SOLITONS AND CONSERVED QUANTITIES OF THE ITO EQUATION
|
|
- Bryce Pierce
- 5 years ago
- Views:
Transcription
1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 3, Number 3/, pp. 5 4 SOLITONS AND CONSERVED QUANTITIES OF THE ITO EQUATION Ghodrt EBADI, A. H. KARA, Mrko D. PETKOVIC 3, Ahmet YILDIRIM 4, Anjn BISWAS 5 University of Tbriz, Tbriz, Fculty of Mthemticl Sciences, ; Irn University of the Witwtersrnd, School of Mthemtics, Wits 5, Johnnesburg, South Afrıc 3 University of Niš, Fculty of Science nd Mthemtics, Višegrdsk 33, 8 Niš, Serbı 4 Ege University, Deprtment of Mthemtics, 35 Bornov, Izmir, Turkey University of South Florid, Deprtment of Mthemtics nd Sttistics, Tmp, FL , USA 5 Delwre Stte University, Deprtment of Mthemticl Sciences, Dover, DE 99-77, USA E-mil: bisws.njn@gmil.com This pper obtins the soliton solutions of the Ito integro-differentil eqution. The GG / method will be used to crry out the solutions of this eqution nd then the solitry wve nstz method will be used to obtin -soliton solution of this eqution. Finlly, the invrince nd multiplier pproch will be pplied to recover few of the conserved quntities of this eqution. Key words: conserved quntities; solitry wve nstz method; solitons; Ito integro-differentil eqution; GG / method.. INTRODUCTION The study of nonliner evolution eqution (NLEE) hs been going on for the pst few decdes [-6]. During this time, there hs been mesurble progress tht hs been mde. There re lots of equtions tht hve been integrted. There re vrious methods of integrbility tht hs been developed so fr. In ddition to NLEEs, there hs been growing interest in the nonliner integro-differentil evolution equtions. Some of these commonly studied integro-differentil evolution equtions re the Ito eqution, the generlized shllow wter wve eqution nd mny others. There re vrious nlyticl methods of solving these NLEEs tht hs lso been developed in the pst couple of decdes. Some of these methods re ep-function method, Fn's F -method, Riccti's eqution method, Adomin decomposition method nd mny others. In this pper, the G' / G method will be developed to solve the Ito eqution. Also, the solitry wve nstz method will integrte the Ito eqution. Finlly, the conserved quntities for the Ito eqution will be computed using the invrince nd multiplier pproch bsed on the well known result tht the Euler-Lgrnge opertor nnihiltes the totl divergence.. THE G / G METHOD In this section, the G'/ G method will be described nd pplied to obtin trveling wve solution of the Ito eqution. This eqution is studied in () dimensions s well s () dimensions. therefore, this work will be done in two following subsections... () Dimensions The ()-dimensionl form of the generlized Ito integro-differentil eqution tht is going to be studied in this subsection is given by tt t ( t t ) t q q q q qq q q d = ()
2 6 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws Here, in (), q is the dependent vrible while nd t re the independent vribles. The coefficient is constnt. The eq. () cn reduced to with using the potentil or equivlently q = v v v v v v v v =, () tt t t t t v.the eq. () is converted to the ODE = (3) 4 ( v) ceu ceu ceu u ceuu ceu u cu e u e u 3 ( v) =, (4) by the wve vribles v= u( ξ), ξ= e ct, where primes denote the derivtives with respect to ξ, nd e, c re rel constnts to be determined lter. The eq. (4) is then integrted twice. This converts it to In cu e u e u 3 ' =, (5) G'/ G method, the solution u( ξ ) of eqution (5) is considered in the finite series form N G ( ξ) i u( ξ)= Ai, (6) G ( ξ) i= where A i re positive integers with AN tht will be determined. N is positive integer tht cn be ccomplished by blncing the liner term of highest order derivtive with the highest order nonliner term in eqution (5). Here blncing u with ( u ) gives N =. Therefore, we cn write the solution of eqution (5) in the form of G ( ξ) u( ξ )= A A, A. G( ξ) The function G( ξ ) in (6) is the solution of the uiliry liner ordinry differentil eqution G ( ξ ) λg ( ξ ) µ G( ξ )=, (8) where λ nd µ re rel constnts to be determined. By using (8) nd the generl solution of (6), we cn find u, u nd ( u ) s polynomils of G'/ G. Substituting the generl solution of (8) together with u, u nd ( u ) s polynomils of G' / G into eqution (5) yield lgebric equtions involving powers of G' / G. Equting the coefficients of ech power of G' / G to zero gives system of lgebric equtions for A, λ, µ, e nd c. Then, solving this system by Mple gives i 3 A =, c= e ( λ 4 µ ), (9) where e nd µ re the rbitrry constnts. Net, depending on the sign of the discriminnt = λ 4µ, the three types of the following trveling wve solutions of the eqution (4) re obtined. When = λ 4 µ >, the hyperbolic function trveling wve solution is (7) or equivlently u( ξ ):= A ( γh( ξ )) ()
3 3 Solitons nd conserved quntities of the Ito eqution 7 λ 4µ 3 =, = e e ( 4 ) t, γ q( ξ) := h( ξ)( h( ξ )), () csinh( γξ ) c cosh( γξ) where γ ξ λ µ h( ξ)=, nd A is n rbitrry c sinh( γξ ) ccosh( γξ) constnt. In () c, c re rbitrry constnts. If them re tken s specil vlues, the vrious known results 6λ in the literture cn be rediscovered, for instnce, setting c =, c =, e=, = nd α = A, c then the solution of () cn be written s v := c tnh ξ α ( ct) tht it is the solution (4) in [6]. 6λ Setting c =, c =, e=, = nd α = A, then the solution of () cn be written s c v := ccoth ξ α ( ct). From (5) we cn etrct other ect solutions of Eq. (5) reported erlier. When = λ 4 µ =, the rtionl function trveling wve solution is or equivlently where ξ = e, h( ξ)=, c cξ c u( ξ ):= A ( h( ξ )) () q( ξ):= h ( ξ ), (3) nd A is n rbitrry constnt. Finlly, when = λ 4 µ <, the trigonometric function trveling wve solution is or equivlently u( ξ ):= A ( γh( ξ )) (4) γ q( ξ ):= ( h ( ξ )), (5) 4µ where γ ξ µ constnt. 3 =, = e e (4 ) t, h c sin( γξ ) c cos( γξ) ( ξ)=, c sin( γξ ) c cos( γξ) nd A is n rbitrry.. () Dimensions The ()-dimensionl form of the generlized Ito integro-differentil eqution tht is going to be studied in this subsection is given by [5] qtt qt ( qqt qqt ) q qt d bqyt dqt =. (6) Here, in (6), q is the dependent vrible while, y nd t re the independent vribles. The coefficients b, nd d re constnts. The eq. (6) cn reduced to
4 8 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws 4 with using the potentil. The eq. (7) is converted to the ODE or equivlently v v v v v v v v bv dv =, (7) tt t t t t yt t ( ) cbef u cde u = (8) 4 ( v) c eu ce u ce u u ce u u ce u u (9) 3 ( v) ( c bf de) u e u e ( u ) =, by the wve vribles v= u( ξ), ξ = e fy ct, where primes denote the derivtives with respect to ξ, nd e, f nd c re rel constnts to be determined lter. The eq. (9) is then integrted twice. This converts it into () 3 ' ( c bf de) u e u e ( u ) =. Here blncing u with ( u ) gives N =. Therefore, we cn write the solution of eqution () in the form of G ( ξ) u( ξ )= A A, A. G( ξ) The function G( ξ ) in () is the solution of the uiliry liner ordinry differentil eqution (). Substituting the generl solution of () together with u, u nd ( u ) s polynomils of G'/ G into eqution () yield lgebric equtions involving powers of G'/ G. Equting the coefficients of ech power of G'/ G to zero gives system of lgebric equtions for Ai, λ, µ, e, f nd c. Then, solving this system by Mple gives where 3 A =, c= bf de e ( λ 4 µ ), () e, f nd µ re the rbitrry constnts. Net, depending on the sign of the discriminnt = λ 4µ, the three types of the following trveling wve solutions of the eqution () re obtined. When = λ 4µ >, the hyperbolic function trveling wve solution is () or equivlently u( ξ ):= A ( γh( ξ )) (3) 3 where ξ ( λ µ ) = e fy bf de e ( 4 ) t, γ q( ξ) := h( ξ)( h( ξ )), (4) h csinh( γξ ) c cosh( γξ) λ µ ( ξ)=, γ =, c sinh( γξ ) ccosh( γξ) 4 nd A is n rbitrry constnt. In (3) c, c re rbitrry constnts. If them re tken s specil vlues, the vrious known results in the literture cn be rediscovered, for instnce, setting 6λ c =, c =, e=, =, f =, c= b d ( λ 4 µ ) nd α = A, then the solution (7) cn be
5 5 Solitons nd conserved quntities of the Ito eqution 9 c ( b d) written s v ( ξ):= α c ( b d) tnh ( y ct), tht it is the solution (73) in [6]. Setting 6λ c =, c =, e=, =, f =, c= b d ( λ 4 µ ) nd α = A, then the solution of (7) cn be c ( b d) written s v ( ξ):= α c ( b d)coth ( y ct). From () we cn rediscover some other ect solutions of eq. (6). When = λ 4 µ =, the rtionl function trveling wve solution is or equivlently where ξ ξ = e fy, h=, c cξ c u( ξ ):= A ( h( ξ )) (5) q( ξ):= h ( ξ ), (6) nd A is n rbitrry constnt. Finlly, when = λ 4 µ <, the trigonometric function trveling wve solution is u( ξ ):= A ( γh( ξ )) (7) or equivlently γ q( ξ ) := ( h ( ξ )), (8) 3 where ξ ( µ ) = e fy bf de e (4 ) t, nd A is n rbitrry constnt. c sin( γξ ) c cos( γξ) 4µ γ c sin( γξ ) c cos( γξ) =, =, h ξ 3. THE ANSATZ METHOD Ito eqution is generliztion of the biliner Korteweg-de Vries (KdV) eqution. It is sometimes referred to s the etension of the NLEE of KdV or modified KdV (mkdv) type to higher order. This eqution is studied in () dimensions s well s () dimensions. This study will therefore be split in the following two subsections. 3. () Dimensions The dimensionless form of the Ito eqution in dimensions is given by [5] tt t t t t d = q q q q qq bq q. (9) Here in (9) nd b re ll constnts. In this section, the serch will be for -soliton solution to (9). The technique tht will be used to crry out the clcultions is the solitry wve nstz method. Therefore, the strting nstz for the -soliton solution to (9) is tken s
6 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws 6 qt (, )= Asech p τ, () where τ = B( vt). (3) Here in (), A is the mplitude of the soliton while B is the inverse width of the soliton. Also, v is the velocity of the soliton while the vlue of the unknown eponent p will be determined while deriving the solution to the Ito eqution. Substituting () into (9) yields p p pvabsech τ pp ( ) vab sech τ 4 4 p 4 p pvab τ pp p p vab τ sech sech p p p p vab τ p p va B τ 4 p p ( 3) sech sech τ τ sech p p p va B p va B sech p p bvp A B τ bvp p A B τ sech sech =. (3) From (38) equting the eponents p nd p gives p =. (33) It needs to be noted tht the sme vlue of p is obtined when the eponents p nd p 4 re equted with ech other. Now, from (3) setting the coefficients of the linerly independent functions sech p j τ for j =,,4 to zero yields v=4 B, (34) 4B A =, b (35) 6B A =. (36) 5 3b Now, equting the two vlues of A from (35) nd (36) yields the condition b =. In such cse, both epressions (35) nd (36) reduce to B A =. (37) Therefore the -soliton solution to Ito eqution is given by qt (, )= Asech [ B ( vt)], (38) where the mplitude-width reltion is given by (37) nd the velocity of the soliton is given by (44). 3.. () Dimensions The dimensionless form of the Ito eqution in () dimensions is given by [5] tt t t t t yt t q q ( q q qq ) bq q d cq dq =. (39) Here,, b, c nd d re ll constnts. In order to seek -soliton solution to (39) the strting nstz is still given by (), where in this cse τ = B B y vt. (4)
7 7 Solitons nd conserved quntities of the Ito eqution Here in (4), A represents the mplitude of the soliton, while B nd B represents the inverse width in the - nd y -directions respectively. Finlly, v represents the velocity of the soliton nd the eponent p, which is unknown t this point, will be determined in terms of n during the course of derivtion of the soliton solution. Substituting () into (39) gin yields pvasech pp va sech p τ τ p p pvabsech τ pp ( ) p p vab sech τ p p( p )( p )( p 3) vabsech τ p( p ) va Bsech τ p p p 4 p p p p va B sech τ p va B sech τ bvp A B sech τ bvp( p ) A B sech τ p p p p cvp AB sech τ cvp( p ) AB sech τ dvp AB sech τ dvp( p ) AB sech τ = Similirly s in the previous sub-section, the sme vlue of p s in (33) is yielded. Proceeding s in the () dimensionl cse, eqution (39) gives 3 (4) v= cb db 4 B, (4) 4B A =, b (4) 6B A =. (44) 5 3b Now, equting the two vlues of A from (43) nd (44) yields to the condition b = s in the previous subsection. In such cse, both epressions (43) nd (44) reduce to B A =. Finlly, the -soliton solution to (39) is given by qyt (,, ) = Asech B By vt (45), (46) where the mplitude A is relted to the width B of the soliton s given by (45) nd the velocity of the soliton is given by (4). 4. CONSERVATION LAWS In the following subsection, more conserved quntities re derived using the multiplier pproch. For this, we resort to the invrince nd multiplier pproch bsed on the well known result tht the Eulert Lgrnge opertor nnihiltes totl divergence. Tht is, if ( T, T ). is conserved vector corresponding to conservtion lw [] DT long the solutions of the differentil eqution in question, then t t D T =, (47) t E [ DT D T ]=, (48) q t where E q is the pproprite Euler-Lgrnge opertors. Moreover, if there eists nontrivil differentil function Q, clled multiplier, such tht for some conserved vector, then t Q.(eqution) = DT D T (49) t
8 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws 8 Eq [ Q.(eqution)] =, (5) from which knowledge of ech multiplier Q leds to conserved vector vi Homotopy opertor. The elborte nd tedious clcultions revel tht number of multipliers nd, hence, nontrivil conservtion lws (nd densities/conserved quntities) re vilble. In this cse, following (), the Euler opertor is 4.. () Dimensions E v nd multipliers obtined re Q = v, Q =, Q3 = t, Q4 = g( t ), where g( t ) is n rbitrry function including the cse Q =. We note tht there re no derivtive dependent multipliers upto third order. The corresponding conserved densities re, respectively, vi the homotopy opertor 3 Tt = ( vt v v 6vvt 9v 6 v (5vv 3 v ) v v 8 vv ), (5) 9 Tt = ( vt 5v vt 5vv 5vv 6v 5vv 6 v ), (5) 3 Tt = ( tvt v 5tv tvt 5tvv 6tv 5(4 v tv tv ) 6 tv ), (53) 4 t t T = ( g v g(v 5v v 5vv 6 v )). (54) The Lie point symmetry lgebr hs bsis X =, X =, X =, X = 3 t v. (55) Hence, the respective conserved quntities re t 3 v 4 t v 3 68AB Tt d =, (56) 5 T t d =, (57) A d =, (58) B 3 Tt T 4 t d =. (59) These conserved qutities re ll computed using the -soliton solution tht is given by (46). 4.. () Dimensions The condition on Q for the multidimensionl cse is bsed on (7) leding the zero th order multipliers Q = α( y) v b4 α ( y), Q = β( y), Q = γ( y bt), Q = δ ( t, y), (6) 3 4
9 9 Solitons nd conserved quntities of the Ito eqution 3 with corresponding densities ( 5 '' 6 6 Tt = b α v α vtv bα vyv dα v αv 36αv 4 v α v 7α v v t y t y 3 bα ( v bv dv v v b v v dv 5vv v 6 v ) 3(5 v bα (4d v v ) '' b α α vt bvy dv vv v 4( 5 ( )))), Tt = ( 5 bβ v 5 v ( bβ β ( v v )) β( v 5bv dv 5v v bv dv 5v v 6v 6 v )), t y t y 3 (5 5 ( ) Tt = bγ v v bγ γ v v γ( vt 5bvy dv 5v vt bvy dv 5v v 6v 6 v )), 4 Tt = ( 5 bδyv δ tv δ ( vt bvy dv 5 vv 5 vv 6 v )). Prticulr cses of these, for e.g., would be from choosing with densities p p p 3 Q = yv, Q = y, Q =( y bt), ( Pt = vtv bvyv dv v 6 vvt 9 bvv dvv vv v 9v v v 8v v 8 vv ), y ( Pt = yvt byvy bv dyv yv yvt byvy dyv 5yv v 6yv 5 v( b yv yv ) 6 yv ), 3 ( 5 Pt = bt y vt b tvy 5 byvy bdtv 5bv dyv 5btv 5yv btv yv b tv t t y byv bdtv dyv 5btv v 5yv v y 6btv 6yv 5 v( b ( bt y) v ( bt y) v ) 6btv 6 yv ). 5. CONCLUSIONS In this pper, the Ito eqution, which is nonliner integro-differentil evolution eqution is studied. First the G ' / G method is used to crry out the integrtion of this eqution nd its solutions re obtined. Subsequently the -soliton solution of the Ito eqution in nd dimensions re obtined. In this contet it ws proved thtthe soliton solution will eist provided the nonlocl term collpses to zero. This -soliton solution is then used to compute the non-trivil conserved quntities where the corresponding conserved densities re computed using the multiplier pproch.
10 4 Ghodrt Ebdi, A.H. Kr, Mrko D. Petkovic, Ahmet Yildirim, Anjn Bisws ACKNOWLEDGMENTS The third uthor (Mrko D. Petkovic) grtefully cknowledge the support from the reserch project 44 of the Serbin Ministry of Science. REFERENCES. A. BISWAS, A. YILDIRIM, T. HAYAT, O. M. ALDOSSARY, R. SASSAMAN, Soliton perturbtion theory for thegenerlized Klein-Gordon eqution with full nonlinerity, Proceedings of the Romnin Acdemy, Series A, 3,, pp. 3 4,.. A. CEASER, S. GOMEZ, New trveling wves solutions to generlized Kup-Kuperschmidt nd Ito equtions, Applied Mthemtics nd Computtion, 6,, pp. 4 5,. 3. C. LI, Y. ZENG, Soliton solutions to higher order Ito eqution: Pfffin technique, Physics Letters A, 363,, pp. 4, D. L. LI, J-X. ZHAO, New ect solutions to the ()-dimensionl Ito eqution; Etended homoclinic test technique, Applied Mthemtics nd Computtion, 5, 5, pp , Q. P. LIU, Hmiltonin structures for Ito's eqution, Physics Letters A, 77,, pp. 3 34,. 6. A. M. WAZWAZ, Multiple-soliton solutions for the generlized ()-dimensionl nd the generlized ()-dimensionl Ito equtions, Applied Mthemtics nd Computtion,,, pp , 8. Received Mrch 5,
Application of Kudryashov method for the Ito equations
Avilble t http://pvmu.edu/m Appl. Appl. Mth. ISSN: 1932-9466 Vol. 12, Issue 1 June 2017, pp. 136 142 Applictions nd Applied Mthemtics: An Interntionl Journl AAM Appliction of Kudryshov method for the Ito
More informationApplication of Exp-Function Method to. a Huxley Equation with Variable Coefficient *
Interntionl Mthemticl Forum, 4, 9, no., 7-3 Appliction of Exp-Function Method to Huxley Eqution with Vrible Coefficient * Li Yo, Lin Wng nd Xin-Wei Zhou. Deprtment of Mthemtics, Kunming College Kunming,Yunnn,
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationLIE SYMMETRY GROUP OF (2+1)-DIMENSIONAL JAULENT-MIODEK EQUATION
M, H.-C., et l.: Lie Symmetry Group of (+1)-Dimensionl Julent-Miodek THERMAL SCIENCE, Yer 01, ol. 18, No. 5, pp. 157-155 157 LIE SYMMETRY GROUP OF (+1)-DIMENSIONAL JAULENT-MIODEK EQUATION by Hong-Ci MA
More informationSolutions of Klein - Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 6 Ver. IV (Nov. - Dec. 2017), PP 19-24 www.iosrjournls.org Solutions of Klein - Gordn equtions, using Finite Fourier
More informationNew exact travelling wave solutions of bidirectional wave equations
Shirz University of Technology From the SelectedWorks of Hbiboll Ltifizdeh June, 0 New exct trvelling wve solutions of bidirectionl wve equtions Hbiboll Ltifizdeh, Shirz University of Technology Avilble
More informationExact solutions for nonlinear partial fractional differential equations
Chin. Phys. B Vol., No. (0) 004 Exct solutions for nonliner prtil frctionl differentil equtions Khled A. epreel )b) nd Sleh Omrn b)c) ) Mthemtics Deprtment, Fculty of Science, Zgzig University, Egypt b)
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationu t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know
More informationNumerical Solutions for Quadratic Integro-Differential Equations of Fractional Orders
Open Journl of Applied Sciences, 7, 7, 57-7 http://www.scirp.org/journl/ojpps ISSN Online: 65-395 ISSN Print: 65-397 Numericl Solutions for Qudrtic Integro-Differentil Equtions of Frctionl Orders Ftheh
More informationChapter 8: Methods of Integration
Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln
More information7. Indefinite Integrals
7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find
More informationSOLUTION OF QUADRATIC NONLINEAR PROBLEMS WITH MULTIPLE SCALES LINDSTEDT-POINCARE METHOD. Mehmet Pakdemirli and Gözde Sarı
Mthemticl nd Computtionl Applictions, Vol., No., pp. 37-5, 5 http://dx.doi.org/.99/mc-5- SOLUTION OF QUADRATIC NONLINEAR PROBLEMS WITH MULTIPLE SCALES LINDSTEDT-POINCARE METHOD Mehmet Pkdemirli nd Gözde
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationSolving the (3+1)-dimensional potential YTSF equation with Exp-function method
Journl of Physics: Conference Series Solving the (3+-dimensionl potentil YTSF eqution with Exp-function method To cite this rticle: Y-P Wng 8 J. Phys.: Conf. Ser. 96 86 View the rticle online for updtes
More informationFredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method
Int. Journl of Mth. Anlysis, Vol. 5, 211, no. 19, 935-94 Fredholm Integrl Equtions of the First Kind Solved by Using the Homotopy Perturbtion Method Seyyed Mhmood Mirzei Deprtment of Mthemtics, Fculty
More informationIn Section 5.3 we considered initial value problems for the linear second order equation. y.a/ C ˇy 0.a/ D k 1 (13.1.4)
678 Chpter 13 Boundry Vlue Problems for Second Order Ordinry Differentil Equtions 13.1 TWO-POINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initil vlue problems for the liner second order eqution
More informationHOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016
HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationResearch Article On Compact and Noncompact Structures for the Improved Boussinesq Water Equations
Mthemticl Problems in Engineering Volume 2013 Article ID 540836 6 pges http://dx.doi.org/10.1155/2013/540836 Reserch Article On Compct nd Noncompct Structures for the Improved Boussinesq Wter Equtions
More informationCLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES
Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED
More informationINEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei
Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV
More informationLinearly Similar Polynomials
Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd
More informationModification Adomian Decomposition Method for solving Seventh OrderIntegro-Differential Equations
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume, Issue 5 Ver. V (Sep-Oct. 24), PP 72-77 www.iosrjournls.org Modifiction Adomin Decomposition Method for solving Seventh OrderIntegro-Differentil
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More information21.6 Green Functions for First Order Equations
21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to
More informationTHIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions
THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, Hui-Hsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationLYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS
Electronic Journl of Differentil Equtions, Vol. 2017 (2017), No. 139, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationAdomian Decomposition Method with Green s. Function for Solving Twelfth-Order Boundary. Value Problems
Applied Mthemticl Sciences, Vol. 9, 25, no. 8, 353-368 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/.2988/ms.25.486 Adomin Decomposition Method with Green s Function for Solving Twelfth-Order Boundry
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationVariational Techniques for Sturm-Liouville Eigenvalue Problems
Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationSolution to Fredholm Fuzzy Integral Equations with Degenerate Kernel
Int. J. Contemp. Mth. Sciences, Vol. 6, 2011, no. 11, 535-543 Solution to Fredholm Fuzzy Integrl Equtions with Degenerte Kernel M. M. Shmivnd, A. Shhsvrn nd S. M. Tri Fculty of Science, Islmic Azd University
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl
More informationAPPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL
ROMAI J, 4, 228, 73 8 APPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL Adelin Georgescu, Petre Băzăvn, Mihel Sterpu Acdemy of Romnin Scientists, Buchrest Deprtment of Mthemtics nd Computer Science, University
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationAike ikx Bike ikx. = 2k. solving for. A = k iκ
LULEÅ UNIVERSITY OF TECHNOLOGY Division of Physics Solution to written exm in Quntum Physics F0047T Exmintion dte: 06-03-5 The solutions re just suggestions. They my contin severl lterntive routes.. Sme/similr
More informationAMATH 731: Applied Functional Analysis Fall Some basics of integral equations
AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)
More informationDYNAMICAL SYSTEMS SUPPLEMENT 2007 pp Natalija Sergejeva. Department of Mathematics and Natural Sciences Parades 1 LV-5400 Daugavpils, Latvia
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 920 926 ON THE UNUSUAL FUČÍK SPECTRUM Ntlij Sergejev Deprtment of Mthemtics nd Nturl Sciences Prdes 1 LV-5400
More informationA Modified ADM for Solving Systems of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 6, 2012, no. 26, 1267-1273 A Modified ADM for Solving Systems of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi nd T. Dmercheli Deprtment of Mthemtics,
More informationPlates on elastic foundation
Pltes on elstic foundtion Circulr elstic plte, xil-symmetric lod, Winkler soil (fter Timoshenko & Woinowsky-Krieger (1959) - Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016) Introduction Winkler
More informationChapter 3 The Schrödinger Equation and a Particle in a Box
Chpter 3 The Schrödinger Eqution nd Prticle in Bo Bckground: We re finlly ble to introduce the Schrödinger eqution nd the first quntum mechnicl model prticle in bo. This eqution is the bsis of quntum mechnics
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationON THE C-INTEGRAL BENEDETTO BONGIORNO
ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives
More informationA Note on a Three-Dimensional Bäcklund Transformation
Interntionl Journl of Applied Science nd Engineering 006., : 5-0 A ote on Three-Dimensionl Bäcklund Trnsformtion Xuncheng Hung * nd Edwrd H. C. Chng Yngzhou Poltechnic Universit-903 Ho Yue Yun, Moon PrkYngzhou,
More informationLYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN
Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR
More informationIntroduction to Finite Element Method
Introduction to Finite Element Method Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pn.pl/ tzielins/ Tble of Contents 1 Introduction 1 1.1 Motivtion nd generl concepts.............
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
More informationOn the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 2, 28, no. 2, 57-62 On the Decomposition Method for System of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi 1 nd M. Mokhtri Deprtment of Mthemtics, Shhr-e-Rey
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationResearch Article Analytical Solution of the Fractional Fredholm Integrodifferential Equation Using the Fractional Residual Power Series Method
Hindwi Compleity Volume 7, Article ID 457589, 6 pges https://doi.org/.55/7/457589 Reserch Article Anlyticl Solution of the Frctionl Fredholm Integrodifferentil Eqution Using the Frctionl Residul Power
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationAn iterative method for solving nonlinear functional equations
J. Mth. Anl. Appl. 316 (26) 753 763 www.elsevier.com/locte/jm An itertive method for solving nonliner functionl equtions Vrsh Dftrdr-Gejji, Hossein Jfri Deprtment of Mthemtics, University of Pune, Gneshkhind,
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationFractional Riccati Equation Rational Expansion Method For Fractional Differential Equations
Appl. Mth. Inf. Sci. 7 No. 4 1575-1584 (2013) 1575 Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dx.doi.org/10.12785/mis/070443 Frctionl Riccti Eqution Rtionl Expnsion Method For
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationUndergraduate Research
Undergrdute Reserch A Trigonometric Simpson s Rule By Ctherine Cusimno Kirby nd Sony Stnley Biogrphicl Sketch Ctherine Cusimno Kirby is the dughter of Donn nd Sm Cusimno. Originlly from Vestvi Hills, Albm,
More informationResonant dispersive Benney and Broer-Kaup systems in 2+1 dimensions
Journl of Physics: Conference Series OPEN ACCESS Resonnt dispersive Benney nd Broer-Kup systems in +1 dimensions To cite this rticle: Jyh-Ho Lee nd Okty K Pshev 014 J. Phys.: Conf. Ser. 48 0106 View the
More informationax bx c (2) x a x a x a 1! 2!! gives a useful way of approximating a function near to some specific point x a, giving a power-series expansion in x
Elementr mthemticl epressions Qurtic equtions b b b The solutions to the generl qurtic eqution re (1) b c () b b 4c (3) Tlor n Mclurin series (power-series epnsion) The Tlor series n n f f f n 1!! n! f
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationMath 5440 Problem Set 3 Solutions
Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 25 1: Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping
More informationResearch Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation
Journl of Applied Mthemtics Volume 2011, Article ID 743923, 7 pges doi:10.1155/2011/743923 Reserch Article On Existence nd Uniqueness of Solutions of Nonliner Integrl Eqution M. Eshghi Gordji, 1 H. Bghni,
More informationMATH , Calculus 2, Fall 2018
MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationAn improvement to the homotopy perturbation method for solving integro-differential equations
Avilble online t http://ijimsrbiucir Int J Industril Mthemtics (ISSN 28-5621) Vol 4, No 4, Yer 212 Article ID IJIM-241, 12 pges Reserch Article An improvement to the homotopy perturbtion method for solving
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More information1 2-D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2-D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
More informationClassical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011
Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,
More informationA General Dynamic Inequality of Opial Type
Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn
More informationGENERALIZED ABSTRACTED MEAN VALUES
GENERALIZED ABSTRACTED MEAN VALUES FENG QI Abstrct. In this rticle, the uthor introduces the generlized bstrcted men vlues which etend the concepts of most mens with two vribles, nd reserches their bsic
More informationGeneralizations of the Basic Functional
3 Generliztions of the Bsic Functionl 3 1 Chpter 3: GENERALIZATIONS OF THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 3.1 Functionls with Higher Order Derivtives.......... 3 3 3.2 Severl Dependent Vribles...............
More informationFRACTIONAL INTEGRALS AND
Applicble Anlysis nd Discrete Mthemtics, 27, 3 323. Avilble electroniclly t http://pefmth.etf.bg.c.yu Presented t the conference: Topics in Mthemticl Anlysis nd Grph Theory, Belgrde, September 4, 26. FRACTONAL
More informationMatrices, Moments and Quadrature, cont d
Jim Lmbers MAT 285 Summer Session 2015-16 Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl
More informationEXAMPLES OF QUANTUM INTEGRALS
EXAMPLES OF QUANTUM INTEGRALS Stn Gudder Deprtment of Mthemtics University of Denver Denver, Colordo 88 sgudder@mth.du.edu Abstrct We first consider method of centering nd chnge of vrible formul for quntum
More informationSturm-Liouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1
Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition
More informationJournal of Computational and Applied Mathematics. On positive solutions for fourth-order boundary value problem with impulse
Journl of Computtionl nd Applied Mthemtics 225 (2009) 356 36 Contents lists vilble t ScienceDirect Journl of Computtionl nd Applied Mthemtics journl homepge: www.elsevier.com/locte/cm On positive solutions
More informationResearch Article On New Inequalities via Riemann-Liouville Fractional Integration
Abstrct nd Applied Anlysis Volume 202, Article ID 428983, 0 pges doi:0.55/202/428983 Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment
More informationSelf-similarity and symmetries of Pascal s triangles and simplices mod p
Sn Jose Stte University SJSU ScholrWorks Fculty Publictions Mthemtics nd Sttistics Februry 2004 Self-similrity nd symmetries of Pscl s tringles nd simplices mod p Richrd P. Kubelk Sn Jose Stte University,
More informationMath 5440 Problem Set 3 Solutions
Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 213 1: (Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
More informationA Modified Homotopy Perturbation Method for Solving Linear and Nonlinear Integral Equations. 1 Introduction
ISSN 1749-3889 (print), 1749-3897 (online) Interntionl Journl of Nonliner Science Vol.13(212) No.3,pp.38-316 A Modified Homotopy Perturbtion Method for Solving Liner nd Nonliner Integrl Equtions N. Aghzdeh,
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationTRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity
More information