Table Of ContentSpringer Optimization and Its Applications 159
Nicholas J. Daras
Themistocles M. Rassias Editors
Computational
Mathematics
and Variational
Analysis
Springer Optimization and Its Applications
Volume 159
SeriesEditors
PanosM.Pardalos ,UniversityofFlorida
MyT.Thai ,UniversityofFlorida
HonoraryEditor
Ding-ZhuDu,UniversityofTexasatDallas
AdvisoryEditors
J.Birge,UniversityofChicago
S.Butenko,TexasA&M
F.Giannessi,UniversityofPisa
S.Rebennack,KarlsruheInstituteofTechnology
T.Terlaky,LehighUniversity
Y.Ye,StanfordUniversity
AimsandScope
Optimizationhascontinuedtoexpandinalldirectionsatanastonishingrate.New
algorithmicandtheoreticaltechniquesarecontinuallydevelopingandthediffusion
into other disciplines is proceeding at a rapid pace, with a spot light on machine
learning, artificial intelligence, and quantum computing. Our knowledge of all
aspects of the field has grown even more profound. At the same time, one of the
most striking trends in optimization is the constantly increasing emphasis on the
interdisciplinary nature of the field. Optimization has been a basic tool in areas
not limited to applied mathematics, engineering, medicine, economics, computer
science,operationsresearch,andothersciences.
The series Springer Optimization and Its Applications (SOIA) aims to publish
state-of-the-art expository works (monographs, contributed volumes, textbooks,
handbooks)thatfocusontheory,methods,andapplicationsofoptimization.Topics
coveredinclude,butarenotlimitedto,nonlinearoptimization,combinatorialopti-
mization,continuousoptimization,stochasticoptimization,Bayesianoptimization,
optimalcontrol,discreteoptimization,multi-objectiveoptimization,andmore.New
totheseriesportfolioincludeWorksattheintersectionofoptimizationandmachine
learning,artificialintelligence,andquantumcomputing.
Volumes from this series are indexed by Web of Science, zbMATH, Mathematical
Reviews,andSCOPUS.
Moreinformationaboutthisseriesathttp://www.springer.com/series/7393
Nicholas J. Daras • Themistocles M. Rassias
Editors
Computational Mathematics
and Variational Analysis
Editors
NicholasJ.Daras ThemistoclesM.Rassias
DepartmentofMathematics DepartmentofMathematics
HellenicMilitaryAcademy NationalTechnicalUniversityofAthens
VariAttikis,Greece Athens,Greece
ISSN1931-6828 ISSN1931-6836 (electronic)
SpringerOptimizationandItsApplications
ISBN978-3-030-44624-6 ISBN978-3-030-44625-3 (eBook)
https://doi.org/10.1007/978-3-030-44625-3
©SpringerNatureSwitzerlandAG2020
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Preface
Computational Mathematics and Variational Analysis discusses a broad spectrum
of computational methods and theories of an interdisciplinary nature. The book
chapters are devoted to the study of research problems from pure and applied
mathematical sciences.Amongthetopics treatedaresubjects ofcalculus ofvaria-
tions,optimizationtheory,operationsresearch,gametheory,differentialequations,
functional analysis, operator theory, approximation theory, numerical analysis,
asymptoticanalysis,andengineering.
Thebookpresents28paperswrittenbyeminentscientistsfromtheinternational
mathematical community. These works are devoted to a number of significant
advances in both classical and modern mathematical problems including algo-
rithms for difference of monotone operators, variational inequalities in semi-inner
product spaces, function variational principles and normed minimizers, equilibria
of parametrized N-player nonlinear games, multi-symplectic numerical schemes
for differential equations, time-delay multi-agent systems, computational methods
in nonlinear design of experiments, unsupervised stochastic learning, asymptotic
statisticalresults,global–localtransformation,scatteringrelationsofelasticwaves,
generalized Ostrowski and trapezoid type rules, numerical approximation, Szász–
Durrmeyer operators and approximation, integral inequalities, behavior of the
solutionsoffunctionalequations,functionalinequalitiesincomplexBanachspaces,
andfunctionalcontractionsinmetricspaces.
Thepresentationofconceptsandmethodsfeaturedinthisbookmakeitaninvalu-
able reference for both professors and researchers in mathematics, engineering,
physics,andeconomics.
Wewouldliketoexpressourwarmthankstothecontributorsofchaptersaswell
astothestaffofSpringerfortheirexcellentcollaborationforthispublication.
VariAttikis,Greece NicholasJ.Daras
Athens,Greece ThemistoclesM.Rassias
v
Contents
Scattering Relations of Elastic Waves by a Multi-Layered
ThermoelasticBody.............................................................. 1
EvageliaS.Athanasiadou,VassiliosSevroglou,andStefaniaZoi
BlindTransferofPersonalDataAchievingPrivacy.......................... 25
AlexisBonnecazeandRobertRolland
Equilibria of Parametrized N-Player Nonlinear Games Using
InequalitiesandNonsmoothDynamics........................................ 33
MonicaG.CojocaruandFatimaEtbaigha
Numerical Approximation of a Class of Time-Fractional
DifferentialEquations........................................................... 55
AleksandraDelic´,BoškoS.Jovanovic´ andSandraŽivanovic´
ApproximatingtheIntegralofAnalyticComplexFunctionson
PathsfromConvexDomainsinTermsofGeneralizedOstrowski
andTrapezoidTypeRules ...................................................... 81
SilvestruSeverDragomir
Szász–DurrmeyerOperatorsandApproximation............................ 107
VijayGupta
Leibniz’sRuleandFubini’sTheoremAssociatedwithaGeneral
QuantumDifferenceOperator ................................................. 121
AlaaE.Hamza,EnasM.Shehata,andPraveenAgarwal
Some New Ostrowski Type Integral Inequalities via General
FractionalIntegrals.............................................................. 135
ArtionKashuriandThemistoclesM.Rassias
SomeNewIntegralInequalitiesviaGeneralFractionalOperators........ 153
ArtionKashuri,ThemistoclesM.Rassias,andRozanaLiko
vii
viii Contents
AsymptoticStatisticalResults:TheoryandPractice......................... 177
ChristosP.KitsosandAmílcarOliveira
OntheComputationalMethodsinNon-linearDesignofExperiments.... 191
ChristosP.KitsosandAmílcarOliveira
GeometricDerivationandAnalysisofMulti-SymplecticNumerical
SchemesforDifferentialEquations............................................ 207
OdysseasKosmas,DimitriosPapadopoulos,andDimitriosVlachos
Additive(ρ ,ρ )-FunctionalInequalitiesinComplexBanachSpaces..... 227
1 2
JungRyeLee,ChoonkilPark,andThemistoclesM.Rassias
FirstStudyforRampSecretSharingSchemesThroughGreatest
CommonDivisorofPolynomials ............................................... 247
GerasimosC.Meletiou,DimitriosS.Triantafyllou,
andMichaelN.Vrahatis
FromRepresentationTheoremstoVariationalInequalities ................ 261
MuhammadAslamNoorandKhalidaInayatNoor
UnsupervisedStochasticLearningforUserProfiles......................... 279
NikolaosK.Papadakis
On the Solution of Boundary Value Problems for Ordinary
DifferentialEquationsofOrdernand2nwithGeneralBoundary
Conditions......................................................................... 299
I.N.Parasidis,E.Providas,andS.Zaoutsos
Additive-QuadraticFunctionalInequalities .................................. 315
ChoonkilPark,ThemistoclesM.Rassias
Time-DelayMulti-AgentSystemsforaMovableCloud..................... 343
RabhaW.Ibrahim
TheGlobal-LocalTransformation ............................................. 357
KonstantinosA.Raftopoulos
OnAlgorithmsforDifferenceofMonotoneOperators...................... 381
MaedeRamazannejad,MohsenAlimohammady,andCarloCattani
A Mathematical Model for Simulation of Intergranular
μ-CapacitanceasaFunctionofNeckGrowthinCeramicSintering....... 403
BranislavM.Randjelovic´ andZoranS.Nikolic´
VariationalInequalitiesinSemi-innerProductSpaces...................... 421
NabinK.Sahu,OuaylChadli,andR.N.Mohapatra
ResultsConcerningCertainLinearPositiveOperators ..................... 441
DanyalSoybas¸andNehaMalik
Contents ix
BehavioroftheSolutionsofFunctionalEquations........................... 465
IoannisP.StavroulakisandMichailA.Xenos
TheIsometryGroupofn-DimensionalEinsteinGyrogroup................ 505
TeerapongSuksumran
FunctionVariationalPrinciplesandNormedMinimizers................... 513
MihaiTurinici
Nadler-LiuFunctionalContractionsinMetricSpaces ...................... 537
MihaiTurinici
Scattering Relations of Elastic Waves by
a Multi-Layered Thermoelastic Body
EvageliaS.Athanasiadou,VassiliosSevroglou,andStefaniaZoi
Abstract The scattering problem of a time-harmonic dependent plane elastic
wave by a multi-layered thermoelastic body in an isotropic and homogeneous
elasticmediumisconsidered.Thedirectscatteringproblemisformulated.Integral
representationsforthetotalexteriorelasticfieldandthetotalinteriorthermoelastic
fields as well as expressions for the far-field patterns are obtained containing the
physicalparametersoftheinteriorthermoelasticlayers.Areciprocitytypetheorem,
a general type scattering theorem and an optical type theorem for plane wave
incidencearepresentedandproved.
MathematicalSubjectClassification 35P25,74J20,35J57,74B05
1 Introduction
Inthispaper,thescatteringproblemoftime-harmonicdependentplaneelasticwaves
by a multi-layered thermoelastic body is considered. A good introduction to the
theoryofthermoelasticitycanbefoundinthebooks[15]and[16].Theexpressions
ofthethermoelasticfar-fieldpatternsandcrosssectionshavebeenpresentedin[11].
The low-frequency theory for thermoelastic wave scattering from an impenetrable
scattererhasbeenstudiedin[12].Uniquenessandexistenceresultsforthescattering
ofthermoelasticwavesfromascreenhavebeenpresentedin[5]andforanisotropic
bodies with cuts in [14]. The scattering problem of elastic waves for a penetrable
thermoelastic body and for a two-layered thermoelastic body has been studied in
E.S.Athanasiadou·S.Zoi((cid:2))
DepartmentofMathematics,NationalandKapodistrianUniversityofAthens,Zographou,
Athens,Greece
e-mail:[email protected];[email protected]
V.Sevroglou
DepartmentofStatistics&InsuranceScience,UniversityofPiraeus,Piraeus,Greece
e-mail:[email protected]
©SpringerNatureSwitzerlandAG2020 1
N.J.Daras,T.M.Rassias(eds.),ComputationalMathematicsandVariational
Analysis,SpringerOptimizationandItsApplications159,
https://doi.org/10.1007/978-3-030-44625-3_1
2 E.S.Athanasiadouetal.
[13]and[6],respectively.Scatteringrelationsforplaneelasticwavesforanelastic
scattererarepresentedin[8].Correspondingresultsforapenetrablethermoelastic
scattererarepresentedin[4].Scatteringrelationsforsphericalelasticwavesforan
elastic scatterer in 2D and 3D are treated in [2] and [1], respectively. Moreover,
scattering theorems for acoustic and electromagnetic waves can be found in [10]
and[3]aswellasforcompletedyadicfieldsin[9].
In Section 2, we formulate the scattering problem. In Section 3, we present
the integral representations of the total exterior elastic and the total interior
thermoelastic fields as well as integral representations for the three normalized
spherical elastic far-field patterns. Moreover, we present and prove alternative
integralrepresentationsforthetotalexteriorelasticfieldaswellasfortheelasticfar-
field patterns, involving volume integrals over all thermoelastic layers and surface
integrals over all surfaces, containing the displacement and temperature parts of
the total interior thermoelastic fields as well as the elastic and thermal parameters
of each thermoelastic layer. Also, we give the formulas of the scattering cross
section,theabsorptioncrosssectionandtheextinctioncrosssection.InSection4,
wepresentandprovescatteringrelationsforourscatteringproblem,corresponding
tothereciprocitytypetheorem,thegeneralscatteringtheoremandtheopticaltype
theoremforplanewaveincidence.
2 Formulation
LetD beabounded,connectedandsmoothsubsetofR3 whoseinteriorisdivided
bymeansofclosedandnon-intersectingsurfacesS (j =1,2,...,N−1)ofclass
j
C2 intolayersDj (j = 1,2,...,N)withSj−1 = ∂Dj−1∩∂Dj whilethesurface
Sj−1 surrounds Sj. We refer to D as the multi-layered thermoelastic scatterer and
wedenoteitsboundaryofclassC2withS .Moreover,wedenotewithD =R3\D
0 0
theelasticmediumofpropagationandwithnˆ theoutwardunitnormalvectorateach
pointofanysurfaceS forj =0,1,...,N −1(Figure1).Therefore:
j
(cid:2) (cid:3) (cid:2) (cid:3)
D = ∪Nj=1Dj ∪ ∪Nj=−11Sj , D0 =R3\D, ∂D =S0, ∂Dj =Sj−1∪Sj,
∂DN =SN−1. (1)
for j = 1,2,...,N −1. The interior of the medium of propagation D is filled
0
up by an elastic material with elastic properties described by the positive Lamé
constants λ(0),μ(0) and the mass density ρ(0) and the interior of the layer D
j
(j = 1,2,...,N) is filled up by a thermoelastic material which is consisted
by elastic properties described by the positive Lamé constants λ(j),μ(j) and the
mass density ρ(j) as well as by thermal properties described by the coefficient of
thermal diffusivity κ(j). We consider that the above domains D and D (j =
0 j
1,2,...,N) are homogeneous and isotropic and that strong ellipticity conditions
μ(0) >0, λ(0)+2μ(0) >0andμ(j) >0, λ(j)+2μ(j) >0,forj =1,2,...,N,
