A Galerkin Finite Element Method for Two-Point Boundary Value Problems of Ordinary Differential Equations
|
|
- Dominick Pope
- 6 years ago
- Views:
Transcription
1 Applied ad Computatioal Matematics 5; 4(: Publised olie Mac 9, 5 (ttp:// doi:.648/j.acm.54.5 ISS: (Pit; ISS: (Olie A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios Getia Zavalai Faculty of Matematics ad Pysics Egieeig Polytecic Uivesity of Tiaa, Albaia addess: zavalaigetia@otmail.com To cite tis aticle: Getia Zavalai. A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios. Applied ad Computatioal Matematics. Vol. 4, o., 5, pp doi:.648/j.acm.54.5 Abstact: I tis pape, we peset a ew metod fo solvig two-poit bouday value poblem fo cetai odiay diffeetial equatio. Te two poit bouday value poblems ave geat impotace i cemical egieeig, deflectio of beams etc. I tis study, Galeki fiite elemet metod is developed fo iomogeeous secod-ode odiay diffeetial equatios. Seveal examples ae solved to demostate te applicatio of te fiite elemet metod. It is sow tat te fiite elemet metod is simple, accuate ad well beaved i te pesece of sigulaities. Keywods: Exact Solutio, Two-Poit Value Bouday Poblem, Fiite Elemet Metod. Itoductio Two-poit bouday-value poblems i odiay diffeetial equatios occu i may baces of pysics; examples iclude te two-dimesioal, icompessible, oedimesioal eat tasfe, bouday laye equatios, etc. Te coespodig odiay diffeetial equatios ca be oliea o liea but wit complex coefficiets. If te diffeetial equatio is oliea o liea but wit complex coefficiets, a closed fom aalytic solutio is, i geeal, difficult to obtai, if ot possible. Teefoe, a umeical solutio is sougt. May eseaces ave developed umeical tecique to study te umeical solutio of two poit bouday value poblems. Villadse ad Stewat [5] poposed solutio of bouday value poblem by otogoal collocatio metod. Jag [6] poposed te solutio of twopoit bouday value poblem by te exteded Adomia decompositio metod. Te Galeki-fiite elemet metod is well kow umeical tecique fo te umeical solutio of diffeetial equatios. Doga [7] poposed te Galekifiite elemet appoac fo te umeical solutios of Buges equatio. Segupta et al. [8] caied out Gakeki fiite elemet metods fo wave poblems. Kaeko et al. [9] discussed te Discotiuous Galeki-fiite elemet metod fo paabolic poblems. EI-Gebeily et al. [] studied te fiite elemet- Galeki metod fo sigula self-adjoit diffeetial equatios. Sama et al. [] poposed Galekifiite Elemet Metods fo umeical solutio of advectio- diffusio equatio. Oa [] poved te asymptotic covegece of te solutio of a paabolic equatio by usig two metods amely, te Galeki metod expessed i tems of liea splies ad te Fiite Elemet Collocatio metod expessed by cubic splie basis fuctios. I tis pape, we coside te followig iomogeeous secod ode diffeetial equatio wee u ( x + p( xu ( x + q( xu ( x = f( x, < x < β u( = u( β = p x ( [, β] (. =C p( x λ > i [, β ], q( x =C [, β], q( x o [, β] ad f x ( =C [, β] We assume tat poblem (. as a uique solutio u( x I te peset wok, we use Galeki-fiite elemet metod fo te umeical solutio of two poit bouday value poblems. Te appoac is simple ad effective. Te emaiig pat of te aticle is ogaised as follows. I Sectio, we sall fist efomulate (. as a vaiatioal poblem i te space vaiablesx.we sall te defie a Galeki appoximatio u( x to te solutio of (. by equiig tat ulie i a fiite-dimesioal space of fuctios, also a eo estimate is give. Te Full Discetized system aisig fom eite of te spatial discetisatios is give i Sectio 3. I sectio 4 of tis pape, we sall make some diect applicatios of appoximatio teoy to some test
2 65 Getia Zavalai: A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios poblems. Fially, Sectio 5 cocludes te aticle wit fial emaks. Fomulatio of te Vaiatioal Poblem ad Galeki Appoximatios Tis poblem may also be stated i weak fom: fid u H ( [, β] suc tat wee β Θ ( u, = ( f, fo w H ([, β ] (. dw du du Θ ( u, = β + wp( x + wq( x u dx dx dx dx, u, w = uwdx (.* Te (stadad Galeki metod fo appoximatig te solutio u of (. amouts to costuctig a family of fiite dimesioal subspaces { S } < <, ad seekig u S satisfyig te liea system of equatios Θ ( u, χ = ( f, χ fo χ S (. We sall assume tat te data ae suc tat te uique solutio u of (. belogs to u ( H ( H ( Ω Ω ad satisfies te elliptic egulaity estimate tat fo some C >, idepedet of f ad uwe ave u C f (.3 Ude ou ypoteses, a uique solutio u of (. exists ad satisfies if u u C u χ x S (.4 fo some costat C idepedet of. Te existeceuiquess of u S is guaateed by Lax-Milgam teoem applied to te Hilbet space ( S, te poof of Lax- Milgam teoem is give i Appedix A. Assumig tat if { ϕ χ + ϕ χ } C ϕ ( H ( H ( x S we obtai fom (.4 te optimal ate ϕ Ω Ω (.5 H ( Ω eo estimate u u C u (.6 Te L eo estimate is obtaied by te itsce tick, by lettig e = u u ad cosideig w ( H ( Ω H ( Ω te solutio of te poblem Θ ( w, ϕ = ( e, ϕ fo ϕ H ( Ω (.7 Te χ S, e = ( ee, = Θ ( we, = Θ ( e, = Θ( e, w χ fo ay By te cotiuity of L i H ( Ω H ( Ω we ave te e = C e w (.8 χ C e w C e e. Hece e C e C u I geeal, assumig tat fo some itege if { w χ + w χ } C w fo ( ( w H ( H x S Ω Ω (.9 s wee. deotes te om i H ( Ω,. =. ad te itsce agumet give et ( + et ( C u ( ( ( H u H Ω Ω 3. Fully Discetized Fiite Elemet Models We sall appoximate te solutio of (. by equiig tat u ad χ lie i { S } < <. Let χκ S fo κ =,,...,. Assume tat te set χ,..., χ is liealy idepedet. Deote by ϒ te subspace spaed by χ,..., χ,let { χ κ } κ = be a basis of S wee = dims.we sall appoximate u of (. by a fuctio u ( x = c χ ( x (3. κ κ κ = Θ ( u, χ = ( f, χ fo χ H ( Ω χ S (3. Substitutig tis expessio fo u i (. ad takig χ = χ, κ =,..., we see tat κ Gc = f (3.3 Wee G is te x matix defied by β ( G = Gκ j = Θ ( χκ, χj = χκ χ + p( x χκ χj + q( x χκ χj dx, κ,j C [ c,, c ] T f (,,,(, T = f χ f χ. Wee G is a positive defiite matice. = ad [ ] 4. umeical Example I tis sectio, some umeical examples ae studied to demostate te accuacy of te peset metod. Te examples ae computed usig MatlabRb. Te vesatility ad accuacy of poposed metod is measued usigl. L = u U = max u ( U j j j Example. Cosideig equatio
3 Applied ad Computatioal Matematics 5; 4(: u ( x + p( x u ( x + q( x u( x = f( x, = x β = (4. wit bouday coditios u( = u( = wee te fuctio p( x ad q( x ae assumed costat, 3, espectively, wile te fuctio f( x is assumed. Te tue solutio of tis poblem is wee / c =, c exp( = + exp( Table. cocetatio eos. Liea elemets Elemets L E E E-6 x x u( x = ce + ce +, Example. Let's coside te same example wit mixed bouday coditios as below u( = u ( = Te tue solutio of tis poblem is x x u( x = ce + ce +, wee c = + exp(, ( + exp( c = ( exp( exp( ( exp( exp( Solutio at ode odes Fig.. Compaiso of umeical ad exact solutio of Example. Liea elemets Example 3. Cosideig equatio x u ( x xu ( x + 4 u( x = x x wit bouday coditios u( = u( = Te tue solutio of tis poblem is.x.6667x x 4 Appoximate solutio Tue solutio Elemets Table. cocetatio eos. Liea elemets L E-4 4.6E E-6 Solutio at ode Appoximate solutio Tue solutio Solutio at ode Appoximate solutio Tue solutio odes Fig. 3. Compaiso of umeical ad exact solutio of Example 3. Liea elemets Table 3. cocetatio eos. Liea elemets odes Fig.. Compaiso of umeical ad exact solutio of Example. Liea elemets Elemets L 6.8E- 4 5.E-.E-
4 67 Getia Zavalai: A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios 5. Cocludig Remaks I tis aticle, Galeki-fiite elemet metod is poposed to fid te appoximate solutios of two poit bouday value poblems. I te solutio pocedue, te fist step is to make weak fomulatio ad te develop fiite elemet fomulatio. Lastly, weigted aveage is used fo fully discetizatio. As test poblem, tee diffeet solutios of tee poit bouday value poblems ae cose. Also, a compaiso of umeical ad aalytical solutios is made ad foud tat te poposed sceme as good accuacy. Appedix A Teoem (Lax Milgam Teoem. Let H be a (eal Hilbet space ad let Θ(, :H H R be a biliea fom o H wic satisfies: Θ φ, ψ c φ ψ φ, ψ H. (. Θ( φ, φ c φ φ H wee c, c ae positive costats idepedet of φ, ψ H. Let F : H R be a (eal valued liea fuctioal o H suc tat. 3. c3 > ψ H F( ψ < c3 ψ Te tee exists a uique u H satisfyig Moeove, Poof. Let evey w H ( Θ u, w = F( w H u F c φ H be fixed. Te Φ:H R defied fo by ( ( φ, Φ = Θ defies a cotiuous liea fuctioal o H. Fo boudedess obseve tat fo eac w H ( φ c Φ ( = Θ, φ ψ Hece Φ c φ < By te Riesz Repesetatio Teoem teefoe, tee exists a uique elemet φ suc tat Φ ( = Θ φ, w = w, φ w H (A. ( ( Hece fo evey φ H we defie a φ H by (A. ad deote te coespodece ϕ φ by φ = Λφ ( φ, ( w, φ Θ = Λ w H φ H (A. ow Λ is a liea opeato defied o H. We claim ow tat Λ, defied by (A. as a age Ra( Λ wic is a closed subspace of H. Let φ = Λφ be a coveget sequece, suc tat φ φ. ow, sice Θ ( φ, = ( w, Λφ w H Θ( φ φ, = ( Λφ Λ φ, w H. Coose m m m w = φ φ ad usig ( get Hece { } φ φ Λφ Λ φ. m m c φ is a Caucy sequece i H,tee exist φ H suc tat φ φ.we ow sow tat φ = Λφ tus sowig tat φ Ra( Λ, tat Ra( Λ is closed. tat Θ φ, w Θ φ, w C φ φ w w H gives ow ( ( ( φ ( φ lim Θ, = Θ, w H Also Λ ( φ, = ( φ, ( φ, ( φ, ( φ, φ φ w Θ φ sice. Sice (, = Λ ( φ, w H Θ ( φ, = ( φ,. Hece Ra( Λ is closed. Also we claim tat Ra( Λ = H Give F o H, by Riesz epesetatio! ξ H suc tat F( = ( ξ, v H tat Λ u = ξ.hece u suc tat.sice Ra( Λ = H F( = ( Λ u, = Θ( u, w H u H suc Fo uiqueess, suppose tat u u suc tat Θ ( u, = F( = Θ( u, w H. Hece Θ( u u, = w H Θ( u u, u u c u u u = u Sice Θ ( u, u = F( u,(,( give tat u c u F( u fom wic u Refeeces F( u c u. Hece F( u sup = F w c w c [] C.w. Cye, Te umeical solutio of bouday value poblems fo secod ode fuctioal diffeetial equatios by fiite diffeeces, ume. Mat. ( [] Y.F. Holt, umeical solutio of oliea two-poit bouday value poblems by fiite diffeece metod, Comm. ACM 7 ( [3] P.G. Cialet, M.H. Scultz ad R.S. Vaga, umeical metods of ig ode accuacy fo oliea bouday value poblems, ume. Mat. 3 ( [4] S Aoa, S S Daliwal, V K Kukeja. Solutio of two poit bouday value poblems usig otogoal collocatio o fiite elemets. Appl. Mat. Comput., 7(5: [5] J Villadse, W E Stewat. Solutio of bouday value poblems by otogoal collocatio. Cem. Eg. Sci., (967:
5 Applied ad Computatioal Matematics 5; 4(: [6] B Jag. Two-poit bouday value poblems by te exteded Adomia decompositio metod. J. Comput. Appl.Mat., 9 ((8: [7] A Doga. A Galeki fiite elemet appoac of Buges equatio. Appl. Mat. Comput., 57 ((4: [8] T K Segupta, S B Talla, S C Pada. Galeki fiite elemet metods fo wave poblems. Sadaa, 3 (5(5: [9] H Kaeko, K S Bey, G J W Hou. Discotiuous Galeki fiite elemet metod fo paabolic poblems. Appl. Mat. Comput., 8 ((6: [] D Sama, R Jiwai, S Kuma. Galeki-fiite Elemet Metods fo umeical Solutio of Advectio- Diffusio Equatio. It. J. Pue ad Appl. Mat., 7 (3(: [] S E Oat. Asymptotic beavio of te Galeki ad te fiite elemet collocatio metods fo a paabolic equatio. Appl. Mat. Comput., 7(:7-3. [3] T Jagveladze, Z Kiguadze, B eta. Galeki fiite elemet metod fo oe oliea itego-diffeetial model. Appl. Mat. Comput., 7 (6(: [] M A EI-Gebeily, K M Fuati, D O Rega. Te fiite elemet- Galeki metod fo sigula self-adjoit diffeetial equatios. J. Comput. Appl. Mat., 3 ((9:
( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to
Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special
More informationLacunary Almost Summability in Certain Linear Topological Spaces
BULLETIN of te MLYSİN MTHEMTİCL SCİENCES SOCİETY Bull. Malays. Mat. Sci. Soc. (2) 27 (2004), 27 223 Lacuay lost Suability i Cetai Liea Topological Spaces BÜNYMIN YDIN Cuuiyet Uivesity, Facutly of Educatio,
More informationDANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD
MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationINVERSE CAUCHY PROBLEMS FOR NONLINEAR FRACTIONAL PARABOLIC EQUATIONS IN HILBERT SPACE
IJAS 6 (3 Febuay www.apapess.com/volumes/vol6issue3/ijas_6_3_.pdf INVESE CAUCH POBLEMS FO NONLINEA FACTIONAL PAABOLIC EQUATIONS IN HILBET SPACE Mahmoud M. El-Boai Faculty of Sciece Aleadia Uivesit Aleadia
More informationLacunary Weak I-Statistical Convergence
Ge. Mat. Notes, Vol. 8, No., May 05, pp. 50-58 ISSN 9-784; Copyigt ICSRS Publicatio, 05 www.i-css.og vailable ee olie at ttp//www.gema.i Lacuay Wea I-Statistical Covegece Haize Gümüş Faculty o Eegli Educatio,
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationA NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS
Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com
More informationON CERTAIN CLASS OF ANALYTIC FUNCTIONS
ON CERTAIN CLASS OF ANALYTIC FUNCTIONS Nailah Abdul Rahma Al Diha Mathematics Depatmet Gils College of Educatio PO Box 60 Riyadh 567 Saudi Aabia Received Febuay 005 accepted Septembe 005 Commuicated by
More informationKEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow
KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you
More informationA Method for Solving Fuzzy Differential Equations using fourth order Runge-kutta Embedded Heronian Means
ISSN (Pit) : 2347-6710 Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 A Metod fo Solvig Fuzzy Diffeetial Equatios usig
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationTHE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 16, Number 2/2015, pp
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seies A, OF THE ROMANIAN ACADEMY Volume 6, Numbe 2/205, pp 2 29 ON I -STATISTICAL CONVERGENCE OF ORDER IN INTUITIONISTIC FUZZY NORMED SPACES Eem
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More informationMinimal order perfect functional observers for singular linear systems
Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationA two-sided Iterative Method for Solving
NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 A two-sided teative Method fo Solvig * A Noliea Matix Equatio X= AX A Saa'a A Zaea Abstact A efficiet ad umeical algoithm is suggested
More informationOn composite conformal mapping of an annulus to a plane with two holes
O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy
More informationSome Integral Mean Estimates for Polynomials
Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationEVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS
EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal
More informationGeneralized Fibonacci-Lucas Sequence
Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationOn Some Fractional Integral Operators Involving Generalized Gauss Hypergeometric Functions
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 5, Issue (Decembe ), pp. 3 33 (Peviously, Vol. 5, Issue, pp. 48 47) Applicatios ad Applied Mathematics: A Iteatioal Joual (AAM) O
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More information= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet
More informationStability analysis of numerical methods for stochastic systems with additive noise
Stability aalysis of umerical metods for stoctic systems wit additive oise Yosiiro SAITO Abstract Stoctic differetial equatios (SDEs) represet pysical peomea domiated by stoctic processes As for determiistic
More informationSome Properties of the K-Jacobsthal Lucas Sequence
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Some Popeties of the K-Jacobsthal Lucas
More informationMultivector Functions
I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed
More informationM A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
More informationSHIFTED HARMONIC SUMS OF ORDER TWO
Commu Koea Math Soc 9 0, No, pp 39 55 http://dxdoiog/03/ckms0939 SHIFTED HARMONIC SUMS OF ORDER TWO Athoy Sofo Abstact We develop a set of idetities fo Eule type sums I paticula we ivestigate poducts of
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More informationLECTURE 2 LEAST SQUARES CROSS-VALIDATION FOR KERNEL DENSITY ESTIMATION
Jauary 3 07 LECTURE LEAST SQUARES CROSS-VALIDATION FOR ERNEL DENSITY ESTIMATION Noparametric kerel estimatio is extremely sesitive to te coice of badwidt as larger values of result i averagig over more
More informationTHE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES
Please cite this atle as: Mhal Matalyck Tacaa Romaiuk The aalysis of some models fo claim pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Sciece 004 Volume 3 Issue pages
More informationAdvanced Physical Geodesy
Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig
More informationy X F n (y), To see this, let y Y and apply property (ii) to find a sequence {y n } X such that y n y and lim sup F n (y n ) F (y).
Modica Mortola Fuctioal 2 Γ-Covergece Let X, d) be a metric space ad cosider a sequece {F } of fuctioals F : X [, ]. We say that {F } Γ-coverges to a fuctioal F : X [, ] if the followig properties hold:
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationSolutions to HW Assignment 1
Solutios to HW: 1 Course: Theory of Probability II Page: 1 of 6 Uiversity of Texas at Austi Solutios to HW Assigmet 1 Problem 1.1. Let Ω, F, {F } 0, P) be a filtered probability space ad T a stoppig time.
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationSolving Fuzzy Differential Equations Using Runge-Kutta Third Order Method for Three Stages Contra-Harmonic Mean
ISSN (Pit): 347-671 Iteatioal Joual of Iovative Reseach i Sciece, Egieeig ad Techology (A High Impact Facto, Mothly Pee Reviewed Joual) Vol. 5, Issue, Febuay 16 Solvig Fuzzy Diffeetial Equatios Usig Ruge-Kutta
More informationEffect of Material Gradient on Stresses of Thick FGM Spherical Pressure Vessels with Exponentially-Varying Properties
M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, 39-46 39 Effect of Mateial Gadiet o Stesses of Thick FGM Spheical Pessue Vessels with Expoetially-Vayig Popeties M. Zamai
More informationOn the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers
O the Explicit Detemiats Sigulaities of -ciculat Left -ciculat Matices with Some Famous Numbes ZHAOLIN JIANG Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA jzh08@siacom JUAN LI Depatmet
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationA note on random minimum length spanning trees
A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu
More informationSolving Fuzzy Differential Equations using Runge-Kutta third order method with modified contra-harmonic mean weights
Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 Solvig Fuzzy Diffeetial Equatios usig Ruge-Kutta thid ode method with modified cota-hamoic mea weights D.Paul Dhayabaa,
More informationApplications of the Dirac Sequences in Electrodynamics
Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe 6-7 6 45 Applicatios of the Diac Sequeces i Electodyamics WILHELM W KECS Depatmet of Mathematics
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationSOLUTION OF THE RADIAL N-DIMENSIONAL SCHRÖDINGER EQUATION USING HOMOTOPY PERTURBATION METHOD
SOLUTIO OF THE ADIAL -DIMESIOAL SCHÖDIGE EQUATIO USIG HOMOTOPY PETUBATIO METHOD SAMI AL-JABE Depatmet of Physics, a-aah atioal Uivesity, ablus, P.O.Box 7, Palestie. E-mail: abe@aah.edu eceived Jue, Homotopy
More information2012 GCE A Level H2 Maths Solution Paper Let x,
GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs
More informationComplementary Dual Subfield Linear Codes Over Finite Fields
1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationOn the convergence, consistence and stability of a standard finite difference scheme
AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 2, Sciece Huβ, ttp://www.sciub.org/ajsir ISSN: 253-649X, doi:.525/ajsir.2.2.2.74.78 O te covergece, cosistece ad stabilit of a stadard fiite differece
More informationThe Advection-Diffusion equation!
ttp://www.d.edu/~gtryggva/cf-course/! Te Advectio-iffusio equatio! Grétar Tryggvaso! Sprig 3! Navier-Stokes equatios! Summary! u t + u u x + v u y = P ρ x + µ u + u ρ y Hyperbolic part! u x + v y = Elliptic
More informationConsider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample
Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied
More informationMinimization of the quadratic test function
Miimizatio of the quadatic test fuctio A quadatic fom is a scala quadatic fuctio of a vecto with the fom f ( ) A b c with b R A R whee A is assumed to be SPD ad c is a scala costat Note: A symmetic mati
More informationOn a Problem of Littlewood
Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995
More information[Dhayabaran*, 5(1): January, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY SOLVING FUZZY DIFFERENTIAL EQUATIONS USING RUNGE-KUTTA
More informationStrong Result for Level Crossings of Random Polynomials
IOSR Joual of haacy ad Biological Scieces (IOSR-JBS) e-issn:78-8, p-issn:19-7676 Volue 11, Issue Ve III (ay - Ju16), 1-18 wwwiosjoualsog Stog Result fo Level Cossigs of Rado olyoials 1 DKisha, AK asigh
More informationON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS
Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics
More informationx x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula
NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat
More informationStructure and Some Geometric Properties of Nakano Difference Sequence Space
Stuctue ad Soe Geoetic Poeties of Naao Diffeece Sequece Sace N Faied ad AA Baey Deatet of Matheatics, Faculty of Sciece, Ai Shas Uivesity, Caio, Egyt awad_baey@yahooco Abstact: I this ae, we exted the
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationInverse Nodal Problems for Differential Equation on the Half-line
Australia Joural of Basic ad Applied Scieces, 3(4): 4498-4502, 2009 ISSN 1991-8178 Iverse Nodal Problems for Differetial Equatio o the Half-lie 1 2 3 A. Dabbaghia, A. Nematy ad Sh. Akbarpoor 1 Islamic
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationMA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions
MA54 : Real Aalysis Tutorial ad Practice Problems - Hits ad Solutios. Suppose that S is a oempty subset of real umbers that is bouded (i.e. bouded above as well as below). Prove that if S sup S. What ca
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationLIMITS AND DERIVATIVES
Capter LIMITS AND DERIVATIVES. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is
More informationA collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called
More informationSums of Involving the Harmonic Numbers and the Binomial Coefficients
Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa
More informationGeneralization of Horadam s Sequence
Tuish Joual of Aalysis ad Nube Theoy 6 Vol No 3-7 Available olie at http://pubssciepubco/tjat///5 Sciece ad Educatio Publishig DOI:69/tjat---5 Geealizatio of Hoada s Sequece CN Phadte * YS Valaulia Depatet
More informationStrong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA
Iteatioal Joual of Reseach i Egieeig ad aageet Techology (IJRET) olue Issue July 5 Available at http://wwwijetco/ Stog Result fo Level Cossigs of Rado olyoials Dipty Rai Dhal D K isha Depatet of atheatics
More informationLIMITS AND DERIVATIVES NCERT
. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is te epected value of f at a give
More informationA Pseudo Spline Methods for Solving an Initial Value Problem of Ordinary Differential Equation
Joural of Matematics ad Statistics 4 (: 7-, 008 ISSN 549-3644 008 Sciece Publicatios A Pseudo Splie Metods for Solvig a Iitial Value Problem of Ordiary Differetial Equatio B.S. Ogudare ad G.E. Okeca Departmet
More informationWeighted Hardy-Sobolev Type Inequality for Generalized Baouendi-Grushin Vector Fields and Its Application
44Æ 3 «Vol.44 No.3 05 5 ADVANCES IN MATHEMATICS(CHINA) May 05 doi: 0.845/sxjz.03075b Weighted Hady-Sobolev Type Ieuality fo Geealized Baouedi-Gushi Vecto Fields ad Its Applicatio ZHANG Shutao HAN Yazhou
More informationModular Spaces Topology
Applied Matheatics 23 4 296-3 http://ddoiog/4236/a234975 Published Olie Septebe 23 (http://wwwscipog/joual/a) Modula Spaces Topology Ahed Hajji Laboatoy of Matheatics Coputig ad Applicatio Depatet of Matheatics
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More information[Dhayabaran*, 5(2): February, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785
IJESRT ITERATIOAL JOURAL OF EGIEERIG SCIECES & RESEARCH TECHOLOGY SOLUTIO FOR FUZZY DIFFERETIAL EQUATIOS USIG FOURTH ORDER RUGE-KUTTA METHOD WITH EMBEDDED HARMOIC MEA DPaul Dhayabaa * JChisty Kigsto *
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
More informationFind quadratic function which pass through the following points (0,1),(1,1),(2, 3)... 11
Adrew Powuk - http://www.powuk.com- Math 49 (Numerical Aalysis) Iterpolatio... 4. Polyomial iterpolatio (system of equatio)... 4.. Lier iterpolatio... 5... Fid a lie which pass through (,) (,)... 8...
More informationGreatest term (numerically) in the expansion of (1 + x) Method 1 Let T
BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value
More informationFurther Methods for Advanced Mathematics (FP2) WEDNESDAY 9 JANUARY 2008
ADVANCED GCE 7/ MATHEMATICS (MEI) Furter Metods for Advaced Matematics (F) WEDNESDAY 9 JANUARY 8 Additioal materials: Aswer Booklet (8 pages) Grap paper MEI Eamiatio Formulae ad Tables (MF) Afteroo Time:
More informationKey wordss Contra-harmonic mean, Fuzzy Differential Equations, Runge-kutta second order method, Triangular Fuzzy Number.
ISO 9:8 Cetified Iteatioal Joual of Egieeig Sciece ad Iovative Techology (IJESIT) Volume 5, Issue, Jauay 6 Solvig Fuzzy Diffeetial Equatios usig Ruge-kutta secod ode method fo two stages cota-hamoic mea
More informationMapping Radius of Regular Function and Center of Convex Region. Duan Wenxi
d Iteatioal Cofeece o Electical Compute Egieeig ad Electoics (ICECEE 5 Mappig adius of egula Fuctio ad Cete of Covex egio Dua Wexi School of Applied Mathematics Beijig Nomal Uivesity Zhuhai Chia 363463@qqcom
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationLower Bounds for Cover-Free Families
Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set
More informationFinite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler
Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio
More informationADDITIONAL INTEGRAL TRANSFORMS
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 897 IX.7 ADDIIONAL INEGRAL RANSFORMS 6.7. Solutio of 3-D Heat Equatio i Cylidical Coodiates 6.7. Melli asfom 6.7.3 Legede asfom
More information