equations have been variously usedby the researchers in science and engineering to model manyreal world problems. If the problems fromvarious branches ofscience and engineering are persistently unresolved throughdifferential equations, the approximate solutions may beobtained through perturbation or numerical methods [1].However, the availability of high speed computers, advancedalgorithm, and a number of sophisticated numerical tech-niques has enabled researchers to solve problems throughdifferential equations of considerable complexity and nonlin-earity. Nonetheless, failures to find exact analytical solutionsto problems have reverted the researchers to various approx-imate analytical methods [2, 3].

Since the power series method fails to provide solutionsto equations at irregular singular points and at infinity, theasymptomatic method, which is an off-shoot of the powerseries method, provides the necessary answers to the singu-larly perturbed equations at infinity as well.The perturbationtechnique involves the expression of the unknown functionof the problem to be solved as a power series expansion andobtain the successive terms of the expansion carefully.

The idea behind the perturbation method is that the ini-tiative is taken with a simplified form of the original problem,and the perturbations or corrections are added gradually sothat the original problem correlates with the solution. Thedifferent perturbation parameters are adopted starting fromthe minimal value taken as zero. The perturbation method

is supplemented by boundary-layer method, multiple-scalemethod, WKB method, and so forth [3].

In problems involving linear or nonlinear differentialequations, a sequence of linear subproblems is devisedcomprised of a number of differential equations; these aresuccessively solved to obtain solution to the original problem.

Researchers have developed nonperturbationmethods aswell in order to solve complicated subproblems involvingperturbed parameters. Among these, Lyapunov’s artificialparameter method and (sigma) expansion method and Ado-mian decomposition method are important. Another mostefficient and nicely developed technique is the homotopyanalysis method proposed by Liao, homotopy perturbationmethod developed by He, optimal homotopy asymptoticmethod and optimal homotopy perturbation methods intro-duced by Marinca, and Herisanu semianalytical techniqueswhich are best suited for nonlinear ordinary and partialdifferential equations [4–7].

However, formal modelings are based on mathematicaltechniques and notations, for describing and analyzing prop-erties of software systems [8, 9].

These mathematical techniques are typically based ondiscrete mathematics such as predicate logic, set theory,relations, functions, and graph theory. The precise mathe-matical notation eliminates the ambiguity, inconsistencies,and incompleteness from any software and hardware system.The aim of formal methods is to demonstrate the process of

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development of a system from requirement to coding. Thisapproach identifies errors and oversights early in the designlife cycle, which are then easy to remove, with consequenthigh quality and cost saving.This improves the informal formof the system but does not replace them. There are varioustypes of formal modeling approaches which include Com-mUnity, Estelle, Esterel, Lotos, Petri Nets, RAISE, Z, VDM,and VDM++ [10–12]. We can divide formal specification intothree components: syntax: grammatical rules to determine ifsentences are well formed, semantics: rules for interpretingthe sentences in a precise,meaningful waywithin the domain,and proof theory: rules for inferring useful information fromthe specification.

Saeed IslamSher Afzal Khan

Gul ZamanIl Hyo Jung

References

[1] A.H.Nayfeh,PerturbationMethods,Wiley,NewYork,NY,USA,1972.

[2] J. H. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of NonlinearSciences and Numerical Simulation, vol. 1, no. 1, pp. 51–70, 2000.

[3] T. S. L. Radhika, T. K. V. Iyengar, and T. R. Rani, ApproximateAnalytical Methods for Solving Ordinary Differential Equations,CRCPress, Taylor& FrancisGroup, BocaRaton, Fla, USA, 2014.

[4] A.-M.Wawaz, “A reliable modification of Adomian decomposi-tion method,” Applied Mathematics and Computation, vol. 102,no. 1, pp. 77–86, 1999.

[5] S. J. Liao, The proposed homotopy analysis technique for thesolutionof nonlinear problems [Ph.D. thesis], Shanghai Jiao TongUniversity, Shanghai, China, 1992.

[6] J.-H. He, “Homotopy perturbation technique,” Computer Meth-ods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp.257–262, 1999.

[7] V. Marica and N. Hersanu, “Application of optimal homotopyasymptoticmethod for solving nonlinear equations arising inheat transfer,” International Journal of Heat and Mass Transfer,vol. 13, pp. 710–715, 2008.

[8] E. A. Strunk, X. Yin, and J. C. Knight, “Echo: a practicalapproach to formal verification,” in Proceedings of the 10thInternational Workshop on Formal Methods for Industrial Crit-ical Systems (FMICS ’05), pp. 44–53, ACM, Lisbon, Portugal,September 2005.

[9] M. C. Edmund and J. M. Winggie, Formal Methods: Stateof the Art and Future Direction, Carnegie Mellon University,Pittsburgh, Pa, USA, 1996.

[10] S. Liu and R. Adams, “Limitations of formal methods and anapproach to implement,” Tech. Rep., Hiroshima City University,Hiroshima, Japan, 1995.

[11] J. Woodcock and J. Davies, Using Z: Specification, Refinement,and Proof, Prentice Hall, 1996.

[12] J. P. Hinchey, Bowen, and G. Michael, Ten Commandments ofFormal Methods, PC World Russia, 1997.

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