Procedia Computer Science

Similar documents
5.1-The Initial-Value Problems For Ordinary Differential Equations

Contraction Mapping Principle Approach to Differential Equations

LAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS

Temperature Rise of the Earth

Convergence of Singular Integral Operators in Weighted Lebesgue Spaces

4.8 Improper Integrals

A Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION

Solutions for Nonlinear Partial Differential Equations By Tan-Cot Method

ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX

e t dt e t dt = lim e t dt T (1 e T ) = 1

GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS

September 20 Homework Solutions

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM

Systems Variables and Structural Controllability: An Inverted Pendulum Case

Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals

On the Pseudo-Spectral Method of Solving Linear Ordinary Differential Equations

Positive and negative solutions of a boundary value problem for a

Green s Functions and Comparison Theorems for Differential Equations on Measure Chains

Application on Inner Product Space with. Fixed Point Theorem in Probabilistic

Mathematics 805 Final Examination Answers

0 for t < 0 1 for t > 0

Asymptotic relationship between trajectories of nominal and uncertain nonlinear systems on time scales

A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES

ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS

IX.2 THE FOURIER TRANSFORM

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 9: The High Beta Tokamak

Chapter Direct Method of Interpolation

FRACTIONAL-order differential equations (FDEs) are

Analytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function

f t f a f x dx By Lin McMullin f x dx= f b f a. 2

RESPONSE UNDER A GENERAL PERIODIC FORCE. When the external force F(t) is periodic with periodτ = 2π

Think of the Relationship Between Time and Space Again

ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX.

..,..,.,

1. Introduction. 1 b b

Chapter 2. First Order Scalar Equations

New Inequalities in Fractional Integrals

SOLUTION FOR A SYSTEM OF FRACTIONAL HEAT EQUATIONS OF NANOFLUID ALONG A WEDGE

ME 391 Mechanical Engineering Analysis

2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )

Physics 2A HW #3 Solutions

How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.

NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory and Application to Heat Transfer Model

The solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.

On Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives

Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Soliton Scattering on the External Potential in Weakly Nonlocal Nonlinear Media

Journal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle

Approximation and numerical methods for Volterra and Fredholm integral equations for functions with values in L-spaces

A new model for solving fuzzy linear fractional programming problem with ranking function

How to prove the Riemann Hypothesis

A NUMERICAL SOLUTION OF THE URYSOHN-TYPE FREDHOLM INTEGRAL EQUATIONS

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function

Abstract. W.W. Memudu 1 and O.A. Taiwo, 2

ON THE STABILITY OF DELAY POPULATION DYNAMICS RELATED WITH ALLEE EFFECTS. O. A. Gumus and H. Kose

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).

3. Renewal Limit Theorems

can be viewed as a generalized product, and one for which the product of f and g. That is, does

ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS OF FOURTH-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH REGULARLY VARYING COEFFICIENTS

Research Article The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

Differential Equations

( ) ( ) ( ) ( ) ( ) ( y )

MTH 146 Class 11 Notes

CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

A Time Truncated Improved Group Sampling Plans for Rayleigh and Log - Logistic Distributions

Fractional operators with exponential kernels and a Lyapunov type inequality

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions

Average & instantaneous velocity and acceleration Motion with constant acceleration

Solutions to Problems from Chapter 2

Research Article Generalized Fractional Integral Inequalities for Continuous Random Variables

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

FURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)

1.0 Electrical Systems

Citation Abstract and Applied Analysis, 2013, v. 2013, article no

On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function

An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples.

FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR

Math 23 Spring Differential Equations. Final Exam Due Date: Tuesday, June 6, 5pm

MAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017

M r. d 2. R t a M. Structural Mechanics Section. Exam CT5141 Theory of Elasticity Friday 31 October 2003, 9:00 12:00 hours. Problem 1 (3 points)

Probability, Estimators, and Stationarity

Solutions to Assignment 1

t + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that

System of Linear Differential Equations

Integral Transform. Definitions. Function Space. Linear Mapping. Integral Transform

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.

A Kalman filtering simulation

HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA

1 jordan.mcd Eigenvalue-eigenvector approach to solving first order ODEs. -- Jordan normal (canonical) form. Instructor: Nam Sun Wang

Chapter 6. Systems of First Order Linear Differential Equations

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon

BU Macro BU Macro Fall 2008, Lecture 4

Ordinary Differential Equations

LAPLACE TRANSFORMS. 1. Basic transforms

Ann. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:

Transcription:

Procedi Compuer Science 00 (0) 000 000 Procedi Compuer Science www.elsevier.com/loce/procedi The Third Informion Sysems Inernionl Conference The Exisence of Polynomil Soluion of he Nonliner Dynmicl Sysems Gunwn Nugroho *, Took Soehrno b, Took R. Biyno c Deprmen of Engineering Physics, Insiu Teknologi Sepuluh Nopember Jl Arief Rhmn Hkim, Surby, Indonesi (60) Absrc The mehod o solve he nonliner differenil euion wih vrible coefficiens is presened in his work. The mehod is bsed on he pplicion of generl Ricci euion, which is subsiued ino he considered nonliner euion nd produce polynomil euion. The polynomil order higher hn four is reduced ino he fourh order nd solved by rdicls. The soluions of he Ricci nd polynomil re hen combined performing he complee soluion which is smooh for ll ime. 0 Published by Elsevier Ld. Selecion nd/or peer-review under responsibiliy of [nme orgnizer] Keywords: Nonliner ordinry differenil euions, he Ricci euion, nlyicl soluions, dynmicl sysems. Inroducion Mny of he more relisic modeling in he dynmicl sysems re bsed on nonliner differenil euions []. The pplicion someimes involve wih vrible coefficiens []. I is well-known h he uliive nlysis of he nonliner differenil euions is sufficien o know he globl behvior on he soluions [3,4]. However, he conceps will no be very useful unil he explici soluions re produced. The explici soluions re cpble o describe he deil feures of he sysems [5]. They my lso help o exend he exisence, uniueness nd regulriy properies of he soluions which re obined from uliive nlysis [6,7]. Therefore, mehod for genering nlyicl soluions of he nonliner differenil euions is imporn from boh physicl nd mhemicl poin of views [8]. The cse of conrol sysem wih nonhomogenous physicl propery such s in vibrion wves is of priculr ineres becuse i resuled in he model wih vrible coefficiens. Since h specific problem rcs mny mhemicins nd *Corresponding uhor gunwn@ep.is.c.id, gunwnzz@yhoo.com, gunwnf3@gmil.com b ooksf@ep.is.c.id, ooksf@yhoo.com c rb@ep.is.c.id, cllrb@gmil.com, cllrb@yhoo.com

Gunwn Gunwn Gunwn Nugroho Nugroho Gunwn e e l./ l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science00 (05) 000 000 physiciss, he mehods o obin exc nd pproxime soluions re ckled sysemiclly nd some ineresing resuls my be produced [6,9,0]. In his pper, he following clss of euion will be discussed, m n m n y y y y y y y m y m n m n p m p p f, y,,..., f, y,,...,... f, y,,..., y 0 () which he mehod for obining exc soluions is conduced by using he Ricci euion. The procedure for obining he soluions is derived in uie simple wy which is bsed on he subsiuion ino he considered euion. The resuling expression is hen produced polynomil euion which hen combined wih he soluion of he Ricci o form finl soluions. The mehod h is explined in his work is new nd he resuls my hve significn conribuions in he re of differenil nd inegrl euions.. Mehod of Genering Soluions Le us consider he following nonliner differenil euions wih vrible coefficiens, y y F, y,,..., 0 () The soluion of () is consruced from he following Ricci euion, y y y 3 (b) wih,, nd 3 re vrible coefficiens. Before subsiue (b) ino (), he following relion is produced, y y y 3 y y y b y y 3 y y 3 3 3 3 3 y 3 4 3 6 3 y 6 y 8 3 5 7 4 y... 3 3 3 3 4 3 8 3 3 y 3 3 3... y... (c) (d) (e) The procedure described in (b d) will produce he following sysem, y y y 3 (f)

r r r r3 r 6 5 4 Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science 00 (05) 000 000 3 y y y... y y 0 (g) where r re he coefficiens from he subsiuion ino (). Firsly, he Ricci euion is rnsformed s in he following, u u 3u 0 (3) Wih u y u. b b Lemm: Consider euion (3), le A nd 3 A A b b, euion (3c) hen hs closed-form exc soluion which b, bnd A re redefined. The soluion will leve A s n rbirry funcion. b Proof: Suppose, A b rnsformed ino, u A bu b Au 0 b b Le, Au Z, he euion cn be rerrnged s, Z A Z b b Z 0 or b A b A Z b Z A A A b A A b nd 3 A A b b b A A A A A A Z 0 A b A b A Le, b b A b A b A A A A A A A 0 3 0 A A A, he bove euion is (3b) (3c), he euion for A is hen, Muliply by he funcion nd differenie (3d) once o perform he non homogenous euion, 3 3 3 A A A 3 3 0 A where will be deermined ler. Suppose h, (3d) (4)

Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 4 Auhor nme / Procedi Compuer Science00 (05) 000 000 b3 x x A3 nd 3 A4 A3 b3 (4b) The euion is hen rewrien s, A3 3 b3 b 4 A4 3 0 (4c) b3 A A A A 3 3 If x A4 3 hen, C exp 3 d, hus euion (4c) cn A be rrnged s, b 3 H H 3 4 A b H H b 3 b3 H b 4 H A4 0 b3 (5) wih, H. The vribles in (5) is reled s in he following, A A3 b3 b 4 3 4 0 nd b b b3 A3 A3 b3 A 3 A4 b3 b3 The relion for b 3 is, 3 b3 A3 A 3 3 3 4 b b b3 x Subsiue ino, A3 b3 o give he expression for b 4 s, 3 r 3 A 3 (5d) x Performing (5d) ino, 3 A4 A3, o produce A 3, A4 3 3 A (5e) 3 3 3 The nex sep is o rele he funcion A 4, s A4 3, which produce he soluion of s, Ce. Thus, by (5b) he euion for A 4 is, (5b) (5c)

r b 4r 3 A4 A3 or Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science 00 (05) 000 000 5 3 A4 A3 or 3 A4 3 3 3 which hen solves s in he following, (6b) where, 3.5 4.5 3.5 (6c) The euion for H becomes, Ce d 3 H H H b 3 5 b 5 0 (7) wih, A3 b3 b5 b3 A 3 A4 A3 b3 b3. Euion (7) cn be rnsformed ino, K K b 5 k b 5 0 H wih, e Kd. The bove relion hs s priculr soluion, he generl soluion is governed by K, which resuled in, l (6) l b5 l 0 or b 5 l l l l l The soluion for A is hen, (7b)

Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 6 Auhor nme / Procedi Compuer Science00 (05) 000 000 Kd d H e d C4 l e d C4 A exp b5d exp b5d d C3 d b5d C e e d C4 C e e d C 3 d C4 (7c) The euion (3c) becomes, Z b Z A A A 0 b A A The soluion for he Ricci euion is hen, u A Z y u Z A d d A C A A e d C C A A e d C A where A is deermined by (7c). (8) (8b) Thus, he procedure leves A s n undefined vrible, his proves lemm Since he soluion of he polynomil by rdicl is limied o he fourh order, he reducion of polynomil order should be performed. The ineresed reder will find he mehod of reducion in []. Theorem : Consider he soluion of he euion (3c) s described by (8b). By combining wih he roo of polynomil, y, he resuling expressions hus complee he soluion of he sysem defined byricci nd polynomil euions. Proof: Le y is he polynomil soluion, he combinion wih (8b) will deermined A s, d d (8c) A C3 e A e A which hen proves heorem. 3. Soluion Properies Now we re sep o nswer nd proof he uesions of exisence nd uniueness of smooh soluion. Since he coefficiens of he Ricci euion re rbirry, hey cn become powerful objecs o jusify he properies under generl iniil-boundry condiions.

Gunwn Nugroho e l. / Informion Sysems Inernionl Conference 05 Auhor nme / Procedi Compuer Science 00 (05) 000 000 7 3.. Uniueness Propery Le us consider he second order ODE which he soluion nd iniil condiion re reled s, Y y y Y 0 y 0 y 0 0. Subsiuing he soluion pirs ino (8b) will hen produce nd uniue soluion for y since i is from liner ODE. As for he polynomil soluion, under he proper selecion of coefficiens will lso produce uniue soluion. 3.. Exisence nd Regulriy Properies Apr from uniueness, he exisence nd regulriy properies depend on he chosen funcion of he vrible coefficiens of Ricci euions, i. Also he proper selecion of coefficiens will produce for ll ime. 4. Conclusions The mehod for he genering soluion of he nonliner differenil euion is proposed in his ricle. The min sregy is o subsiue he Ricci euion ino he considered euion. The resuled polynomil is hen solved by rdicls nd combined wih he soluion of he Ricci euion. I is shown h he mehod cn obin he soluions of rbirry coefficiens nd rbirry order in closed-form. The soluion is exis nd smooh for ll ime. We pln o conduc he pplicions in our fuure works. References [] Brrio RA nd Vre C, Non-Liner Sysems, Physic A 37, 006, pp. 0 3. [] Sdri S, Rveshi MR nd Amiri S, Efficiency Anlysis of Srigh Fin wih Vrible He Trnsfer Coefficien nd Therml Conduciviy, Journl of Mechnicl Science nd Technology 6 (4), 0, pp. 83 90. [3] Dohery MFnd Ollio JM, Chos in Deerminisic Sysems: Srnge Arcors, Turbulence nd Applicions in Chemicl Engineering, Chemicl Engineering Science, Vol. 43, No., 988, pp. 39 83. [4] Clogero F, Gomez-Ulle D, Snini PM nd Sommcl M, Towrds Theory of Chos Explined s Trvel on Riemnn Surfces, J. Phy. A: Mh. Theor. 4, 009. [5] Abdel-Gwd HI, On he Behvior of Soluions of Clss of Nonliner Pril Differenil Euions, Journl of Sisicl Physics, 999, Vol. 97, Nos. /. [6] Rmos JI, Anlyicl nd Approxime Soluions o Auonomous, Nonliner, Third-Order Ordynry Differenil Euions, Nonliner Anlysis: Rel World Applicions, 00, pp. 63 636. [7] Heywood JG, Ng W nd Xie W, A Numericlly Bsed Exisence Theorem for he Nvier-Sokes Euions, Journl of Mhemicl Fluid Mechnics, 999, pp. 5 3. [8] Glkonov VA nd Svirshchevskii AR, Exc Soluions nd Invrin Subspces of Nonliner Pril Differenil Euions in Mechnics nd Physics, Tylor & Frncis Group, Boc Ron, 007. [9] Bougoff L, On he Exc Soluions for Iniil Vlue Problems of Second Order Differenil Euions, Applied Mhemics Leers, 009, pp. 48 5. [0] Adomin G, Solving Fronier Problem of Physics: The Decomposiion Mehod, Kluwer Acdemic Publishers, Dordrech, The Neherlnds, 994. [] Nugroho G, Applicion of Firs Order Polynomil Differenil Euion ofr Genering Anlyicl Soluions o he Three-Dimensionl Incompressible Nvier-Sokes Euions, Europen Journl of Mhemicl Sciences, Vo., N0., 03, pp. 7 40.

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback