Springer Proceedings in Mathematics & Statistics Sergei Silvestrov Milica Rančić Editors Engineering Mathematics I Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering Springer Proceedings in Mathematics & Statistics Volume 178 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 č ć Sergei Silvestrov Milica Ran i (cid:129) Editors Engineering Mathematics I Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering 123 Editors SergeiSilvestrov Milica Rančić Division of AppliedMathematics Division of AppliedMathematics The Schoolof Education, Culture The Schoolof Education, Culture andCommunication andCommunication Mälardalen University Mälardalen University Västerås Västerås Sweden Sweden ISSN 2194-1009 ISSN 2194-1017 (electronic) SpringerProceedings in Mathematics& Statistics ISBN978-3-319-42081-3 ISBN978-3-319-42082-0 (eBook) DOI 10.1007/978-3-319-42082-0 LibraryofCongressControlNumber:2016951670 Mathematics Subject Classification (2010): 00A69, 60-XX, 65-XX, 74-XX, 78-XX, 78A25, 78A50, 91Gxx,00B15,00B10 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This book highlights the latest advances in engineering mathematics with a main focus on the mathematical models, structures, concepts, problems and computa- tional methods and algorithms most relevant for applications in modern technolo- giesandengineering.Inparticular,itfeaturesmathematicalmethodsandmodelsof applied analysis, probability theory, differential equations, tensor analysis and computational modelling used in applications to important problems concerning electromagnetics, antenna technologies, fluid dynamics, material and continuum physics and financial engineering. Theindividualchapterscoverboththeoryandapplications,andincludeawealth offigures, schemes, algorithms, tables and results of data analysis and simulation. Presenting new methods and results, reviews of cutting-edge research, and open problems for future research, they equip readers to develop new mathematical methodsandconceptsoftheirown,andtofurthercompareandanalysethemethods and results discussed. Chapters 1–10 are concerned with applied mathematics methods and models applied in electrical engineering, electromagnetism and antenna technologies. Chapter 1 by Dragan Poljak is concerned with applications of integro-differential equations and numerical analysis methods to the analysis of grounding systems important in the design of lightning protection systems. The analysis of horizontal groundingelectrodeshasbeencarriedoutusingtheantennatheoryapproachinthe frequency and time domain respectively. The formulation is based on the corre- spondingspace-frequencyandspace-timePocklingtonintegro-differentialequations. The integro-differential relationships are numerically handled via the Galerkin– BubnovschemeoftheIndirectBoundaryElementMethod.Frequencydomainand timedomainanalysisisillustratedbycomputationalexamples.Chapter2bySilvestar ŠesnićandDraganPoljakdealswiththeuseofanalyticalmethodsforsolvingvarious integro-differentialequationsinelectromagneticcompatibility,withtheemphasison thefrequencyandtimedomainsolutionsofthethin-wireconfigurationsburiedina lossy ground. Solutions in the frequency domain are carried out via certain mathe- maticalmanipulationswiththecurrentfunctionappearingincorrespondingintegral equations.Ontheotherhand,analyticalsolutionsinthetimedomainareundertaken v vi Preface usingtheLaplacetransformandCauchyresiduetheorem.Obtainedanalyticalresults are compared to those calculated using the numerical solution of the frequency domain Pocklington equation, where applicable. Also, an overview of analytical solutionstotheGrad-Shafranovequationfortokamakplasmaisprovided.InChap.3 byMilicaRančić,RadoslavJankoski,SergeiSilvestrovandSlavoljubAleksić,anew simpleapproximationthatcanbeusedformodellingonetypeofSommerfeldinte- gralstypicallyoccurringintheexpressionsthatdescribesourcesburiedinthelossy ground, is proposed. The proposed approximation has a form of a weighted expo- nentialfunctionwithanadditionalcomplexconstantterm.Thederivationprocedure for this approximation is explained in detail, and the validation is supplied by applyingittotheanalysisofabareconductorfedinthecentreandimmersedinthe lossy ground at arbitrary depth. In Chap. 4 by Radoslav Jankoski, Milica Rančić, Vesna Arnautovski-Toseva and Sergei Silvestrov, high frequency analysis of a horizontaldipoleantennaburiedinlossygroundisperformed.Thesoilistreatedasa homogenous half-space of known electrical parameters. The authors compare the range of applicability of two forms of transmission line models, a hybrid circuit method,andapoint-matchingmethodinthiscontext.Chapter5byPushpanjaliG. Metri pertains to an experimental implementation and evaluation of geometrically designedantennas.Anoveldesignforanequilateraltriangularmicrostripantennais proposed and tested. The antenna is designed, fabricated and tested for single and multibandoperation.Atheoryforsuchantennasbasedontheexperimentalresultsis also considered. Chapter 6 by Nenad Cvetković, Miodrag Stojanović, Dejan Jovanović,AleksaRistić,DraganVučkovićandDejanKrstićprovidesabriefreview of the derivation of two groups of approximate closed form expressions for the electrical scalar potential Green’s functions that originates from the current of the pointgroundelectrodeinthepresenceofasphericalgroundinhomogeneity,proposes approximatesolutionsandconsidersknownexactsolutionsinvolvinginfiniteseries sums.Theexactsolutionisreorganizedinordertofacilitatecomparisontotheclosed formsolutions,andtoestimatetheerrorintroducedbytheapproximatesolutions,and errorestimationisperformedcomparingtheresultsfortheelectricalscalarpotential obtainedapplyingtheapproximateexpressionsandtheaccuratecalculations.Thisis illustratedbyanumberofnumericalexperiments.InChap.7byMarioCvetkovićand Dragan Poljak, the electromagnetic thermal dosimetry model for the human brain exposedtoelectromagneticradiationisdeveloped.Theelectromagneticmodelbased onthesurfaceintegralequationformulationisderivedusingtheequivalencetheorem forthecaseofalossyhomogeneousdielectricbody.Thethermaldosimetrymodel of the brain is based on the form of Pennes’ equation of heat transfer in biological tissue.Thenumericalsolutionoftheelectromagneticmodeliscarriedoutusingthe Method of Moments, while the bioheat equation is solved using the finite element method. The electromagnetic thermal model developed here has been applied in internaldosimetryofthehumanbraintoassesstheabsorbedelectromagneticenergy andconsequenttemperaturerise.InChap.8byMirjanaPerić,SašaIlićandSlavoljub Aleksić, multilayered shielded structures are analysed using the hybrid boundary elementmethod.Theapproachisbasedontheequivalentelectrodesmethod,onthe point-matchingmethodforthepotentialoftheperfectelectricconductorelectrodes Preface vii andforthenormalcomponentofelectricfieldattheboundarysurfacebetweenany twodielectriclayers.Inordertoverifytheobtainedresults,theyhavebeencompared with the finite element method and results that have already been reported in the literature. In Chap. 9 by Vesna Javor, new engineering modified transmission line models of lightning strokes are presented. The computational results for lightning electromagnetic field at various distances from lightning discharges are in good agreement with experimental results that are usually employed for validating elec- tromagnetic, engineering and distributed-circuit models. Electromagnetic theory relations,thin-wireantennaapproximationofalightningchannelwithouttortuosity andbranching,aswellastheassumptionofaperfectlyconductingground,areused for electric and magnetic field computation. An analytically extended function, suitableforapproximatingchannel-basecurrentsinthesemodels,isalsoconsidered. Chapter10byKarlLundengård,MilicaRančić,VesnaJavorandSergeiSilvestrov explores the properties of the multi-peaked analytically extended function for approximationoflightningdischargecurrents.Accordingtoexperimentalresultsfor lightningdischargecurrents,theyareclassifiedintowaveshapesrepresentingthefirst positive,firstandsubsequentnegativestrokes,andlong-strokes.Aclassofanalyti- callyextendedfunctionsispresentedandusedforthemodellingoflightningcurrents. The basic properties of this function with a finite number of peaks are examined. A general framework for estimating the parameters of the analytically extended functionusingtheMarquardtleast-squaresmethodforawaveformwithanarbitrary (finite)numberofpeaksaswellasforthegivenchargetransferandspecificenergyis describedandusedtofindparametersforsomecommonsingle-peakwaveforms. In turn, Chaps. 11–15 address the mathematical modelling and optimisation of technological processes with applications of partial differential equations, ordinary differential equations, numerical analysis, perturbation methods and special func- tions in fluid mechanics models that are important in engineering applications and technologies. Chapter 11 by Jüri Olt, Olga Liivapuu, Viacheslav Maksarov, Alexander Liyvapuu and Tanel Tärgla, is devoted to the mathematical modelling of the process system which paves the way for research on the selection and optimisation of machining conditions. The subject of this chapter is the method of dynamic process approximation method, which makes it possible to analyse the behaviour of the machining process system in the process of chip formation at a sufficient level of accuracy. In Chap. 12 by Prashant G. Metri, Veena M. Bablad, Pushpanjali G. Metri, M. Subhas Abel and Sergei Silvestrov, a mathematical analysis is carried out to describe mixed convection heat transfer in magnetohy- drodynamicnon-Darcianflowduetoanexponentialstretchingsheetembeddedina porous medium in the presence of a non-uniform heat source/sink. Approximate analytical similarity solutions of the highly nonlinear momentum and energy equationsareobtained.Thegoverningsystemofpartialdifferentialequationsisfirst transformed into a system of nonlinear ordinary differential equations using simi- larity transformation. The transformed equations are nonlinear coupled differential equationsandare solved veryefficiently by employinga fifth order Runge–Kutta– Fehlberg method with shooting technique for various values of the governing parameters. The numerical solutions are obtained by considering an exponential viii Preface dependent stretching velocity and prescribed boundary temperature on the flow directional coordinate. The computed results are compared with the previously publishedworkonvariousspecialcasesoftheproblemandareingoodagreement with the earlier studies. The effects of various physical parameters, such as the Prandtl number, the Grashof number, the Hartmann number, porous parameter, inertia coefficient and internal heat generation on flow and heat transfer charac- teristics are presented graphically to reveal a number of interesting aspects of the physical parameter. Chapter 13 by Prashant G. Metri, M. Subhas Abel and Sergei Silvestrov presents an analysis of the boundary layer flow and heat transfer over a stretching sheet due to nanofluids with the effects of the magnetic field, Brownian motion, thermophoresis, viscous dissipation and convective boundary conditions. The transport equations used in the analysis take into account the effect of Brownian motion and thermophoresis parameters. The highly nonlinear partial differential equations governing flow and heat transport are simplified using simi- larity transformation, and the ordinary differential equations obtained are solved numericallyusingtheRunge–Kutta–FehlbergandNewton–Raphsonschemesbased on the shooting method. The solutions for velocity temperature and nanoparticle concentration depend on parameters such as Brownian motion, thermophoresis parameter, magnetic field and viscous dissipation, which have a significant influ- ence on controlling of the dynamics. In Chap. 14 by Jawali C. Umavathi, KuppalapalleVajravelu,PrashantG.MetriandSergeiSilvestrov,thelinearstability of Maxwell fluid-nanofluid flow in a saturated porous layer is examined theoreti- callywhenthewallsoftheporouslayersaresubjectedtotime-periodictemperature modulations. A modified Darcy-Maxwell model is used to describe the fluid motion,andthenanofluidmodelusedincludestheeffectsoftheBrownianmotion. The thermal conductivity and viscosity are considered to be dependent on the nanoparticle volume fraction. A perturbation method that is based on a small amplitudeofanappliedtemperaturefieldisusedtocomputethecriticalvalueofthe Rayleigh number and the wave number. The stability of the system, characterized by a critical Rayleigh number, is calculated as a function of the relaxation parameter, the concentration Rayleigh number, the porosity parameter, the Lewis number, the heat capacity ratio, the Vadász number, the viscosity parameter, the conductivity variation parameter, and the frequency of modulation. Three types of temperaturemodulationsareconsidered,andtheeffectsofallthreetypesarefound todestabilizethesystemascomparedtotheunmodulatedsystem.Chapter15byJ. Pratap Kumar, Jawali C. Umavathi, Prashant G. Metri and Sergei Silvestrov is devoted to a study of magneto-hydrodynamic flow in a vertical double passage channel taking into account the presence of the first order chemical reaction. The governing equations are solved by using a regular perturbation technique valid for smallvaluesoftheBrinkmannumberandadifferentialtransformmethodvalidfor all values of the Brinkman number. The results are obtained for velocity, temper- ature and concentration. The effects of various dimensionless parameters such as the thermal Grashof number, mass Grashof number, Brinkman number, first order chemical reaction parameter, and Hartman number on the flow variables are dis- cussed and presented graphically for open and short circuits. The validity of Preface ix solutions obtained by the differential transform method and regular perturbation method are in good agreement for small values of the Brinkman number. Further, the effects of governing parameters on the volumetric flow rate, species concen- tration, total heat rate, skin friction and Nusselt number are also observed and tabulated. Chapters 16–18 are concerned with mathematical methods of stochastic pro- cesses, probability theory, differential geometry, tensor analysis, representation theory, differential equations, algebra and computational mathematics for applica- tions in materials science and financial engineering. In Chap. 16 by Anatoliy Malyarenko and Martin Ostoja-Starzewski, a random field model of the 21-dimensional elasticity tensor is considered, and representation theory is used to obtain the spectral expansion of the model in terms of stochastic integrals with respecttorandommeasures.Themotivationfortreatingthistensorasarandomfield isthatnearlyallthematerialsencounteredinnatureaswellthoseproducedbyman, except for the purest crystals, possess some degree of disorder or inhomogeneity. At the same time, elasticity is the starting point for any solid mechanics model. Chapter17byAnatoliyMalyarenko,JanRömanandOskarSchybergisdevotedto mathematicalmodelsforcatastrophebondswhichareanimportantinstrumentinthe fieldsoffinance,insuranceandreinsurance,wherethenaturalriskindexisdescribed by the Merton jump-diffusion while the risk-free interest rate is governed by the Hull–Whitestochasticdifferentialequation.Thesensitivitiesofthebondpricewith respect to the initial condition, volatility of the diffusion component, and jump amplitudearecalculatedusingtheMalliavincalculusapproach.Lastly,inChap.18 byBetuelCanhanga,AnatoliyMalyarenko,Jean-PaulMuraraandSergeiSilvestrov, stochastic volatilities models for pricing European options are considered as a response to the weakness of the constant volatility models, which have not suc- ceeded in capturing the effects of volatility smiles and skews. A model with two-factorstochasticvolatilitieswherethecorrelationbetweentheunderlyingasset priceandthevolatilitiesvariesrandomlyisconsidered,andthefirstorderasymptotic expansionmethods areused todetermine the price of European options. The book consists of carefully selected and refereed contributed chapters cov- ering research developed as a result of a focused international seminar series on mathematics and applied mathematics, as well as three focused international research workshops on engineering mathematics organised by the Research Environment in Mathematics and Applied Mathematics at Mälardalen University from autumn 2014 to autumn 2015: the International Workshop on Engineering Mathematics for Electromagnetics and Health Technology; the International Workshop on Engineering Mathematics, Algebra, Analysis and Electromagnetics; andthe1stSwedish-EstonianInternationalWorkshoponEngineeringMathematics, Algebra, Analysis and Applications. This book project has been realised thanks to the strategic support offered by Mälardalen University for the research and research education in Mathematics, which is conducted by the research environment Mathematics and Applied Mathematics (MAM), in theestablished researchareaof Educational Sciences and MathematicsattheSchoolofEducation,CultureandCommunicationatMälardalen