Laplace Adomian Decomposition Method for Solving the Nonlinear Volterra Integral Equation with Weakly Kernels
|
|
- Britney Miles
- 6 years ago
- Views:
Transcription
1 Studie in Nonlinear Science (4): 9-4, ISSN -9 IDOSI Publication, Laplace Adomian Decompoition Method for Solving the Nonlinear Volterra Integral Equation with Weakly Kernel F.A. Hendi Department of Mathematic Faculty of Science, King Abdul Aziz Univerity, Jeddah, K.S.A. Abtract: A computational method (The Comb ined Laplace Adomian decompoition method) applied for olving linear and nonlinear Volterra integral equation with weakly kernel, we dicu ome eample, when the kernel take a Carleman and logarithmic form to demontrate the high accuracy and implicity of our method.a comparion with the Toeplitz matri method i made. MSC: (45D5) Key word: Adomian decompoition method nonlinear volterra integral equation laplace tranform carleman and logarithmic kernel INTRODUCTION In practical application one frequently encounter the Volterra integral equation of the econd kind with a weakly ingular kernel of the form y() = g() +λ p(,t)k(,y(t))dt () a where the kernel p(,t) i weakly ingular and the given function g() and k(,y(t)) are aumed to be ufficiently mooth in order to guarantee the eitence and uniquene of a olution y C[a,], [-]. Typical form of p(,t) are (i) p(,t) = t, <α< (ii) p(,t) = ln t For volterra equation with bounded kernel, the moothne of the kernel and of the forcing function g() determine the moothne of the olution on the cloed interval [a,] with >a. If we allow weakly ingular kernel, then the reulting olution are typically non-mooth at the initial point of the interval of integration, where their derivative become unbounded. Some reult concerning the behavior of the eact olution of equation of type () are given in [4, 5]. It i well known that the nonlinear Volterra integral equation () i uually handled by many method, uch a Toeplitz matri method [6] and other. The traditional Laplace method by itelf cannot be ued in thi cae becaue of the nonlinearity of thi equation, then from [7] there eit a method combining the Laplace tranform method with Adomian decompoition method for analytic treatment of nonlinear ingular integral equation that decribe heat tranfer. If p(,t) of() i a difference kernel p(-t). The nonlinear Volterra integral equation () can be epreed a: y() = g() +λ p( t)k(y(t))dt () o it i poible to overcome thi difficulty of nonlinearity by the powerful Adomian decompoition method. Correponding Author: F.A. Hendi, Department of Mathematic, Faculty of Science, King Abdul Aziz Univerity, Jeddah, K.S.A. 9
2 Studie in Nonlinear Sci., (4): 9-4, In thi paper we conider the nonlinear Volterra integral equation with weakly kernel of the form (i) and (ii). The combined algorithm i capable of handling the two type ingular Volterra integral equation, then we compare the reult with the numerical olution for the ingular Volterra integral equation by uing Toeplitz matri method [6]. CARLEMAN KERNEL We aume that the kernel p(,t) of () take the form (i), then nonlinear Volterra integral equation() can be epreed a: Applying the Laplace tranform to both ide of () give: y() = g() +λ t k(y(t))dt () { } { } Y() = G() +λl t L k(y(t)) (4) The Adomian decompoition method can be ued to handle (4). We repreent the linear term Y() by So Y() = Y() (5) n= n= n y() = y() (6) n The nonlinear term k(y(t)) will be repreented by the Adomian polynomial A n in the form: where A n, n are given by k(y()) = A() (7) n= n n i n n i λ= n! d i= n d A = [ k( λ y)],n =,,,... (8) λ where the o called Adomian polynomial A n can be evaluated for all form of nonlinearity. Subtituting (5) and (6) into (4) lead to { } The Adomian decompoition method introduce the recurive relation Yn = G() +λ L t L A() n (9) n= n= { } Y() = G(),Y k () α + =λ L A k(),k () Appling the invere Laplace tranform to the firt part of (9) give y (), that will define A. Uing A () will enable u to evaluate y (). A k () are the Adomian for the nonlinear term. The Adomian decompoition method aume that the linear term y() can be decompoed by the erie. The obtained erie olution may converge to an eact olution if uch a olution eit. Eample (Linear cae) [6]: Conider the nonlinear volterra integral equation with Carleman kernel
3 Studie in Nonlinear Sci., (4): 9-4, Table : y E y LA E LA y TM E TM.7 4.9E- 4.94E- 4.E E-.95E-.56.6E-.585E-.848E-.548E-.58E E-.775E-.58E E- 4.89E E E-.968E E-.64E E-.964E- 4.8E E-.4E-5..E-.66E-.658E E-.59E E E E E- 5.49E-6.6.6E-.649E E E-.58E-6.4.6E-.66E-.55E E- 8.6E-7. 4.E-4 4.8E-4.78E E-4.55E-7 () y() = +λ t k(y(t))dt where the eact olution i y() =, α = /, λ =.. By taking Laplace tranform we obtain Γ[ ] Y() = +λ Y[] () The Adomian decompoition method introduce the recurive relation Γ[ ] Y() =,Y k+ () =λ Y(),k k () o that, taking the invere Laplace tranform of both ide of relation () give 8 Γ = = =λ n n= y() y(),y(),y() Γ ( Γ ) λ y() = λ,y() = [ ( Γ ),y y + y + y Γ 4 We denote by y E to the eact olution and y LA, y TM to Laplace Adomian and Toeplitz matri olution.a comparion of the reult hown in Table. Eample (Nonlinear cae) [8] 4 y() = ( ) + ( t) y(t)dt (4) where the eact olution y() = /. By taking Laplace tranform we obtain π( + ) π Y() = + L 5 { y[t] } (5)
4 Studie in Nonlinear Sci., (4): 9-4, Table : y E y LA E LA E E E E E E E E E E-7 The Adomian decompoition method admit the ue of where the Adomian polynomial are: π( + ) π Y(),Y () L A (),k 5 k+ k = = { } (6) A() = y,a() = y y,a () = ( y + 4y y ) (7) and o on for other Adomian polynomial. Uing the recurrence relation (7), we find y() = ( ),y() = ( ) 5 8 = y() ( ),y y y y 7445 (8) Comparing the reult with the eact olution given in Table. LOGARITHMIC KERNEL If the kernel p(,t) of () take the form (ii), then nonlinear Volterra integral equation() can be epreed a: Applying the Laplace tranform to both ide of (9) give: y() = g() +λ ln t k(y(t))dt (9) γ+ ln() Y() G() ( )Lk(y(t)) = +λ { } () where γ i Euler' contant. Uing the ame method we hall find at the end the required olution by the invere of Laplace tranform. Eample (Linear cae) [6] y() = +λ ln(t )y(t)dt < t< ()
5 Studie in Nonlinear Sci., (4): 9-4, Table : y E y LA E LA y TM E TM.7 4.9E E-.569E- 4,964E- 6.47E-.56.6E-.75E E-4.689E-.95E E-.7595E E E-.68E-.4.96E-.9574E-.957E E E-5..E E-.8E-5.8E-.87E E- 6.98E- 6.79E E E-6.6.6E-.5969E-.898E-6.69E-.959E-6.4.6E E-.8E-6.65E-.54E-6. 4.E E E-7 4.4E-4.976E-7 where the eact olution y() =, λ =.. By taking Laplace tranform, we obtain γ+ ln() Y() ( )Lk(y(t)) The Adomian decompoition method introduce the recurive relation = +λ { } () o that γ+ ln Y() =,Y k+ () =λ Y(), k k λ y() =,y() = [ + 6ln], 8 λ y() = [45 π ln + 7(ln) ],y y + y + y () Comparing thi reult by the Toeplitz matri method given in Table. Eample 4 (Nonlinear cae)[6] (4) y() = +λ ln( t)y(t)dt where the eact olution y() =, λ = By taking Laplace tranform,we obtain γ+ ln() Y() ( )Lk(y(t)) The Adomian decompoition method introduce the recurive relation = +λ { } (5) o that γ+ ln Y() =,Y k+ () =λ Y(), k k (6) 7 y() =,y() = [ 6 4 ln] 98 y() = π 977 ln + 58(ln),y y +λ y + y ( ) 8 Reult in Table 4.
6 Studie in Nonlinear Sci., (4): 9-4, Table 4: y E y LA E LA y TM E TM..E E E E-.E E E-.864E-7 6.4E-.656E-9.6.6E E-.5958E-8.6E-.9E-.4.6E-.6E-.667E-9.6E- 9.E-. 4.E-4 4.E E- 4.E-4 4.E- CONCLUSION In thi work we howed the accuracy and implicity of Combined Laplace-Adomian method applied to linear and nonlinear Volterra integral equation with weakly kernel. Thi method preent a ueful way to develop computational method for olving thee kind of nonlinear ingular integral equation. We made a comparion with Toeplitz matri method for olving the ame problem. REFERENCES. Atkinon, K.E., 974. An eitence theorem for Abel integral equation. SIAM J. Math. Anal., 5: Baker, C.T.H., 977. The numerical treatment of integral equation. Clarendon Pre Oford.. Cahlon, B., 98. Numerical Solution of nonlinear Volterra integral equation. J. Comput. Appl. Math., 7: Logan, J.E., 976. The approimate olution of Volterra integral equation of the econd kind. PhD. Thei, Univerity of Iowa, Iowa City. 5. Lubich, Ch., 98. Runge-Kutta theory for Volterra and Able integral equation of the econd kind. Math. Comp., 4: Hendi, F.A. and Sh. Al-Hazmi,. The Nonlinear Volterra Integral Equation with Weakly Kernal and Toeplitz Matri Method. IJRRAS, (): Wazwaz, A. and M.S. Mehanna,. The Combined Laplace-Adomian method for handling ingular integral equation of Heat tranfer. I.J. of Nonlinear Science, (): Brunner, H., A. Peda and G. Vainikko, 997. The piecewie polynomial collocation method for nonlinear weakly ingular Volterra equation. Helinki Univerity of technology, reearch report, A9: Epoo, Finland. 4
Application of Laplace Adomian Decomposition Method on Linear and Nonlinear System of PDEs
Applied Mathematical Science, Vol. 5, 2011, no. 27, 1307-1315 Application of Laplace Adomian Decompoition Method on Linear and Nonlinear Sytem of PDE Jaem Fadaei Mathematic Department, Shahid Bahonar Univerity
More informationConvergence of an Approach for Solving Fredholm Functional Integral Equations
Iranian Journal of Mathematical Science and Informatic Vol. 11, No. 1 (2016), pp 35-46 DOI: 10.7508/imi.2016.01.004 Convergence of an Approach for Solving Fredholm Functional Integral Equation Naer Aghazadeh,
More informationApproximate Analytical Solution for Quadratic Riccati Differential Equation
Iranian J. of Numerical Analyi and Optimization Vol 3, No. 2, 2013), pp 21-31 Approximate Analytical Solution for Quadratic Riccati Differential Equation H. Aminikhah Abtract In thi paper, we introduce
More informationChapter 4. The Laplace Transform Method
Chapter 4. The Laplace Tranform Method The Laplace Tranform i a tranformation, meaning that it change a function into a new function. Actually, it i a linear tranformation, becaue it convert a linear combination
More information7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281
72 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 28 and i 2 Show how Euler formula (page 33) can then be ued to deduce the reult a ( a) 2 b 2 {e at co bt} {e at in bt} b ( a) 2 b 2 5 Under what condition
More informationDigital Control System
Digital Control Sytem Summary # he -tranform play an important role in digital control and dicrete ignal proceing. he -tranform i defined a F () f(k) k () A. Example Conider the following equence: f(k)
More informationLAPLACE TRANSFORM REVIEW SOLUTIONS
LAPLACE TRANSFORM REVIEW SOLUTIONS. Find the Laplace tranform for the following function. If an image i given, firt write out the function and then take the tranform. a e t inh4t From #8 on the table:
More informationLaplace Homotopy Analysis Method for Solving Fractional Order Partial Differential Equations
IOSR Journal of Mathematic (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume 3, Iue 5 Ver. I (Sep. - Oct. 207), PP 49-54 www.iorjournal.org Laplace Homotopy Analyi Method for Solving Fractional Order
More informationL 2 -transforms for boundary value problems
Computational Method for Differential Equation http://cmde.tabrizu.ac.ir Vol. 6, No., 8, pp. 76-85 L -tranform for boundary value problem Arman Aghili Department of applied mathematic, faculty of mathematical
More informationChapter 7: The Laplace Transform Part 1
Chapter 7: The Laplace Tranform Part 1 王奕翔 Department of Electrical Engineering National Taiwan Univerity ihwang@ntu.edu.tw November 26, 213 1 / 34 王奕翔 DE Lecture 1 Solving an initial value problem aociated
More informationA Constraint Propagation Algorithm for Determining the Stability Margin. The paper addresses the stability margin assessment for linear systems
A Contraint Propagation Algorithm for Determining the Stability Margin of Linear Parameter Circuit and Sytem Lubomir Kolev and Simona Filipova-Petrakieva Abtract The paper addree the tability margin aement
More informationMath 201 Lecture 17: Discontinuous and Periodic Functions
Math 2 Lecture 7: Dicontinuou and Periodic Function Feb. 5, 22 Many example here are taken from the textbook. he firt number in () refer to the problem number in the UA Cutom edition, the econd number
More informationReading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions
Chapter 4 Laplace Tranform 4 Introduction Reading aignment: In thi chapter we will cover Section 4 45 4 Definition and the Laplace tranform of imple function Given f, a function of time, with value f(t
More informationReading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions
Chapter 4 Laplace Tranform 4 Introduction Reading aignment: In thi chapter we will cover Section 4 45 4 Definition and the Laplace tranform of imple function Given f, a function of time, with value f(t
More informationAnalysis of Step Response, Impulse and Ramp Response in the Continuous Stirred Tank Reactor System
ISSN: 454-50 Volume 0 - Iue 05 May 07 PP. 7-78 Analyi of Step Repone, Impule and Ramp Repone in the ontinuou Stirred Tank Reactor Sytem * Zohreh Khohraftar, Pirouz Derakhhi, (Department of hemitry, Science
More informationPractice Problems - Week #7 Laplace - Step Functions, DE Solutions Solutions
For Quetion -6, rewrite the piecewie function uing tep function, ketch their graph, and find F () = Lf(t). 0 0 < t < 2. f(t) = (t 2 4) 2 < t In tep-function form, f(t) = u 2 (t 2 4) The graph i the olid
More informationLinear System Fundamentals
Linear Sytem Fundamental MEM 355 Performance Enhancement of Dynamical Sytem Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Content Sytem Repreentation Stability Concept
More informationChapter 7: The Laplace Transform
Chapter 7: The Laplace Tranform 王奕翔 Department of Electrical Engineering National Taiwan Univerity ihwang@ntu.edu.tw November 2, 213 1 / 25 王奕翔 DE Lecture 1 Solving an initial value problem aociated with
More informationDesign By Emulation (Indirect Method)
Deign By Emulation (Indirect Method he baic trategy here i, that Given a continuou tranfer function, it i required to find the bet dicrete equivalent uch that the ignal produced by paing an input ignal
More informationSolutions for homework 8
Solution for homework 8 Section. Baic propertie of the Laplace Tranform. Ue the linearity of the Laplace tranform (Propoition.7) and Table of Laplace tranform on page 04 to find the Laplace tranform of
More informationName: Solutions Exam 2
Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer
More informationLaplace Transformation
Univerity of Technology Electromechanical Department Energy Branch Advance Mathematic Laplace Tranformation nd Cla Lecture 6 Page of 7 Laplace Tranformation Definition Suppoe that f(t) i a piecewie continuou
More informationBogoliubov Transformation in Classical Mechanics
Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How
More informationThe Riemann Transform
The Riemann Tranform By Armando M. Evangelita Jr. armando78973@gmail.com Augut 28, 28 ABSTRACT In hi 859 paper, Bernhard Riemann ued the integral equation f (x ) x dx to develop an explicit formula for
More informationSOME RESULTS ON INFINITE POWER TOWERS
NNTDM 16 2010) 3, 18-24 SOME RESULTS ON INFINITE POWER TOWERS Mladen Vailev - Miana 5, V. Hugo Str., Sofia 1124, Bulgaria E-mail:miana@abv.bg Abtract To my friend Kratyu Gumnerov In the paper the infinite
More informationIntroduction to Laplace Transform Techniques in Circuit Analysis
Unit 6 Introduction to Laplace Tranform Technique in Circuit Analyi In thi unit we conider the application of Laplace Tranform to circuit analyi. A relevant dicuion of the one-ided Laplace tranform i found
More informationTRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL
GLASNIK MATEMATIČKI Vol. 38583, 73 84 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National
More informationName: Solutions Exam 2
Name: Solution Exam Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will
More informationA SIMPLE NASH-MOSER IMPLICIT FUNCTION THEOREM IN WEIGHTED BANACH SPACES. Sanghyun Cho
A SIMPLE NASH-MOSER IMPLICIT FUNCTION THEOREM IN WEIGHTED BANACH SPACES Sanghyun Cho Abtract. We prove a implified verion of the Nah-Moer implicit function theorem in weighted Banach pace. We relax the
More informationME 375 EXAM #1 Tuesday February 21, 2006
ME 375 EXAM #1 Tueday February 1, 006 Diviion Adam 11:30 / Savran :30 (circle one) Name Intruction (1) Thi i a cloed book examination, but you are allowed one 8.5x11 crib heet. () You have one hour to
More informationSTABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION OF FIRST ORDER WITH VARIABLE DELAYS
Bulletin of Mathematical Analyi and Application ISSN: 1821-1291, URL: http://bmathaa.org Volume 1 Iue 2(218), Page 19-3. STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION OF FIRST ORDER WITH VARIABLE
More informationPosition. If the particle is at point (x, y, z) on the curved path s shown in Fig a,then its location is defined by the position vector
34 C HAPTER 1 KINEMATICS OF A PARTICLE 1 1.5 Curvilinear Motion: Rectangular Component Occaionall the motion of a particle can bet be decribed along a path that can be epreed in term of it,, coordinate.
More informationUnbounded solutions of second order discrete BVPs on infinite intervals
Available online at www.tjna.com J. Nonlinear Sci. Appl. 9 206), 357 369 Reearch Article Unbounded olution of econd order dicrete BVP on infinite interval Hairong Lian a,, Jingwu Li a, Ravi P Agarwal b
More informationReformulation of Block Implicit Linear Multistep Method into Runge Kutta Type Method for Initial Value Problem
International Journal of Science and Technology Volume 4 No. 4, April, 05 Reformulation of Block Implicit Linear Multitep Method into Runge Kutta Type Method for Initial Value Problem Muhammad R., Y. A
More informationCalculation of the temperature of boundary layer beside wall with time-dependent heat transfer coefficient
Ŕ periodica polytechnica Mechanical Engineering 54/1 21 15 2 doi: 1.3311/pp.me.21-1.3 web: http:// www.pp.bme.hu/ me c Periodica Polytechnica 21 RESERCH RTICLE Calculation of the temperature of boundary
More informationinto a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get
Lecture 25 Introduction to Some Matlab c2d Code in Relation to Sampled Sytem here are many way to convert a continuou time function, { h( t) ; t [0, )} into a dicrete time function { h ( k) ; k {0,,, }}
More informationCorrection for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002
Correction for Simple Sytem Example and Note on Laplace Tranform / Deviation Variable ECHE 55 Fall 22 Conider a tank draining from an initial height of h o at time t =. With no flow into the tank (F in
More informationINITIAL VALUE PROBLEMS OF FRACTIONAL ORDER HADAMARD-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equation, Vol. 205 205), No. 77, pp. 9. ISSN: 072-669. URL: http://ejde.math.txtate.edu or http://ejde.math.unt.edu ftp ejde.math.txtate.edu INITIAL VALUE PROBLEMS OF
More informationCHAPTER 4 DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL
98 CHAPTER DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL INTRODUCTION The deign of ytem uing tate pace model for the deign i called a modern control deign and it i
More informationAdvanced D-Partitioning Analysis and its Comparison with the Kharitonov s Theorem Assessment
Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 Advanced D-Partitioning Analyi and it Comparion with the haritonov Theorem Aement amen M. Yanev Profeor,
More informationCollege of Engineering Mechanics and Soft Materials, Hohai University, Jiangning, Nanjing, China b
THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 S65 A MODIFICATION FRACTIONA VARIATIONA ITERATION METHOD FOR SOVING NON-INEAR GAS DYNAMIC AND COUPED KdV EQUATIONS INVOVING OCA FRACTIONA OPERATORS
More informationChapter 13. Root Locus Introduction
Chapter 13 Root Locu 13.1 Introduction In the previou chapter we had a glimpe of controller deign iue through ome imple example. Obviouly when we have higher order ytem, uch imple deign technique will
More informationTAYLOR POLYNOMIALS FOR NABLA DYNAMIC EQUATIONS ON TIME SCALES
TAYLOR POLYNOMIALS FOR NABLA DYNAMIC EQUATIONS ON TIME SCALES DOUGLAS R. ANDERSON Abtract. We are concerned with the repreentation of polynomial for nabla dynamic equation on time cale. Once etablihed,
More informationPacific Journal of Mathematics
Pacific Journal of Mathematic OSCILLAION AND NONOSCILLAION OF FORCED SECOND ORDER DYNAMIC EQUAIONS MARIN BOHNER AND CHRISOPHER C. ISDELL Volume 230 No. March 2007 PACIFIC JOURNAL OF MAHEMAICS Vol. 230,
More informationMarch 18, 2014 Academic Year 2013/14
POLITONG - SHANGHAI BASIC AUTOMATIC CONTROL Exam grade March 8, 4 Academic Year 3/4 NAME (Pinyin/Italian)... STUDENT ID Ue only thee page (including the back) for anwer. Do not ue additional heet. Ue of
More informationName: Solutions Exam 3
Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer
More informationNULL HELIX AND k-type NULL SLANT HELICES IN E 4 1
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 57, No. 1, 2016, Page 71 83 Publihed online: March 3, 2016 NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 JINHUA QIAN AND YOUNG HO KIM Abtract. We tudy
More informationChapter 5 Consistency, Zero Stability, and the Dahlquist Equivalence Theorem
Chapter 5 Conitency, Zero Stability, and the Dahlquit Equivalence Theorem In Chapter 2 we dicued convergence of numerical method and gave an experimental method for finding the rate of convergence (aka,
More informationEstimation of Peaked Densities Over the Interval [0,1] Using Two-Sided Power Distribution: Application to Lottery Experiments
MPRA Munich Peronal RePEc Archive Etimation of Peaed Denitie Over the Interval [0] Uing Two-Sided Power Ditribution: Application to Lottery Experiment Krzyztof Konte Artal Invetment 8. April 00 Online
More informationSOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5
SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5 Problem : For each of the following function do the following: (i) Write the function a a piecewie function and ketch it graph, (ii) Write the function a a combination
More informationManprit Kaur and Arun Kumar
CUBIC X-SPLINE INTERPOLATORY FUNCTIONS Manprit Kaur and Arun Kumar manpreet2410@gmail.com, arun04@rediffmail.com Department of Mathematic and Computer Science, R. D. Univerity, Jabalpur, INDIA. Abtract:
More informationRESEARCH ON THE THEORIES OF BEM BASED CYCLOSTATIONARY NEAR FIELD ACOUSTIC HOLOGRAPHY. Haibin Zhang, Weikang Jiang and Quan Wan
ICV14 Cairn Autralia 9-12 July, 2007 REEARCH ON THE THEORIE OF BEM BAED CYCLOTATIONARY NEAR FIELD ACOUTIC HOLOGRAPHY Haibin Zhang, Weikang Jiang and Quan Wan tate Key Laboratory of Mechanical ytem and
More information[Saxena, 2(9): September, 2013] ISSN: Impact Factor: INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY
[Saena, (9): September, 0] ISSN: 77-9655 Impact Factor:.85 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY Contant Stre Accelerated Life Teting Uing Rayleigh Geometric Proce
More informationThe Power Series Expansion on a Bulge Heaviside Step Function
Applied Mathematical Science, Vol 9, 05, no 3, 5-9 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/0988/am054009 The Power Serie Expanion on a Bulge Heaviide Step Function P Haara and S Pothat Department of
More informationRao Transforms: A New Approach to Integral and Differential Equations
Rao Tranform: A New Approach to Integral and Differential Equation Dr. Muralidhara SubbaRao (Rao) SUNY at Stony Brook, murali@ece.unyb.edu,, rao@integralreearch.net Rao Tranform (RT) provide a brand new
More informationDemonstration of Riemann Hypothesis
Demontration of Riemann Hypothei Diego arin June 2, 204 Abtract We define an infinite ummation which i proportional to the revere of Riemann Zeta function ζ(). Then we demontrate that uch function can
More informationChapter 4 Interconnection of LTI Systems
Chapter 4 Interconnection of LTI Sytem 4. INTRODUCTION Block diagram and ignal flow graph are commonly ued to decribe a large feedback control ytem. Each block in the ytem i repreented by a tranfer function,
More informationSolving Differential Equations by the Laplace Transform and by Numerical Methods
36CH_PHCalter_TechMath_95099 3//007 :8 PM Page Solving Differential Equation by the Laplace Tranform and by Numerical Method OBJECTIVES When you have completed thi chapter, you hould be able to: Find the
More informationAnalysis and Design of a Third Order Phase-Lock Loop
Analyi Deign of a Third Order Phae-Lock Loop DANIEL Y. ABRAMOVITCH Ford Aeropace Corporation 3939 Fabian Way, MS: X- Palo Alto, CA 94303 Abtract Typical implementation of a phae-lock loop (PLL) are econd
More informationAvoiding Forbidden Submatrices by Row Deletions
Avoiding Forbidden Submatrice by Row Deletion Sebatian Wernicke, Jochen Alber, Jen Gramm, Jiong Guo, and Rolf Niedermeier Wilhelm-Schickard-Intitut für Informatik, niverität Tübingen, Sand 13, D-72076
More informationCopyright 1967, by the author(s). All rights reserved.
Copyright 1967, by the author(). All right reerved. Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or
More informationLecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas)
Lecture 7: Analytic Function and Integral (See Chapter 4 in Boa) Thi i a good point to take a brief detour and expand on our previou dicuion of complex variable and complex function of complex variable.
More informationON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION. Xiaoqun Wang
Proceeding of the 2008 Winter Simulation Conference S. J. Maon, R. R. Hill, L. Mönch, O. Roe, T. Jefferon, J. W. Fowler ed. ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION Xiaoqun Wang
More informationAn Inequality for Nonnegative Matrices and the Inverse Eigenvalue Problem
An Inequality for Nonnegative Matrice and the Invere Eigenvalue Problem Robert Ream Program in Mathematical Science The Univerity of Texa at Dalla Box 83688, Richardon, Texa 7583-688 Abtract We preent
More informationQuestion 1 Equivalent Circuits
MAE 40 inear ircuit Fall 2007 Final Intruction ) Thi exam i open book You may ue whatever written material you chooe, including your cla note and textbook You may ue a hand calculator with no communication
More informationPhysics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014
Phyic 7 Graduate Quantum Mechanic Solution to inal Eam all 0 Each quetion i worth 5 point with point for each part marked eparately Some poibly ueful formula appear at the end of the tet In four dimenion
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Laplace Tranform Paul Dawkin Table of Content Preface... Laplace Tranform... Introduction... The Definition... 5 Laplace Tranform... 9 Invere Laplace Tranform... Step Function...4
More informationMidterm Test Nov 10, 2010 Student Number:
Mathematic 265 Section: 03 Verion A Full Name: Midterm Tet Nov 0, 200 Student Number: Intruction: There are 6 page in thi tet (including thi cover page).. Caution: There may (or may not) be more than one
More informationMacromechanical Analysis of a Lamina
3, P. Joyce Macromechanical Analyi of a Lamina Generalized Hooke Law ij Cijklε ij C ijkl i a 9 9 matri! 3, P. Joyce Hooke Law Aume linear elatic behavior mall deformation ε Uniaial loading 3, P. Joyce
More informationPusan National University
Chapter 12. DESIGN VIA STATE SPACE Puan National Univerity oratory Table of Content v v v v v v v v Introduction Controller Deign Controllability Alternative Approache to Controller Deign Oberver Deign
More informationHELICAL TUBES TOUCHING ONE ANOTHER OR THEMSELVES
15 TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS 0 ISGG 1-5 AUGUST, 0, MONTREAL, CANADA HELICAL TUBES TOUCHING ONE ANOTHER OR THEMSELVES Peter MAYRHOFER and Dominic WALTER The Univerity of Innbruck,
More informationINITIAL-VALUE PROBLEMS FOR HYBRID HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equation, Vol. 204 204, No. 6, pp. 8. ISSN: 072-669. URL: http://ejde.math.txtate.edu or http://ejde.math.unt.edu ftp ejde.math.txtate.edu INITIAL-VALUE PROBLEMS FOR
More informationReliability Analysis of Embedded System with Different Modes of Failure Emphasizing Reboot Delay
International Journal of Applied Science and Engineering 3., 4: 449-47 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay Deepak Kumar* and S. B. Singh Department
More informationFOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS
FOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS Nguyen Thanh Lan Department of Mathematic Wetern Kentucky Univerity Email: lan.nguyen@wku.edu ABSTRACT: We ue Fourier erie to find a neceary
More informationA BATCH-ARRIVAL QUEUE WITH MULTIPLE SERVERS AND FUZZY PARAMETERS: PARAMETRIC PROGRAMMING APPROACH
Mathematical and Computational Application Vol. 11 No. pp. 181-191 006. Aociation for Scientific Reearch A BATCH-ARRIVA QEE WITH MTIPE SERVERS AND FZZY PARAMETERS: PARAMETRIC PROGRAMMING APPROACH Jau-Chuan
More informationEvolutionary Algorithms Based Fixed Order Robust Controller Design and Robustness Performance Analysis
Proceeding of 01 4th International Conference on Machine Learning and Computing IPCSIT vol. 5 (01) (01) IACSIT Pre, Singapore Evolutionary Algorithm Baed Fixed Order Robut Controller Deign and Robutne
More informationwhere F (x) (called the Similarity Factor (SF)) denotes the function
italian journal of pure and applied mathematic n. 33 014 15 34) 15 GENERALIZED EXPONENTIAL OPERATORS AND DIFFERENCE EQUATIONS Mohammad Aif 1 Anju Gupta Department of Mathematic Kalindi College Univerity
More informationMinimal state space realization of MIMO systems in the max algebra
KULeuven Department of Electrical Engineering (ESAT) SISTA Technical report 94-54 Minimal tate pace realization of MIMO ytem in the max algebra B De Schutter and B De Moor If you want to cite thi report
More information5.5 Application of Frequency Response: Signal Filters
44 Dynamic Sytem Second order lowpa filter having tranfer function H()=H ()H () u H () H () y Firt order lowpa filter Figure 5.5: Contruction of a econd order low-pa filter by combining two firt order
More informationWe are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors
We are IntechOpen, the world leading publiher of Open Acce book Built by cientit, for cientit 3,5 8,.7 M Open acce book available International author and editor Download Our author are among the 5 Countrie
More informationThe combined Laplace-homotopy analysis method for partial differential equations
Available online at wwwir-publicationcom/jmc J Math Computer Sci 6 (26), 88 2 Reearch Article The combined Laplace-homotopy analyi method for partial differential equation Javad Vahidi Department of Mathematic,
More informationThe Laplace Transform , Haynes Miller and Jeremy Orloff
The Laplace Tranform 8.3, Hayne Miller and Jeremy Orloff Laplace tranform baic: introduction An operator take a function a input and output another function. A tranform doe the ame thing with the added
More informationDesign of Digital Filters
Deign of Digital Filter Paley-Wiener Theorem [ ] ( ) If h n i a caual energy ignal, then ln H e dω< B where B i a finite upper bound. One implication of the Paley-Wiener theorem i that a tranfer function
More informationThe Laplace Transform
Chapter 7 The Laplace Tranform 85 In thi chapter we will explore a method for olving linear differential equation with contant coefficient that i widely ued in electrical engineering. It involve the tranformation
More informationON THE SMOOTHNESS OF SOLUTIONS TO A SPECIAL NEUMANN PROBLEM ON NONSMOOTH DOMAINS
Journal of Pure and Applied Mathematic: Advance and Application Volume, umber, 4, Page -35 O THE SMOOTHESS OF SOLUTIOS TO A SPECIAL EUMA PROBLEM O OSMOOTH DOMAIS ADREAS EUBAUER Indutrial Mathematic Intitute
More informationThe Hassenpflug Matrix Tensor Notation
The Haenpflug Matrix Tenor Notation D.N.J. El Dept of Mech Mechatron Eng Univ of Stellenboch, South Africa e-mail: dnjel@un.ac.za 2009/09/01 Abtract Thi i a ample document to illutrate the typeetting of
More informationMAE140 Linear Circuits Fall 2012 Final, December 13th
MAE40 Linear Circuit Fall 202 Final, December 3th Intruction. Thi exam i open book. You may ue whatever written material you chooe, including your cla note and textbook. You may ue a hand calculator with
More informationChapter 2 Sampling and Quantization. In order to investigate sampling and quantization, the difference between analog
Chapter Sampling and Quantization.1 Analog and Digital Signal In order to invetigate ampling and quantization, the difference between analog and digital ignal mut be undertood. Analog ignal conit of continuou
More informationin a circular cylindrical cavity K. Kakazu Department of Physics, University of the Ryukyus, Okinawa , Japan Y. S. Kim
Quantization of electromagnetic eld in a circular cylindrical cavity K. Kakazu Department of Phyic, Univerity of the Ryukyu, Okinawa 903-0, Japan Y. S. Kim Department of Phyic, Univerity of Maryland, College
More informationThen C pid (s) S h -stabilizes G(s) if and only if Ĉpid(ŝ) S 0 - stabilizes Ĝ(ŝ). For any ρ R +, an RCF of Ĉ pid (ŝ) is given by
9 American Control Conference Hyatt Regency Riverfront, St. Loui, MO, USA June -, 9 WeC5.5 PID Controller Synthei with Shifted Axi Pole Aignment for a Cla of MIMO Sytem A. N. Gündeş and T. S. Chang Abtract
More informationME 375 FINAL EXAM Wednesday, May 6, 2009
ME 375 FINAL EXAM Wedneday, May 6, 9 Diviion Meckl :3 / Adam :3 (circle one) Name_ Intruction () Thi i a cloed book examination, but you are allowed three ingle-ided 8.5 crib heet. A calculator i NOT allowed.
More informationMathematical modeling of control systems. Laith Batarseh. Mathematical modeling of control systems
Chapter two Laith Batareh Mathematical modeling The dynamic of many ytem, whether they are mechanical, electrical, thermal, economic, biological, and o on, may be decribed in term of differential equation
More informationControl of Delayed Integrating Processes Using Two Feedback Controllers R MS Approach
Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) Control of Delayed Integrating Procee Uing Two Feedback Controller R MS Approach LIBOR
More informationFeedback Control Systems (FCS)
Feedback Control Sytem (FCS) Lecture19-20 Routh-Herwitz Stability Criterion Dr. Imtiaz Huain email: imtiaz.huain@faculty.muet.edu.pk URL :http://imtiazhuainkalwar.weebly.com/ Stability of Higher Order
More informationPower Technology Issue 102. Comparison of Conventional and Elementary Block Method of Handling Transfer Function Blocks in PSS E Dynamic Models
SEMENS Power Tranmiion and itribution Power Technology ue 102 Comparion of Conventional and Elementary Block Method of Handling Tranfer Function Block in PSS E ynamic Model Jay Senthil Senior Staff Software
More informationWeek 3 Statistics for bioinformatics and escience
Week 3 Statitic for bioinformatic and escience Line Skotte 28. november 2008 2.9.3-4) In thi eercie we conider microrna data from Human and Moue. The data et repreent 685 independent realiation of the
More informationonline learning Unit Workbook 4 RLC Transients
online learning Pearon BTC Higher National in lectrical and lectronic ngineering (QCF) Unit 5: lectrical & lectronic Principle Unit Workbook 4 in a erie of 4 for thi unit Learning Outcome: RLC Tranient
More informationGiven the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is
EE 4G Note: Chapter 6 Intructor: Cheung More about ZSR and ZIR. Finding unknown initial condition: Given the following circuit with unknown initial capacitor voltage v0: F v0/ / Input xt 0Ω Output yt -
More informationAN EXAMPLE FOR THE GENERALIZATION OF THE INTEGRATION OF SPECIAL FUNCTIONS BY USING THE LAPLACE TRANSFORM
Journal of Inequalitie Special Function ISSN: 7-433, URL: http://www.iliria.com Volume 6 Iue 5, Page 5-3. AN EXAMPLE FOR THE GENERALIZATION OF THE INTEGRATION OF SPECIAL FUNCTIONS BY USING THE LAPLACE
More informationInvariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Transformations
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 7, 8197 1998 ARTICLE NO AY986078 Invariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Tranformation Evelyn
More information