research article approximation for transient of nonlinear...

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2013 Article ID 973813 6 pageshttpdxdoiorg1011552013973813

Research ArticleApproximation for Transient of Nonlinear Circuits UsingRHPM and BPES Methods

H Vazquez-Leal1 K Boubaker2 L Hernandez-Martinez3 and J Huerta-Chua4

1 Electronic Instrumentation and Atmospheric Sciences School Universidad VeracruzanaCircuito Gonzalo Aguirre Beltran SN Xalapa 91000 VER Mexico

2 Ecole Superieure de Sciences et Techniques de Tunis (ESSTT) Universite de Tunis63 rue Sidi Jabeur 5100 Mahdia Tunisia

3 National Institute for Astrophysics Optics and Electronics Luis Enrique Erro No 1 72840 Santa Marıa Tonantzintla PUE Mexico4 Facultad de Ingenieria Civil Universidad Veracruzana Venustiano Carranza SN Col RevolucionCP 93390 Poza Rica VER Mexico

Correspondence should be addressed to H Vazquez-Leal hvazquezuvmx

Received 20 December 2012 Accepted 19 January 2013

Academic Editor Esteban Tlelo-Cuautle

Copyright copy 2013 H Vazquez-Leal et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The microelectronics area constantly demands better and improved circuit simulation tools Therefore in this paper rationalhomotopy perturbation method and Boubaker Polynomials Expansion Scheme are applied to a differential equation from anonlinear circuit Comparing the results obtained by both techniques revealed that they are effective and convenient

1 Introduction

Industrial competition constantly pushes the area of elec-tronic circuit design to the limits of technology This hascaused a rapid growth in the levels of integration for inte-grated circuits and the emergence of novel devices suchas single-electron transistors and memristors Because ofthis the development and improvement of mathematicaland numerical tools applied to circuit simulation for thetransient domain are important In the dynamic domain(transient) the circuit analysis is carried out only numeri-cally because the resulting differential equations are highlynonlinear Nevertheless several methods are focused to findapproximate solutions to nonlinear differential equationslike homotopy perturbation method (HPM) [1ndash12] rationalhomotopy perturbation method (RHPM) [5 6] variationaliteration method (VIM) [13ndash16] and Boubaker PolynomialsExpansion Scheme (BPES) [17ndash36] among many othersTherefore we propose the comparison between RHPM andBPES methods by solving the nonlinear differential equa-tion that represents the dynamics of a nonlinear circuitThe results should be a meaningful supply for monitoringcomplex nonlinear circuits behaviours and responses In

fact the used protocols try to embed boundary conditionsinstead of direct solving as preceded in spectral or limit-cyclebifurcations approaches

This paper is arranged as follows In Section 2 we presentthe differential equation of a nonlinear circuit Sections 3 and4 present the fundamentals of RHPM and BPES methodsrespectively The solutions obtained using both methodsare explained in Section 5 Comparisons between the twomethods and some other results presented in the recentliterature have been illustrated in Section 6 Conclusions willbe discussed in Section 7

2 Nonlinear Circuit

The rapid increase in the number of transistors by integratedcircuit and the increase of complexity for the models (a resultof lowering the dimension of the components) results in acomplex calculation for the transient Furthermore the taskof tracing the transient for nonlinear circuits is a criticaland difficult task In fact commercial circuit simulators donot provide any symbolicanalytic solution for the transientof any given circuit Instead the simulator provides only

2 Journal of Electrical and Computer Engineering

119894(119905)119877

119862+

minus

Figure 1 Nonlinear RC circuit

numerical data that allows circuit designers to explore alimited range of dynamics of nonlinear circuits

Consider the analysis of the nonlinear circuit depicted inFigure 1 as a case study [37] Let the branch relationship of thenonlinear capacitor to be

119907 (119902) = 1205721199023 (1)

where 119907 is the voltage 119902 is the charge of the capacitor and 120572 isa parameter of the capacitor

By applying Kirchhoff Laws we obtain the equation forthe transient

119889119902 (119905)

119889119905

+

1

119877

119907 (119902) = 119894 (119905) (2)

Therefore119889119902 (119905)

119889119905

+

120572

119877

1199023(119905) = 119894 (119905) (3)

If we consider the case for DC excitation then 119894(119905) = 119868resulting in

119889119902 (119905)

119889119905

+

120572

119877

1199023(119905) = 119868 119902 (0) = 0 (4)

3 Fundamentals of the Rational HomotopyPerturbation Method

The rational homotopy perturbation method RHPM [5 6]can be considered as a combination of the classical perturba-tion technique [38 39] and the homotopy (whose origin is inthe topology) [40ndash42] but not restricted to a small parameterlike traditional perturbation methods For example RHPMrequires neither small parameter nor linearization but onlyfew iterations to obtain accurate solutions

To figure out how RHPM method works consider ageneral nonlinear equation in the form

119860 (119906) minus 119891 (119903) = 0 119903 isin Ω (5)

with the following boundary conditions

119861(119906

120597119906

120597119906

) = 0 119903 isin Γ (6)

where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) is a known analytical function and Γ is thedomain boundary for Ω119860 can be divided into two operators119871 and 119873 where 119871 is linear and 119873 nonlinear from this laststatement (5) can be rewritten as

119871 (119906) + 119873 (119906) minus 119891 (119903) = 0 (7)

Generally a homotopy can be constructed in the form [1ndash3]119867(119907 119901) = (1 minus 119901) [119871 (119907) minus 119871 (1199060

)]

+ 119901 [119871 (119907) + 119873 (119907) minus 119891 (119903)] = 0

119901 isin [0 1] 119903 isin Ω

(8)

where119901 is a homotopy parameter whose values arewithin therange of 0 and 1 119906

0is the first approximation for the solution

of (6) that satisfies the boundary conditionsWhen 119901 rarr 0 (8) is reduced to

119871 (119907) minus 119871 (1199060) = 0 (9)

where operator 119871 possesses trivial solutionFor 119901 rarr 1 (8) is reduced to the original problem

119873(119907) + 119871 (119907) minus 119891 (119903) = 0 (10)Assuming that the solution for (8) can be written as a powerseries of 119901

119907 =

1199070+ 1199011199071+ 11990121199072+ sdot sdot sdot

1199080+ 1199011199081+ 11990121199082+ sdot sdot sdot

(11)

where 1199070 1199071 1199072 are unknown functions to be determined

by the RHPM and 1199080 1199081 1199082 are known analytic func-

tions of the independent variableSubstituting (11) into (8) and equating identical powers

of 119901 terms it is possible to obtain values for the sequence1199070 1199071 1199072

When119901 rarr 1 in (11) it yields in the approximate solutionfor (5) in the form

119906 = lim119901rarr1

(119907) =

1199070+ 1199071+ 1199072+ sdot sdot sdot

1199080+ 1199081+ 1199082+ sdot sdot sdot

(12)

Convergence of RHPMmethod is studied in [5 6]

4 Fundamentals of the Boubaker PolynomialsExpansion Scheme BPES

The Boubaker Polynomials Expansion Scheme BPES [17ndash36] is a resolution protocol which has been successfullyapplied to several applied-physics and mathematical prob-lems The BPES protocol ensures the validity of the relatedboundary conditions regardless of main equation featuresThe Boubaker Polynomials Expansion Scheme BPES is basedon the Boubaker polynomials first derivatives properties

119873

sum

119902=1

1198614119902(119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=0

= minus 2119873 = 0

119873

sum

119902=1

1198614119902 (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=119903119902

= 0

119873

sum

119902=1

1198891198614119902(119909)

119889119909

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=0

= 0

119873

sum

119902=1

1198891198614119902(119909)

119889119909

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=119903119902

=

119873

sum

119902=1

119867119902

(13)

Journal of Electrical and Computer Engineering 3

with

119867119899= 1198611015840

4119899(119903119899) = (

4119903119899[2 minus 119903

2

119899] times sum119899

119902=11198612

4119902(119903119899)

1198614(119899+1)

(119903119899)

+ 41199033

119899)

(14)

Several solutions have been proposed through the BPESin many fields like numerical analysis [17ndash20] theoreticalphysics [21ndash24] mathematical algorithms [25] heat transfer[26] homodynamic [27 28] material characterization [29]fuzzy systems modelling [30ndash34] and biology [35 36]

5 Application of RHPM and BPES

51 SolutionUsing RHPMMethod Using (8) we establish thefollowing RHPM homotopy map

(1 minus 119901) (1199071015840minus 1199061015840

0) + 119901(119907

1015840+

120572

119877

1199073minus 119868) = 0 (15)

where the trial function 1199060= 0

Using (11) we propose the following rational solution

119907 = (1199070+ 1199011199071+ 11990121199072+ 11990131199073+ 11990141199074

+ 11990151199075+ 11990161199076+ 11990171199077)

times (1 + 11990111989611199093+ 119901211989621199096)

minus1

(16)

where 1199080= 1 119908

1= 11989611199093 and 119908

2= 11989621199096

We substitute (16) into (15) regroup and equate termswith identical powers of 119901 In order to fulfil boundarycondition of (16) it follows that 119907

0(0) = 0 119907

1(0) for the

homotopy mapThe results are recast in the following systems of differen-

tial equations

1199010 119907

1015840

0= 0 119907

0 (0) = 0

1199011 1199071015840

1minus 119868 +

120572

119877

1199073+ 21199071015840

011987011198833minus 3119883211990701198701= 0 119907

1 (0) = 0

(17)

Solving (17) yields

1199070= 1199060= 0

1199071= 119868119909

1199072= 11986811989611199094

(18)

Substituting (18) into (16) and calculating the limitwhen 119901 rarr 1 we obtain the seventh-order approximation

119902 (119905) = lim119901rarr1

(119907)

= (119868119909 + (1198681198961minus (14) (120572119868

3119877)) 119909

4

+ (1198681198962minus (14) (1205721198961

1198683119877) + (328) (120572

211986851198772)) 1199097

minus (14) (12057211989621198683119877) 11990910)

times (1 + 11989611199093+ 11989621199096)

minus1

(19)

If we consider 119877 = 120 119868 = 10 and 120572 = 40 as reported in[37] it is possible to obtain the adjustment parameters usingthe procedure reported in [6 12] resulting in 119896

1= 36289 and

1198962= 4471843

52 Solution Using the Boubaker Polynomials ExpansionScheme BPES The Boubaker Polynomials ExpansionScheme BPES is applied to (4) using the setting expression

119902 (119905) =

1

21198730

1198730

sum

119896=1

120582119896times

1198891198614119896(119903119896119905)

119889119905

(20)

Using the properties provided by (13) boundary conditionsare verified in advance of the resolution process The systemin (16) is reduced to

1

21198730

1198730

sum

119896=1

120582119896119903119896

11988921198614119896(119903119896119905)

1198892119905

+

120572

81198733

0119877

[

1198730

sum

119896=1

120582119896

1198891198614119896(119903119896119905)

119889119905

]

3

= 119868

(21)

Boundary conditions become redundant since they arealready verified by the proposed expansion consecutivelyand thus majoring and integrating along the given inter-val for the time variable 119905 transform the problem in alinear system with unknown real variables 120582

119896|119896=11198730

Cal-culations are reduced to approximately (8119873

0)3arithmetical

operations Solutions are obtained by using the Householder[39 40] algorithm detailed elsewhere and are denotedby 120582(sol)119896|119896=11198730

The final solution is given as

119902 (119905) =

1

21198730

1198730

sum

119896=1

120582(sol)119896times

1198891198614119896(119903119896119905)

119889119905

(22)

6 Results and Discussion

From Table 1 we can observe that RHPM solution (19) andBPES solution (22) are in good agreement with the numericalresults obtained using Fehlberg fourth-fifth-order Runge-Kutta method with degree four interpolant (RKF45) [43 44](built-in function of Maple software) In order to guaranteea good numerical reference RKF45 is configured using anabsolute error of 10minus7and a relative error of 10minus6 The power


Table 1 Numerical comparison of proposed solutions and RKF45 solution of (4)

119905 q(t) (RKF45) RHPM BPES1198730= 37 119873

0= 107 119873

0= 113

000 000000 000000 000000 000000 000000001 0098066 0098070 009436 009456 009827002 17475 017481 017123 017395 017513003 21314 021388 021051 021266 021394004 022656 022802 022669 022557 022775005 023054 023005 022678 023009 023076006 023165 023235 022346 023056 023246

of RHPM method is based on the capability of rationalexpressions containing a huge amount of information ofdynamics from asymptotic problems

Moreover convergence of the BPES algorithm has beenobtained for moderate values of 119873

0(1198730lt 120) since

as mentioned above boundary conditions were verified inadvance of the resolution process Both methods generatedanalytical expressions useful for other analysis like circuitpower consumption such analytical expressions can providemore information about the nature and behaviour of cir-cuits than numerical integration schemes with variable stepsize [41 44ndash46] Nonetheless semianalytical techniques likeRHPM and BPESmay be combined with numerical methods[43ndash46] to improve the simulation tools of VLSI circuits

7 Conclusion

In this paper powerful analytical methods like rationalhomotopy perturbation method (RHPM) and BoubakerPolynomials Expansion Scheme (BPES) are presented toconstruct semianalytical solutions for the transient of anonlinear circuit The results exhibited that both techniquesare powerful obtaining highly accurate analytical expressionsfor the transient of a simple test circuit While RHPM yieldedaccurate and reliable results BPES exhibited the advantageof ensuring the validity of boundary conditions regardlessof main equation features This feature made the protocolyielding faster and provided more convergent solutions thanmany numerical integration schemes with variable step sizeFurther work is necessary to extend the use of both methodsfor larger circuits

AcknowledgmentsTheauthors gratefully acknowledge the financial support pro-vided by the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024The authors would like to express their gratitude to Rogelio-AlejandroCallejas-Molina andRobertoRuiz-Gomez for theircontribution to this project

References

[1] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[2] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[3] J H He ldquoHomotopy perturbation method a new nonlinearanalytical techniquerdquo Applied Mathematics and Computationvol 135 no 1 pp 73ndash79 2003

[4] H Vazquez-Leal Y Khan G Fernandez-Anaya et al ldquoA generalsolution for Troeschrsquos problemrdquo Mathematical Problems inEngineering vol 2012 Article ID 208375 14 pages 2012

[5] H Vazquez-Leal A Sarmiento-Reyes Y Khan U Filobello-Nino and A Diaz-Sanchez ldquoRational biparameter homotopyperturbation method and laplace-pade coupled versionrdquo Jour-nal of Applied Mathematics vol 2012 Article ID 923975 21pages 2012

[6] H Vazquez-Leal ldquoRational homotopy perturbation methodrdquoJournal of Applied Mathematics vol 2012 Article ID 490342 14pages 2012

[7] Y Khan H Vazquez-Leal and L Hernandez-MartınezldquoRemoval of noise oscillation term appearing in the nonlinearequation solutionrdquo Journal of Applied Mathematics vol 2012Article ID 387365 9 pages 2012

[8] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematical Sciences vol 6 no 85ndash88 pp 4331ndash4344 2012

[9] H Vazquez-Leal R Castaneda-Sheissa A Yıldırım et alldquoBiparameter homotopy-based direct current simulation ofmultistable circuitsrdquoBritish Journal ofMathematicsampComputerScience vol 2 no 3 pp 137ndash150 2012

[10] U Filobello-Nino Hector Vazquez-Leal R Castaneda-Sheissaet al ldquoAn approximate solution of Blasius equation by usingHPM methodrdquo Asian Journal of Mathematics amp Statistics vol5 no 2 Article ID 103923 pp 50ndash59 2012

[11] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[12] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J S Orea ldquoHigh accurate simple approxi-mation of normal distribution integralrdquoMathematical Problemsin Engineering vol 2012 Article ID 124029 22 pages 2012

[13] Y Khan H Vazquez-Leal L Hernandez-Martınez and NFaraz ldquoVariational iteration algorithm-II for solving linear andnon-linear ODEsrdquo International Journal of the Physical Sciencesvol 7 no 25 pp 3099ndash4002 2012

[14] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999


[15] J H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[16] J H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[17] M Agida andA S Kumar ldquoA Boubaker polynomials expansionscheme solution to random Loversquos equation in the case of arational Kernelrdquo Electronic Journal of Theoretical Physics vol 7no 24 pp 319ndash326 2010

[18] A Yildirim S T Mohyud-Din and D H Zhang ldquoAnalyticalsolutions to the pulsed Klein-Gordon equation using ModifiedVariational Iteration Method (MVIM) and Boubaker Polyno-mials Expansion Scheme (BPES)rdquo Computers and Mathematicswith Applications vol 59 no 8 pp 2473ndash2477 2010

[19] J Ghanouchi H Labiadh and K Boubaker ldquoAn attempt tosolve the heat transfer equation in a model of pyrolysis sprayusing 4q-orderm-boubaker polynomialsrdquo International Journalof Heat and Technology vol 26 no 1 pp 49ndash53 2008

[20] S Slama J Bessrour K Boubaker and M Bouhafs ldquoAdynamical model for investigation of A3 point maximal spatialevolution during resistance spot welding using Boubaker poly-nomialsrdquoTheEuropean Physical Journal Applied Physics vol 44no 3 pp 317ndash322 2008

[21] S Slama M Bouhafs and K B Mahmoud ldquoA boubakerpolynomials solution to heat equation for monitoring A3 pointevolution during resistance spot weldingrdquo International Journalof Heat and Technology vol 26 no 2 pp 141ndash146 2008

[22] S Lazzez K B Ben Mahmoud S Abroug F Saadallah and MAmlouk ldquoA Boubaker polynomials expansion scheme (BPES)-related protocol for measuring sprayed thin films thermalcharacteristicsrdquo Current Applied Physics vol 9 no 5 pp 1129ndash1133 2009

[23] T Ghrib K Boubaker and M Bouhafs ldquoInvestigation of ther-mal diffusivitymicrohardness correlation extended to surface-nitrured steel using Boubaker polynomials expansionrdquoModernPhysics Letters B vol 22 no 29 pp 2893ndash2907 2008

[24] S Fridjine K B Ben Mahmoud M Amlouk and M BouhafsldquoA study of sulfurselenium substitution effects on physi-cal and mechanical properties of vacuum-grown ZnS1-xSexcompounds using Boubaker polynomials expansion scheme(BPES)rdquo Journal of Alloys and Compounds vol 479 no 1-2 pp457ndash461 2009

[25] C Khelia K Boubaker T B Nasrallah M Amlouk and SBelgacem ldquoMorphological and thermal properties of 120573-SnS2sprayed thin films using Boubaker polynomials expansionrdquoJournal of Alloys and Compounds vol 477 no 1-2 pp 461ndash4672009

[26] K B Mahmoud and M Amlouk ldquoThe 3D Amlouk-Boubakerexpansivity-energy gap-Vickers hardness abacus a new toolfor optimizing semiconductor thin film materialsrdquo MaterialsLetters vol 63 no 12 pp 991ndash994 2009

[27] M Dada O B Awojoyogbe and K Boubaker ldquoHeat transferspray model an improved theoretical thermal time-response touniform layers deposit using Bessel and Boubaker polynomi-alsrdquo Current Applied Physics vol 9 no 3 pp 622ndash624 2009

[28] S A H A E Tabatabaei T Zhao O B Awojoyogbe and FO Moses ldquoCut-off cooling velocity profiling inside a keyholemodel using the Boubaker polynomials expansion schemerdquoHeat and Mass Transfer vol 45 no 10 pp 1247ndash1251 2009

[29] A Belhadj J Bessrour M Bouhafs and L Barrallier ldquoExper-imental and theoretical cooling velocity profile inside laser

welded metals using keyhole approximation and Boubakerpolynomials expansionrdquo Journal of Thermal Analysis andCalorimetry vol 97 no 3 pp 911ndash915 2009

[30] A Belhadj O F Onyango and N Rozibaeva ldquoBoubaker poly-nomials expansion scheme-related heat transfer investigationinside keyhole modelrdquo Journal of Thermophysics and HeatTransfer vol 23 no 3 pp 639ndash640 2009

[31] P Barry and A Hennessy ldquoMeixner-type results for Riordanarrays and associated integer sequencesrdquo Journal of IntegerSequences vol 13 no 9 pp 1ndash34 2010

[32] A S Kumar ldquoAn analytical solution to applied mathematics-related Loversquos equation using the Boubaker polynomials expan-sion schemerdquo Journal of the Franklin Institute vol 347 no 9 pp1755ndash1761 2010

[33] S Fridjine and M Amlouk ldquoA new parameter an ABACUS foroptimizig functional materials using the Boubaker polynomialsexpansion schemerdquoModern Physics Letters B vol 23 no 17 pp2179ndash2191 2009

[34] M Benhaliliba C E Benouis K Boubaker M Amlouk andA Amlouk ldquoA new guide to thermally optimized doped oxidesmonolayer spray-grown solar cells the amlouk-boubakeroptothermal expansivity Ψabrdquo in Solar CellsmdashNew Aspects andSolutions L A Kosyachenko Ed pp 27ndash41 InTech 2011

[35] A Milgram ldquoThe stability of the Boubaker polynomials expan-sion scheme (BPES)-based solution to Lotka-Volterra problemrdquoJournal of Theoretical Biology vol 271 no 1 pp 157ndash158 2011

[36] H Rahmanov ldquoA solution to the non lLinear korteweg-de-vriesequation in the particular case dispersion-adsorption problemin porous media using the spectral boubaker polynomialsexpansion scheme (BPES)rdquo Studies in Nonlinear Sciences vol2 no 1 pp 46ndash49 2011

[37] M Koksal and S Herdem ldquoAnalysis of nonlinear circuits byusing differential Taylor transformrdquo Computers and ElectricalEngineering vol 28 no 6 pp 513ndash525 2002

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoPerturba-tion methodand Laplace-Pade approximation to solve nonlin-ear problemsrdquoMiskolc Mathematical Notes In press

[39] H Vazquez-Leal U Filobello-Nino A Yildirim et al ldquoTran-sient andDC approximate expressions for diode circuitsrdquo IEICEElectronics Express vol 9 no 6 pp 522ndash530 2012

[40] H Vazquez-Leal L Hernandez-Martinez A Sarmiento-ReyesR Castaneda-Sheissa and A Gallardo-Del-Angel ldquoHomotopymethod with a formal stop criterion applied to circuit simu-lationrdquo IEICE Electronics Express vol 8 no 21 pp 1808ndash18152011

[41] H Vazquez-Leal L Hernandez-Martinez and A Sarmiento-Reyes ldquoDouble-bounded homotopy for analysing nonlinearresistive circuitsrdquo inProceeding of the IEEE International Sympo-sium on Circuits and Systems (ISCAS rsquo05) pp 3203ndash3206 KobeJapan May 2005

[42] H Vazquez-Leal L Hernandez-Martinez A Sarmiento-Reyesand R Castaneda-Sheissa ldquoNumerical continuation scheme fortracing the double bounded homotopy for analysing nonlinearcircuitsrdquo in Proceedings of the International Conference onCommunications Circuits and Systems pp 1122ndash1126 HongKong China May 2005

[43] W H Enright K R Jackson S P Norsett and P G ThomsenldquoInterpolants for runge-kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986

[44] E Fehlberg ldquoKlassischerunge-kutta-formelnvierter und nied-rigererordnungmitschrittweiten-kontrolle und ihreanwendung


auf waermeleitungsproblemerdquoComputing vol 6 no 1-2 pp 61ndash71 1970

[45] E Tlelo-Cuautle J M Munoz-Pacheco and J Martınez-Carballido ldquoFrequency scaling simulation of Chuarsquos circuitby automatic determination and control of step-sizerdquo AppliedMathematics and Computation vol 194 no 2 pp 486ndash4912007

[46] L Portero A Arraras and J C Jorge ldquoVariable step-size frac-tional step Runge-Kutta methods for time-dependent partialdifferential equationsrdquo Applied Numerical Mathematics vol 62no 10 pp 1463ndash1476 2012

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119894(119905)119877

119862+

minus

Figure 1 Nonlinear RC circuit

numerical data that allows circuit designers to explore alimited range of dynamics of nonlinear circuits

Consider the analysis of the nonlinear circuit depicted inFigure 1 as a case study [37] Let the branch relationship of thenonlinear capacitor to be

119907 (119902) = 1205721199023 (1)

where 119907 is the voltage 119902 is the charge of the capacitor and 120572 isa parameter of the capacitor

By applying Kirchhoff Laws we obtain the equation forthe transient

119889119902 (119905)

119889119905

+

1

119877

119907 (119902) = 119894 (119905) (2)

Therefore119889119902 (119905)

119889119905

+

120572

119877

1199023(119905) = 119894 (119905) (3)

If we consider the case for DC excitation then 119894(119905) = 119868resulting in

119889119902 (119905)

119889119905

+

120572

119877

1199023(119905) = 119868 119902 (0) = 0 (4)

3 Fundamentals of the Rational HomotopyPerturbation Method

The rational homotopy perturbation method RHPM [5 6]can be considered as a combination of the classical perturba-tion technique [38 39] and the homotopy (whose origin is inthe topology) [40ndash42] but not restricted to a small parameterlike traditional perturbation methods For example RHPMrequires neither small parameter nor linearization but onlyfew iterations to obtain accurate solutions

To figure out how RHPM method works consider ageneral nonlinear equation in the form

119860 (119906) minus 119891 (119903) = 0 119903 isin Ω (5)

with the following boundary conditions

119861(119906

120597119906

120597119906

) = 0 119903 isin Γ (6)

where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) is a known analytical function and Γ is thedomain boundary for Ω119860 can be divided into two operators119871 and 119873 where 119871 is linear and 119873 nonlinear from this laststatement (5) can be rewritten as

119871 (119906) + 119873 (119906) minus 119891 (119903) = 0 (7)

Generally a homotopy can be constructed in the form [1ndash3]119867(119907 119901) = (1 minus 119901) [119871 (119907) minus 119871 (1199060

)]

+ 119901 [119871 (119907) + 119873 (119907) minus 119891 (119903)] = 0

119901 isin [0 1] 119903 isin Ω

(8)

where119901 is a homotopy parameter whose values arewithin therange of 0 and 1 119906

0is the first approximation for the solution

of (6) that satisfies the boundary conditionsWhen 119901 rarr 0 (8) is reduced to

119871 (119907) minus 119871 (1199060) = 0 (9)

where operator 119871 possesses trivial solutionFor 119901 rarr 1 (8) is reduced to the original problem

119873(119907) + 119871 (119907) minus 119891 (119903) = 0 (10)Assuming that the solution for (8) can be written as a powerseries of 119901

119907 =

1199070+ 1199011199071+ 11990121199072+ sdot sdot sdot

1199080+ 1199011199081+ 11990121199082+ sdot sdot sdot

(11)

where 1199070 1199071 1199072 are unknown functions to be determined

by the RHPM and 1199080 1199081 1199082 are known analytic func-

tions of the independent variableSubstituting (11) into (8) and equating identical powers

of 119901 terms it is possible to obtain values for the sequence1199070 1199071 1199072

When119901 rarr 1 in (11) it yields in the approximate solutionfor (5) in the form

119906 = lim119901rarr1

(119907) =

1199070+ 1199071+ 1199072+ sdot sdot sdot

1199080+ 1199081+ 1199082+ sdot sdot sdot

(12)

Convergence of RHPMmethod is studied in [5 6]

4 Fundamentals of the Boubaker PolynomialsExpansion Scheme BPES

The Boubaker Polynomials Expansion Scheme BPES [17ndash36] is a resolution protocol which has been successfullyapplied to several applied-physics and mathematical prob-lems The BPES protocol ensures the validity of the relatedboundary conditions regardless of main equation featuresThe Boubaker Polynomials Expansion Scheme BPES is basedon the Boubaker polynomials first derivatives properties

119873

sum

119902=1

1198614119902(119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=0

= minus 2119873 = 0

119873

sum

119902=1

1198614119902 (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=119903119902

= 0

119873

sum

119902=1

1198891198614119902(119909)

119889119909

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=0

= 0

119873

sum

119902=1

1198891198614119902(119909)

119889119909

10038161003816100381610038161003816100381610038161003816100381610038161003816119909=119903119902

=

119873

sum

119902=1

119867119902

(13)


with

119867119899= 1198611015840

4119899(119903119899) = (

4119903119899[2 minus 119903

2

119899] times sum119899

119902=11198612

4119902(119903119899)

1198614(119899+1)

(119903119899)

+ 41199033

119899)

(14)




(1 minus 119901) (1199071015840minus 1199061015840

0) + 119901(119907

1015840+

120572

119877

1199073minus 119868) = 0 (15)



119907 = (1199070+ 1199011199071+ 11990121199072+ 11990131199073+ 11990141199074

+ 11990151199075+ 11990161199076+ 11990171199077)

times (1 + 11990111989611199093+ 119901211989621199096)

minus1

(16)

where 1199080= 1 119908

1= 11989611199093 and 119908

2= 11989621199096


0(0) = 0 119907

1(0) for the


tial equations

1199010 119907

1015840

0= 0 119907

0 (0) = 0

1199011 1199071015840

1minus 119868 +

120572

119877

1199073+ 21199071015840

011987011198833minus 3119883211990701198701= 0 119907

1 (0) = 0

(17)

Solving (17) yields

1199070= 1199060= 0

1199071= 119868119909

1199072= 11986811989611199094

(18)


119902 (119905) = lim119901rarr1

(119907)

= (119868119909 + (1198681198961minus (14) (120572119868

3119877)) 119909

4

+ (1198681198962minus (14) (1205721198961

1198683119877) + (328) (120572

211986851198772)) 1199097

minus (14) (12057211989621198683119877) 11990910)

times (1 + 11989611199093+ 11989621199096)

minus1

(19)


1= 36289 and

1198962= 4471843


119902 (119905) =

1

21198730

1198730

sum

119896=1

120582119896times

1198891198614119896(119903119896119905)

119889119905

(20)


1

21198730

1198730

sum

119896=1

120582119896119903119896

11988921198614119896(119903119896119905)

1198892119905

+

120572

81198733

0119877

[

1198730

sum

119896=1

120582119896

1198891198614119896(119903119896119905)

119889119905

]

3

= 119868

(21)


119896|119896=11198730


0)3arithmetical



119902 (119905) =

1

21198730

1198730

sum

119896=1

120582(sol)119896times

1198891198614119896(119903119896119905)

119889119905

(22)





119905 q(t) (RKF45) RHPM BPES1198730= 37 119873

0= 107 119873

0= 113

000 000000 000000 000000 000000 000000001 0098066 0098070 009436 009456 009827002 17475 017481 017123 017395 017513003 21314 021388 021051 021266 021394004 022656 022802 022669 022557 022775005 023054 023005 022678 023009 023076006 023165 023235 022346 023056 023246



0(1198730lt 120) since


7 Conclusion



References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










with

119867119899= 1198611015840

4119899(119903119899) = (

4119903119899[2 minus 119903

2

119899] times sum119899

119902=11198612

4119902(119903119899)

1198614(119899+1)

(119903119899)

+ 41199033

119899)

(14)




(1 minus 119901) (1199071015840minus 1199061015840

0) + 119901(119907

1015840+

120572

119877

1199073minus 119868) = 0 (15)



119907 = (1199070+ 1199011199071+ 11990121199072+ 11990131199073+ 11990141199074

+ 11990151199075+ 11990161199076+ 11990171199077)

times (1 + 11990111989611199093+ 119901211989621199096)

minus1

(16)

where 1199080= 1 119908

1= 11989611199093 and 119908

2= 11989621199096


0(0) = 0 119907

1(0) for the


tial equations

1199010 119907

1015840

0= 0 119907

0 (0) = 0

1199011 1199071015840

1minus 119868 +

120572

119877

1199073+ 21199071015840

011987011198833minus 3119883211990701198701= 0 119907

1 (0) = 0

(17)

Solving (17) yields

1199070= 1199060= 0

1199071= 119868119909

1199072= 11986811989611199094

(18)


119902 (119905) = lim119901rarr1

(119907)

= (119868119909 + (1198681198961minus (14) (120572119868

3119877)) 119909

4

+ (1198681198962minus (14) (1205721198961

1198683119877) + (328) (120572

211986851198772)) 1199097

minus (14) (12057211989621198683119877) 11990910)

times (1 + 11989611199093+ 11989621199096)

minus1

(19)


1= 36289 and

1198962= 4471843


119902 (119905) =

1

21198730

1198730

sum

119896=1

120582119896times

1198891198614119896(119903119896119905)

119889119905

(20)


1

21198730

1198730

sum

119896=1

120582119896119903119896

11988921198614119896(119903119896119905)

1198892119905

+

120572

81198733

0119877

[

1198730

sum

119896=1

120582119896

1198891198614119896(119903119896119905)

119889119905

]

3

= 119868

(21)


119896|119896=11198730


0)3arithmetical



119902 (119905) =

1

21198730

1198730

sum

119896=1

120582(sol)119896times

1198891198614119896(119903119896119905)

119889119905

(22)





119905 q(t) (RKF45) RHPM BPES1198730= 37 119873

0= 107 119873

0= 113

000 000000 000000 000000 000000 000000001 0098066 0098070 009436 009456 009827002 17475 017481 017123 017395 017513003 21314 021388 021051 021266 021394004 022656 022802 022669 022557 022775005 023054 023005 022678 023009 023076006 023165 023235 022346 023056 023246



0(1198730lt 120) since


7 Conclusion



References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119905 q(t) (RKF45) RHPM BPES1198730= 37 119873

0= 107 119873

0= 113

000 000000 000000 000000 000000 000000001 0098066 0098070 009436 009456 009827002 17475 017481 017123 017395 017513003 21314 021388 021051 021266 021394004 022656 022802 022669 022557 022775005 023054 023005 022678 023009 023076006 023165 023235 022346 023056 023246



0(1198730lt 120) since


7 Conclusion



References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article approximation for transient of nonlinear...

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