nonlocal conditions.


 Reynard Andrews
 1 years ago
 Views:
Transcription
1 ISSN prin, online Inernaional Journal of Nonlinear Science Vol No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed M Maar Mahemaics Deparmen, AlAzhar UniversiyGaza, Gaza Srip, PA. Received 17 November 29, acceped 2 January 211 Absrac: In his paper, he exisence problems of he soluion of some boundary value fracional inegrodifferenial equaions wih nonlocal condiions are invesigaed. he resuls are obained using Banach and Krasnoselkii fixed poin heorems. Keywords: Fracional calculus; exisence and uniqueness; inegrodifferenial equaions; fixed poin heorems; nonlocal condiions. 1 Inroducion Fracional differenial equaions are emerged as a new branch of applied mahemaics by which many physical and engineering approaches can be modeled. he fac ha fracional differenial equaions are considered as alernaive models o nonlinear differenial equaions which induced exensive researches in various fields including he heoreical par see [1][11] and references herein. he exisence and uniqueness problems of fracional nonlinear differenial and inegrodifferenial equaions as a basic heoreical par of some applicaions are invesigaed by many auhors see for examples [3], [6], and [7]. In [1], he auhors obained sufficien condiions for he exisence of soluions for a class of boundary value problem of fracional differenial equaions in he case of 1 < 2 involving he Capuo fracional derivaive and nonlocal condiions using he Banach and Schaefer s fixed poins heorems. he Cauchy problems for some fracional absrac differenial equaions in he case of < 1 wih nonlocal condiions are invesigaed by he auhors in [2] and [1] using he Banach and Krasnoselkii fixed poin heorems. he Banach fixed poin heorem is used in [4] and [8] o invesigae he exisence problems of fracional inegrodifferenial equaions in he case of < 1 in Banach spaces. Moivaed by hese works we sudy in his paper he exisence of soluion of boundary value problem for fracional inegrodifferenial equaions in he case of 1 < 2 in Banach spaces by using Banach and Krasnoselkii fixed poin heorems. 2 Preliminaries We need some basic definiions and properies of fracional calculus which will be used in his paper. Definiion 1 A real funcion f is said o be in he space C μ, μ R if here exiss a real number p > μ, such ha f p f 1, where f 1 C[, ; and i is said o be in he space C n μ if and only if f n C μ, n N. Definiion 2 A funcion f C μ, μ 1 is said o be fracional inegrable of order > if and if, hen I f f. I f I f 1 s 1 fsds <, Γ Corresponding auhor. address: mohammed Copyrigh c World Academic Press, World Academic Union IJNS /434
2 4 Inernaional Journal of Nonlinear Science, Vol.11211, No.1, pp. 39 Nex, we inroduce he Capuo fracional derivaive. Definiion 3 he fracional derivaive in he Capuo sense is defined as D f d D f I n n f 1 d n s n 1 d n fs Γ n ds n ds for n 1 < n, n N, >, f C 1. n In paricular, if 1 < 2, hen D f 1 Γ2 s 1 f sds, where f s d2 fs ds, and f C is a funcion wih values in absrac space X. he ideniy I D f f + a + b, 1 where J, a, b are consans and oher properies of he fracional operaors used in he general heory of fracional differenial equaions can be found in [5], [9], and [11]. e Y CJ, X be a Banach space of all coninuous funcions x from a compac inerval J [, ] ino a Banach space X. e D {, s : s } be subse of R 2. Consider he fracional nonlinear inegrodifferenial equaion D x f, x, g, s, xsds, x hx, x kx, where 1 < 2, and he nonlinear funcions f, g, h, and k saisfy he following hypoheses: H1 f : J Y Y Y, g : D Y Y are joinly coninuous funcions and here exiss posiive consans B, C such ha { f, x, u f, y, v Cx y + u v g, s, x g, s, y B x y for any J,, s D, x, y, u, v Y. Moreover, le A sup J f,, H2 h : Y Y, and k : Y Y are coninuous funcions such ha for any x, y Y. { hx hy C x y kx ky C x y 2 g, s, ds, and max{a, B, C}. emma 1 he inegrodifferenial equaion D x f, g, sds, x hx, x kx, 3 is equivalen o he inegral equaion where y f, x hx + kx I y + I y 4 g, sds is a fracional inegrable of order funcion. IJNS for conribuion:
3 Mohammed M Maar: Boundary Value Problem for Some Fracional Inegrodifferenial Equaions 5 Proof. Applying he fracional inegral operaor I o boh sides of equaion 3, and using he ideniy 1, we ge I D x I f, x + a + b 1 Now, if, we have a hx, and if, we have g, sds s 1 fs, gs, rdrds. which implies ha herefore kx hx + b 1 b kx + hx + 1 s 1 fs, s 1 fs, gs, rdrds gs, rdrds. x hx + kx hx s 1 fs, gs, rdrds hx + kx s 1 fs, s 1 fs, gs, rdrds s 1 fs, gs, rdrds gs, rdrds which is equaion 4. On he oher hand, applying he fracional differenial operaor D o boh sides of equaion 4, i is easily o ge equaion 3. In view of emma 1, equaion 2 is equivalen o he inegral equaion x hx + kx I F x + I F x where F x fs, xs, saisfies he following esimaes and gs, r, xrdr is a fracional inegrable of order nonlinear operaor. he operaor F I F x I F x F + I F C s 1 x + Bs x + Ads C x + CB+1 x + CA Γ x + 1 IJNS homepage: hp://www.nonlinearscience.org.uk/
4 6 Inernaional Journal of Nonlinear Science, Vol.11211, No.1, pp. 39 I F x F y 1 + x y + 1 for every x, y Y, J. 3 Exisence problems We prove he exisence of he fracional nonlinear inegrodifferenial equaion 2 by using he wellknown Banach fixed poin heorem. he following condiion is essenial o ge he conracion propery. H3 e < q < 1, and r be a posiive finie real number such ha 2 q 1 + Γ r 1 q 1 h + k + 22 Γ+1. Moreover, le B r {y Y : y r}. heorem 2 If he hypoheses H1H3 are saisfied, hen he fracional inegrodifferenial equaion 2 has a unique soluion on J. Proof. Define he operaor Ψ : Y Y by Ψx hx + kx I F x + I F x. We show ha Ψ has a fixed poin on B r. his fixed poin is hen a soluion of equaion 2. Firsly, we show ha ΨB r B r. e x B r, hen Ψx hx + kx + I F x + I F x hx + kx + I F x + I F x h + C x + C x + k x x h + k q r + qr r x IJNS for conribuion:
5 Mohammed M Maar: Boundary Value Problem for Some Fracional Inegrodifferenial Equaions 7 Hence, he operaor Ψ maps B r ino iself. Nex, we prove ha Ψ is a conracion mapping on B r. e x, y B r, hen Ψx Ψy hx + kx I F x + I F x hy ky + I F y I F y hx hy + kx ky + I F y F x + I F x F y x y + x y x y x y x y q x y. + 1 Hence, he operaor Ψ has a unique fixed poin which is a soluion o equaion 2. he nex resul is based on he following wellknown fixed poin heorem. heorem 3 Krasnoselkii e S be a closed convex and nonempy subse of a Banach space X. e P and Q be wo operaors such ha i P x + Qy S whenever x, y S; ii P is a conracion mapping; iii Q is compac and coninuous. hen here exiss z S such ha z P z + Qz. o apply he above heorem we need he following condiion insead of he condiion H1. H4 he funcions f : J Y Y, and g : D Y Y are joinly coninuous and here exiss a posiive consan such ha for all, x, y J Y Y. f, x, y heorem 4 If he hypoheses H2 and H4 are saisfied, and if C < 1, hen he fracional inegrodifferenial equaion 2 has a soluion on J. Proof. e r 1 C 1 2 h + k + Γ+1. Define he operaors P, and Q on he compac se B r {y Y : y r} Y by { P x hx + kx Qy I F y I F y. We observe ha hence I F y Qy, 2 5 IJNS homepage: hp://www.nonlinearscience.org.uk/
6 8 Inernaional Journal of Nonlinear Science, Vol.11211, No.1, pp. 39 and P x + Qy P x + Qy hx + kx + I F y I F y h + k C x + C herefore, if x, y B r, hen P x + Qy B r. On he oher hand, i is easily o show ha he operaor P is a conracion. Indeed, since P x P y hx hy + kx ky C x y + C x y C x y. By he hypohesis H4, he operaor Q is coninuous and by he inequaliy 5, i is uniformly bounded on B r. For he equiconinuiy of Qy, le 1, 2 in J, and y B r, we have x. hence I F y 1 I F y s 1 fs, ys, gs, r, yrdrds s 1 fs, ys, gs, r, yrdrds 1 2 s 1 1 s 1 ds s 1 ds Qy 1 Qy 2 I F y 1 I F y I F y 1 I F y As 2 1, he righhand side of he above inequaliy ends o zero which gives he equiconinuiy of Qy. So QB r is relaively compac. By he Arzela Ascoli heorem, Q is compac. Hence by he Krasnoselkii heorem here exiss a soluion o equaion 2. References [1] M. Benchohra, S. Hamania, S. K. Nouyas.Boundary value problems for differenial equaions wih fracional order and nonlocal condiions. Nonlinear Analysis, 7129: IJNS for conribuion:
7 Mohammed M Maar: Boundary Value Problem for Some Fracional Inegrodifferenial Equaions 9 [2] K. Balachandran, J. Y. Park.Nonlocal Cauchy problem for absrac fracional semilinear evoluion equaions.nonlinear Analysis, 7129: [3] D. Delbosco,. Rodino.Exisence and uniqueness for a fracional differenial equaion. Journal of Mahemaical Analysis and Applicaions, : [4] O. K. Jarada, A. AlOmari, S. Momani.Exisence of he mild soluion for fracional semilinear iniial value problem. Nonlinear Analysis, 6928: [5] A. A. Kilbas, H. M. Srivasava, J. J. rujillo. heory and applicaions of fracional differenial equaions. Elsevier, Amserdam,26. [6] V. akshmikanham.heory of fracional funcional differenial equaions.nonlinear Analysis, 6928: [7] V. akshmikanham, A. S. Vasala.Basic heory of fracional differenial equaions. Nonlinear Analysis, 6928: [8] M Maar.On exisence and uniqueness of he mild soluion for fracional semilinear inegrodifferenial equaions. Journal of Inegral Equaions and Applicaions, acceped. [9] K. S. Miller, B Ross.An inroducion o he fracional calculus and fracional differenial equaions. J.Wiley & Sons, New York, [1] G. M. N guerekaa. A Cauchy problem for some fracional absrac differenial equaion wih nonlocal condiion. Nonlinear Analysis, [11] I. Podlubny. Fracional differenial equaions. Academic Press, New York, IJNS homepage: hp://www.nonlinearscience.org.uk/
A NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationA UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS
A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationA Note on Renewal Theory for T iid Random Fuzzy Variables
Applied Mahemaical Sciences, Vol, 6, no 6, 97979 HIKARI Ld, wwwmhikaricom hp://dxdoiorg/988/ams6686 A Noe on Renewal Theory for T iid Rom Fuzzy Variables Dug Hun Hong Deparmen of Mahemaics, Myongji
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationDynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract
Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationResearch Article Optimal Geometric Mean Returns of Stocks and Their Options
Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationConvexity theory for term structure equation: an extension to the jumpdiffusion case
U.U.D.M. Projec Repor :5 Convexiy heory for erm srucure equaion: an exension o he jumpdiffusion case Kailin Zeng Examensarbee i maemaik, 3 hp Handledare och examinaor: Johan Tysk Maj Deparmen of Mahemaics
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationThe Heisenberg group and Pansu s Theorem
The Heisenberg group and Pansu s Theorem July 31, 2009 Absrac The goal of hese noes is o inroduce he reader o he Heisenberg group wih is Carno Carahéodory meric and o Pansu s differeniaion heorem. As
More informationResearch Article Solitary Wave Solutions for a TimeFraction Generalized HirotaSatsuma Coupled KdV Equation by a New Analytical Technique
Hindawi Publishing Corporaion Inernaional Journal of Differenial Equaions Volume, Aricle ID 954674, pages doi:.55//954674 Research Aricle Soliary Wave Soluions for a TimeFracion Generalized HiroaSasuma
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationIntroduction to Stochastic Calculus
IEOR E477: Financial Engineering: Coninuousime Models Fall 21 c 21 by Marin Haugh Inroducion o Sochasic Calculus hese noes provide an inroducion o sochasic calculus, he branch of mahemaics ha is mos idenified
More informationImproper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].
Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationEndpoint Strichartz estimates and global solutions for the nonlinear Dirac equation 1
Endpoin Sricharz esimaes and global soluions for he nonlinear Dirac equaion 1 Shuji Machihara, Makoo Nakamura, Kenji Nakanishi, and Tohru Ozawa Absrac. We prove endpoin Sricharz esimaes for he KleinGordon
More informationTWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE
Annals of he Academy of Romanian Scieniss Series on Mahemaics and is Applicaions ISSN 2666594 Volume 3, Number 2 / 211 TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE Werner Krabs
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationMean Field Games. Math 581 Project
Mean Field Games Tiago Miguel Saldanha Salvador Mah 58 Projec April 23 Conens Inroducion 2 2 Analysis of second order MFG 3 2. On he FokkerPlank equaion................................ 4 2.2 Exisence
More informationOPTIMAL PRODUCTION SALES STRATEGIES FOR A COMPANY AT CHANGING MARKET PRICE
REVISA DE MAEMÁICA: EORÍA Y APLICACIONES 215 22(1) : 89 112 CIMPA UCR ISSN: 1492433 (PRIN), 22153373 (ONLINE) OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY A CHANGING MARKE PRICE ESRAEGIAS ÓPIMAS DE
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationAND BACKWARD SDE. Nizar Touzi nizar.touzi@polytechnique.edu. Ecole Polytechnique Paris Département de Mathématiques Appliquées
OPIMAL SOCHASIC CONROL, SOCHASIC ARGE PROBLEMS, AND BACKWARD SDE Nizar ouzi nizar.ouzi@polyechnique.edu Ecole Polyechnique Paris Déparemen de Mahémaiques Appliquées Chaper 12 by Agnès OURIN May 21 2 Conens
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationChapter 2: Principles of steadystate converter analysis
Chaper 2 Principles of SeadySae Converer Analysis 2.1. Inroducion 2.2. Inducor volsecond balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer
More informationANALYTIC PROOF OF THE PRIME NUMBER THEOREM
ANALYTIC PROOF OF THE PRIME NUMBER THEOREM RYAN SMITH, YUAN TIAN Conens Arihmeical Funcions Equivalen Forms of he Prime Number Theorem 3 3 The Relaionshi Beween Two Asymoic Relaions 6 4 Dirichle Series
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationSMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY. J. E. Palomar Tarancón. A.M.S. (MOS) Subject Classification Codes. 44A05, 34A99, 18B99
Elecronic Journal: Souhwes Journal of Pure an Applie Mahemaics Inerne: hp://raler.cameron.eu/swjpam.hml ISSN 10830464 Issue 2 December 2003 pp. 36 48. Submie: February 2003. Publishe: December 31 2003.
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationMalliavin Calculus. Matheus Grasselli Tom Hurd Department of Mathematics & Statistics, McMaster University Hamilton, Ontario, Canada L8S 4K1
Malliavin Calculus Maheus Grasselli Tom Hurd Deparmen of Mahemaics & Saisics, McMaser Universiy Hamilon, Onario, Canada L8S 4K1 April, 25 1 Inroducion 2 Malliavin Calculus 2.1 The Derivaive Operaor Consider
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationStochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions
Sochasic inegraion wih respec o mulifracional Brownian moion via angen fracional Brownian moions Eric Herbin, Joachim Lebovis, Jacques Lévy Véhel To cie his version: Eric Herbin, Joachim Lebovis, Jacques
More informationLinear Quadratic Optimal Control of. Problem æ. Birgit Jacob. Abstract. general does not process a solution on the whole interval.
Journal of Mahemaical Sysems, Esimaion, and Conrol Vol. 5, No. 1, 1995, pp. 1í28 cæ 1995 BirkhíauserBoson Linear Quadraic Opimal Conrol of TimeíVarying Sysems wih Indeænie Coss on Hilber Spaces: The Finie
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Wholerange Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationViscosity Solution of Optimal Stopping Problem for Stochastic Systems with Bounded Memory
Viscosiy Soluion of Opimal Sopping Problem for Sochasic Sysems wih Bounded Memory MouHsiung Chang Tao Pang Mousapha Pemy April 5, 202 Absrac We consider a finie ime horizon opimal sopping problem for
More informationSteps for D.C Analysis of MOSFET Circuits
10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.
More informationModeling Stock Price Dynamics with Fuzzy Opinion Networks
Modeling Sock Price Dynamics wih Fuzzy Opinion Neworks LiXin Wang Deparmen of Auomaion Science and Technology Xian Jiaoong Universiy, Xian, P.R. China Email: lxwang@mail.xju.edu.cn Key words: Sock price
More informationON THE APPROXIMATION CAPABILITY OF NEURAL NETWORKS DYNAMIC SYSTEM MODELING AND CONTROL
122 Asian Journal of Conrol, Vol. 3, No. 2, pp. 12213, June 21 ON THE APPROXIMATION CAPABILITY OF NEURAL NETWORKS DYNAMIC SYSTEM MODELING AND CONTROL Chu Kwong Chak, Gang Feng and Jian Ma ABSTRACT This
More informationThe Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations
Universal Jornal of Engineering Science ( 46, OI.89/jes.. hp//www.hrpb.org The Linear, Nonlinear and Parial ifferenial Eqaions are no Fracional Order ifferenial Eqaions Ali Karci eparmen of Comper Engineering,
More informationOn the paper Is Itô calculus oversold? by A. Izmailov and B. Shay
On he paper Is Iô calculus oversold? by A. Izmailov and B. Shay M. Rukowski and W. Szazschneider March, 1999 The main message of he paper Is Iô calculus oversold? by A. Izmailov and B. Shay is, we quoe:
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationUse SeDuMi to Solve LP, SDP and SCOP Problems: Remarks and Examples*
Use SeDuMi o Solve LP, SDP and SCOP Problems: Remarks and Examples* * his file was prepared by WuSheng Lu, Dep. of Elecrical and Compuer Engineering, Universiy of Vicoria, and i was revised on December,
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationA DYNAMIC PROGRAMMING APPROACH TO THE PARISI FUNCTIONAL
A DYNAMIC PROGRAMMING APPROACH TO THE PARISI FUNCTIONAL AUKOSH JAGANNATH AND IAN TOBASCO Absrac. G. Parisi prediced an imporan variaional formula for he hermodynamic limi of he inensive free energy for
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationContents. 1. The simplest operator whose average is the Hilbert. transform WHY THE RIESZ TRANSFORMS ARE AVERAGES OF THE DYADIC SHIFTS?
Publ. Ma. (22), 29 228 Proceedings of he 6 h Inernaional Conference on Harmonic Analysis and Parial Differenial Equaions. El Escorial, 2. WHY THE RIESZ TRANSFORMS ARE AVERAGES OF THE DYADIC SHIFTS? S.
More informationOn the Role of the Growth Optimal Portfolio in Finance
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 14418010 www.qfrc.us.edu.au
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationSolution of a differential equation of the second order by the method of NIGAM
Tire : Résoluion d'une équaion différenielle du second[...] Dae : 16/02/2011 Page : 1/6 Soluion of a differenial equaion of he second order by he mehod of NIGAM Summarized: We presen in his documen, a
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More information1 The basic circulation problem
2WO08: Graphs and Algorihms Lecure 4 Dae: 26/2/2012 Insrucor: Nikhil Bansal The Circulaion Problem Scribe: Tom Slenders 1 The basic circulaion problem We will consider he maxflow problem again, bu his
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and
More informationAnalysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer
Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationAn Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price
An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor
More information