MACsolutions of the nonexistent solutions of mathematical physics


 Rosamond Blake
 1 years ago
 Views:
Transcription
1 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE University of Dodom Deprtment of Mthemtics nd Sttistics P.O.Box 259, Dodom TANZANIA Abstrct: The method of dditionl conditions or MAC is pplied to the boundry vlue problems of mthemticl physics, where the clssicl solution does not exist or nonphysicl generlized solution is obtined. The Dirichlet problem for circulr elstic membrne is considered s the simplest exmple of the problem with nonexistent solution. The constnt boundry conditions re given in the center of the membrne nd on the finite rdius. The scheme of DugdleBrrenblt in Griffith s liner elstic crck problem is used to obtin the MACsolution. The dditionl pressure on membrne is pplied ner the center, the conditions of smoothness re used, the lineintegrl, which follows from the second Green s identity is considered nd it is tken s mesure of the introduced correction of the initil Dirichlet problem for plce eqution. The pplied conditions crete n unique solution, which t lest corresponds to the physicl sitution nd esily cn be proofed experimentlly. The similr situtions re briefly considered, where MAC pproch seems to be useful: circulr elstic plte, symmetric vibrtions of membrne, Kirsch problem, me problem. Key Words: Singulrity, Mthemticl Physics Equtions, Method of Additionl Conditions 8 Mrch 20 Introduction Some clssicl boundry vlue problems from elsticity will be considered. It is esy to solve these problems nd obtin the generl solution of the differentil equtions. The problem is to stisfy the prescribed boundry conditions. We will see tht the solutions of some problems does not exist. In this cse it is possible to crete generlized solution, using limit of existing solutions. These solutions cn be esily verified in experiments. Experiment show, tht the physicl solution exist nd differs from the obtined generlized solution. Then the MAC or the method of dditionl conditions cn be pplied. This method llows to trnsform the obtined generlized solution to the physiclly cceptble form. The MAC ws introduced bout yers go in scheme of DugdleBrrenblt [, 3] in frcture mechnics. This scheme ws pplied to the liner elstic crck problem. The liner elstic solution hs singulrity ner the tip of crck. To void this singulrity Dugdle nd Brrenblt introduced dditionl yield stresses ner the tip. The pplied nonsingulr condition gve the size of the zone, there the stresses re pplied. This scheme ws developed in [5], where the second dditionl condition of zero Jintegrl ws introduced. The MAC solution for the plce eqution in n ngle ws considered. The principle of superposition ws used to crete the MAC model for the membrne in [6]. This new condition gve the vlue of the pplied dditionl stresses, which re 6 times more then the given stresses t infinity. This stress concentrtion fctor corresponds to the experiments of Griffith s nd Inglis [3]. The obtined the liner elstic field ner the tip of crck mkes it possible to use lot of nlyticl, numericl methods to nlyze the stress field in different situtions. The usul strength criteri, which re used in the elstic stress fields without singulrities, cn be used here. et us pply the method of dditionl conditions MAC to consider the displcements of n elstic membrne. ISBN:
2 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements 2 Circulr elstic membrne 2. Nonexistent solution et us consider circulr elstic membrne with symmetric Dirichlet boundry conditions. Then it is resonble to consider the symmetric solution. The plce s eqution in this 2D cse is: r We consider domin Ω: r u The boundry conditions re: = 0; 0 r ; 2 u0 = u 0 0; 3 u = 0; 4 The generl solution of the eqution is u = A + B ln r ; 5 where A, B  re rbitrry constnts. It follows from the first boundry condition t r = 0, tht u0 = A = u 0 ; B = 0 ; 6 Then the solution cn not stisfy the second boundry condition, becuse we hve u = A = u 0 0; 7 It is esy to see in the physicl model of membrne, tht the solution must exist. 2.2 Generlized solution We cn obtin the generlized solution in the following wy. et us consider the membrne s ring: 0 < r ; 8 The plce eqution in symmetric 2D cse will be r r u = 0 9 The boundry conditions re u = u 0 0; 0 u = 0 The solution of the boundry vlue problem 9, 0, is u = u 0 ln ln + u 0 ln lnr 2 If 0, then the generlized solution of the initilly given problem, 2, 3, 4 is u = u 0 for r = 0, u = 0 for 0 < r. This solution is not physicl, becuse s we cn esy see in experiment, tht the solution hs finite derivtive t r = 0 nd is not zero in the domin. 2.3 MACsolution 2.3. The chnged boundry conditions Now we will consider the MAC solution of our problem with nonexistent solution. It is not importnt, but convenient, to chnge slightly the boundry conditions 3, 4 to the following ones u0 = 0; 3 u = u 0; Generlized solution The the generlized solution in this cse will tke the form u0 = 0, 5 ur = u, 0 < r Algorithm to obtin the MACsolution To crete the MACsolution we use the following lgorithm: The second Green s identity gives the invrint integrl like Jintegrl in frcture mechnics. The generlized solution is used. The pplied externl pressure ner the center of the membrne in the domin Ω : 0 r is constnt. Solution hs not singulrity in grdient t r = 0. The domin Ω is divided in two prts: Domin Ω : 0 r ; Domin Ω 2 : < r. Solution u is from the clss C. The mximum of the J integrl in the first domin must be equl to tht constnt vlue, which is obtined on the clssicl generlized solution. ISBN:
3 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements Additionl condition in the form of the line integrl To obtin the dditionl condition in the form of line integrl let us consider the clssicl second Green s identity [4]: Ω v u u v dω = Ω v u u v ds, 7 where the functions u nd v ɛ C 2 Ω. Consider ny two solutions of plce eqution u, v then we hve: u = 0; 8 v = 0. 9 Then we obtin for ny closed contour which is piecewise smooth nd belongs to Ω v u u v ds = Consider = + 2, where the contours, 2, re : r = ; 2 where 2 : r = 2 ; 22 0 < 2 ; 23 nd the contour follows in the opposite clockwise direction. We cn split our integrl 20 in two prts f ds = f ds + f ds = 0, 24 2 where f is the following function From 24 we obtin f ds = f = v u u v f ds. 26 If we chnge the direction of integrtion in the integrl long, then we obtin n invrint integrl long ny curve : r =, where the rdius is rbitrry nd stisfies the condition 0 <. We hve J = f ds = v u u v ds = C, 27 where C is some constnt. The formul 27 is right for ny closed piecewise smooth curve round the origin O. It tkes the following form in cse of circumference : r = : J = v u u v r dϕ = :r= 2 π r v u u v = C The vlue of the Jintegrl et us clculte this constnt, tking two solutions of the plce eqution. The first solution u = u is prt of the generlized solution 5, 6. The second solution is the generl solution of the plce eqution : v = A + B lnr, 29 where A, B re rbitrry constnts nd they will be determined below. Then we get from 28: J = 2 π u B = C. 30 And invrint integrl J is equl in this cse J = 2 π u B = C. 3 The constnt B will be determined lter The vlue J0 for physicl solution The experiments with membrne show tht it is resonble to suggest tht the rel physicl solution hs the following properties: the deflection is smooth nd its derivtive is bounded t the origin nd in its vicinity. If we denote the MACsolution s ũ, then we obtin in the smll enough vicinity of the origin r = 0: ũ C, 32 ũ C r, 33 where C is some positive constnt. It is esy now to estimte the vlue of function Jr ccording to 27 or 28 t r 0, where the generlized solution u is replced by the physicl MACsolution ũ. We obtin in the vicinity of the origin Jr = 2 π r v ũ ũ v 2 π r v ũ + ũ v. 34 ISBN:
4 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements Using 32, 33 the inequlity 34 tkes the form Jr 2 π r A + B ln r C + C r B r, or we cn write 35 Jr 2 π r C A + B ln r + B. 36 It follows from the inequlity 36 tht we cn define the vlue of J0 s J0 = lim r 0 Jr = If we tke function v ccording to 29 such tht B 0. Then we obtin from 3 tht the constnt C 0. It mens tht the condition 37 is not fulfilled in cse of generlized solution 5, 6. emrk on the J integrl The vlue of the J integrl obtined from the second Green s identity depends on the solution being considered: the generlized solution or the physicl MACsolution. It will be shown below. The first Green s identity does not possess this property. et us consider this identity [4]: u v u dω + Ω x v x + u y v dω = y Ω = Ω v u ds. 38 It is esy to see tht if the generlized solution u 7 is tken then ll integrls in 39 re zero becuse the function u = const for ny smooth enough function v. As we hve seen the second Green s identity hs creted the nonzero line integrl MACsolution in the domin Ω Consider the domin Ω : 0 r nd the Poisson eqution: r r u = p, 39 where the constnt p is unknown pressure, tht must help to trnsform the nonphysicl generlized solution to the physicl MACsolution. The prmeter p will be determined some lter. The generl solution of the eqution 39 is u = C 2 + C 3 ln r + p r2 4, 40 where C 2 nd C 3 re rbitrry constnts. We hve to stisfy the boundry condition 3, therefore we obtin u0 = C 2 = 0, 4 C 3 = 0, 42 becuse we hve to void singulrity t r = 0. Then the MACsolution in the domin Ω is u = p r The prmeter p will be determined lter MACsolution in the domin Ω 2 Consider the domin Ω 2 : r nd plce eqution: r r u = The generl solution of the eqution 44 is u = A + B lnr, 45 where A, B re rbitrry constnts. The boundry condition 3 gives the reltion between the constnts A nd B : u = A + B ln. 46 Then the MACsolution 45 cn be written in the form: r u = u + B ln. 47 There re three prmeters to determine: p, B,. We will use two smoothness conditions nd the properties of the function Jr 28 to obtin the unique vlues of these prmeters Smoothness conditions Consider the smoothness conditions t r = on the boundry between domins Ω nd Ω 2. We require tht the MACsolution is smooth, therefore u 0 = u + 0, 48 u u 0 = If we use the MACsolutions 43 nd 47, then the smoothness conditions 48 nd 49 give the reltions: p 2 4 = u + B ln, 50 p 2 = B. 5 ISBN:
5 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements The equtions 50 nd 5 llow to get the prmeters p, B s the functions of prmeter. We obtin 2 u B = + 2 ln 52, 4 u p = ln 53. The remining prmeter will be found using the function Jr Extremum of Jr Consider the function Jr ccording to 28, where the function u is the MACsolution: 43 in Ω nd 47 in Ω 2 nd the function v in the whole domin Ω equls to the function 47. Then it is esy to see tht J0 = 0 nd the property 37 is stisfied. Moreover the function Jr = 0 in the domin Ω 2 becuse the functions u nd v coincide in this domin ccording to the choice of these functions. This function Jr is not zero in the domin Ω nd there we cn consider its extremum. If the function u is the generlized solution of the stted membrne problem 6 for 0 < r then the function Jr is constnt ccording to 3 nd equls Jr = 2 π u B, 54 Introducing the MACsolution we men to connect the physicl zero vlue J0 with the vlue 54. Then we continue the dditionl pressure to the position where it will be possible smoothly to connect the MACsolutions in two domins Ω nd Ω 2. Therefore it is resonble to require the following condition: for 0 < r < extrjr = 2 π u B, 55 The function Jr from 28 hs the expression r Jr = 2πr u + B ln p r 2 B pr2. 4 r 56 Or we hve Jr = p π r 2 r u + B ln B We re looking for the points of extremum. Therefore we need the derivtive of the function Jr. The derivtive of the function Jr?? is J r r = 2 p π r u + B ln. 58 The necessry condition for n extremum is J r = 0 59 nd cretes two points. The first one is r = 0 nd Jr = 0. And the second point of extremum stisfies the eqution: r2 u + B ln = 0, 60 where the constnt B is tken from 52. Then we obtin from 52 nd 60 the eqution 2 u r2 u + ln = ln It follows form 6 tht r2 + 2 ln = 0, 62 nd this eqution 62 hs solution for the second extremum r 2 = e. 63 Then the vlue of Jr 2 ccording to 57 nd 63 will be Jr 2 = p π 2 e u + B ln e The expression 64 cn be simplified using 52 s follows nd finlly we obtin the extremum of the function Jr inside the domin Ω : Jr 2 = 2 p u π 2 e + 2 ln dius Consider the condition 55. It mens the equlity of extremum of the function Jr inside the domin Ω 65to the vlue of the Jintegrl for the generlized solution 54. This condition is very importnt if we consider the vibrtion of the membrne. The condition 55 is 2 p u π 2 e + 2 ln = 2 π u p 2 2, 66 where the expression for B from 5 is pplied. The nonzero solution of the eqution 66 is = e 2 e. 67 The vlue 67 of is outside the intervl 0 < nd the mximum of Jr 2 65 will be t = nd we hve to ccept this vlue of. It mens tht we hve only domin Ω nd the domin Ω 2 does not exist in considered problem. ISBN:
6 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements Expression for MACsolution The MACsolution of the problem, 2, 3 nd 4 is given in the form 43 nd 53 t = s follows r 2 u = u 68 in the domin Ω emrks to the MACsolution The obtined MACsolution 68 shows tht if we mesure the function nd its derivtive on the boundry then we cn determine the inclusion with given potentil of ny size. It is evident if we consider n experiment with membrne: ny nonzero displcement in the center of membrne cn be determined through the mesurements on the boundry. If the generlized solution 5, 6 is used then we hve to conclude tht it is impossible to determine the point inclusion in the middle of membrne. It could be interesting to consider this effect of two theories in experiment with n electrosttic field. The clssicl pproch nd the theory is presented in []. 3 Eigenfrequencies of circulr membrne Consider the symmetric vibrtions of circulr membrne in the domin 0 r. The differentil eqution of membrne is r r u = c 2 2 u t The Dirichlet boundry conditions re u0 = 0, u = It is esy to see tht the eigenfrequencies in this problem does not exist. It is clssicl solution of the problem. If we consider the generlized solution tking the domin r nd the first boundry condition in 70 s u = 0, 7 then the eigenfrequencies exist. If 0 then tht eigenfrequencies re the sme s in membrne which does not hve ny hole nd the first boundry condition in 70. Similr result is obtined in more generl problem bout eigenvlues of the plce opertor [2], where the miniml eigenvlue of bll does not depend on very smll internl hole in form of bll. The clssicl nd generlized solutions does not correspond to relity. It cn be esily shown experimentlly. The MACsolution cn be obtined using similr dditionl condition in form of line integrl 27 nd the sme lgorithm s before. It is evident tht we hve possibility to stisfy the experimentl dt. 4 Conclusion Some problems of mthemticl physics with nonexistent solutions cn hve the MACsolutions. These MACsolutions re corresponding nd explining the rel physicl situtions. Therefore it is importnt to consider not only clssicl solutions, which cn hve nonphysicl singulrities or even sometimes do not exist, but lso the MACsolutions of tht problems. The problems in liner elsticity bout stress concentrtion fctors, contct problems, crck problems cn crete very interesting MACsolutions. The similr problems cn be esily found in other boundry vlue problems for the different prtil differentil equtions. eferences: [] H. Ammri, H. Kng, Polriztion nd Moment Tensores With Applictions to Inverse Problems nd Effective Medium Theory, Springer, 2007, pp. 32. [2] M. H. C. Anis, A.. Aithl, On two functionls connected to the plcin in clss of doubly connected domins in spceforms, Proc. Indin Acd. Sci. Mth. Sci., Vol. 5, No., 2005, pp [3] V. I. Astfjev, J. N. dev,. V. Stepnov, Nonliner frcture mechnics, Smr, Publisher Smr University, [4] G. Gilbrg, N. S. Trudinger, Elliptic prtil differentil equtions of second order, SpringerVerlg, Berlin, Heidelberg, New York, Tokyo, 983. [5] I. Neygebuer, The method to obtin the finite solutions in the continuum mechnics, Fundmentl reserch t the Sint Petersburg Stte Polytechnic University, [6] I. Neygebuer, MAC solution for rectngulr membrne, Journl of Concrete nd Applicble Mthemtics, Vol.8,No.2, 200, pp ISBN:
New Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationPressure Wave Analysis of a Cylindrical Drum
Pressure Wve Anlysis of Cylindricl Drum Chris Clrk, Brin Anderson, Brin Thoms, nd Josh Symonds Deprtment of Mthemtics The University of Rochester, Rochester, NY 4627 (Dted: December, 24 In this pper, hypotheticl
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant 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 informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville 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 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 informationGreen s functions. f(t) =
Consider the 2nd order liner inhomogeneous ODE Green s functions d 2 u 2 + k(t)du + p(t)u(t) = f(t). Of course, in prctice we ll only del with the two prticulr types of 2nd order ODEs we discussed lst
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
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 informationSolutions of Klein  Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSRJM) eissn: 22785728, pissn: 2319765X. Volume 13, Issue 6 Ver. IV (Nov.  Dec. 2017), PP 1924 www.iosrjournls.org Solutions of Klein  Gordn equtions, using Finite Fourier
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More information4. Calculus of Variations
4. Clculus of Vritions Introduction  Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the
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 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 informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationInternational Jour. of Diff. Eq. and Appl., 3, N1, (2001),
Interntionl Jour. of Diff. Eq. nd Appl., 3, N1, (2001), 3137. 1 New proof of Weyl s theorem A.G. Rmm Mthemtics Deprtment, Knss Stte University, Mnhttn, KS 665062602, USA rmm@mth.ksu.edu http://www.mth.ksu.edu/
More information1.3 The Lemma of DuBoisReymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBoisReymond We needed extr regulrity to integrte by prts nd obtin the Euler Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More information1 2D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2D 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 informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics ENotes 8(8)  c IN 675 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 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 informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationPlates on elastic foundation
Pltes on elstic foundtion Circulr elstic plte, xilsymmetric lod, Winkler soil (fter Timoshenko & WoinowskyKrieger (1959)  Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016) Introduction Winkler
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationResearch Article Harmonic Deformation of Planar Curves
Interntionl Journl of Mthemtics nd Mthemticl Sciences Volume, Article ID 9, pges doi:.55//9 Reserch Article Hrmonic Deformtion of Plnr Curves Eleutherius Symeonidis MthemtischGeogrphische Fkultät, Ktholische
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationTHE INTERVAL LATTICE BOLTZMANN METHOD FOR TRANSIENT HEAT TRANSFER IN A SILICON THIN FILM
ROMAI J., v.9, no.2(2013), 173 179 THE INTERVAL LATTICE BOLTZMANN METHOD FOR TRANSIENT HEAT TRANSFER IN A SILICON THIN FILM Alicj PiseckBelkhyt, Ann Korczk Institute of Computtionl Mechnics nd Engineering,
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationIntroduction to the Calculus of Variations
Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is selfcontined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More informationDETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE
Determintion RevAdvMterSci of mechnicl 0(009) 7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING
More informationWhen a force f(t) is applied to a mass in a system, we recall that Newton s law says that. f(t) = ma = m d dt v,
Impulse Functions In mny ppliction problems, n externl force f(t) is pplied over very short period of time. For exmple, if mss in spring nd dshpot system is struck by hmmer, the ppliction of the force
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationGreen function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr SturmLiouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
More informationModule 1. Energy Methods in Structural Analysis
Module 1 Energy Methods in Structurl Anlysis Lesson 4 Theorem of Lest Work Instructionl Objectives After reding this lesson, the reder will be ble to: 1. Stte nd prove theorem of Lest Work.. Anlyse stticlly
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationSome estimates on the HermiteHadamard inequality through quasiconvex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13693 Some estimtes on the HermiteHdmrd inequlity through qusiconvex functions Dniel Alexndru Ion Abstrct. In this pper
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
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 informationMath 124A October 04, 2011
Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model
More informationLecture 13  Linking E, ϕ, and ρ
Lecture 13  Linking E, ϕ, nd ρ A Puzzle... InnerSurfce Chrge Density A positive point chrge q is locted offcenter inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationGreen s function. Green s function. Green s function. Green s function. Green s function. Green s functions. Classical case (recall)
Green s functions 3. G(t, τ) nd its derivtives G (k) t (t, τ), (k =,..., n 2) re continuous in the squre t, τ t with respect to both vribles, George Green (4 July 793 3 My 84) In 828 Green privtely published
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationLine Integrals. Chapter Definition
hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xyplne. It
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics ENotes, 5(005), 5360 c ISSN 1607510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview IntroductionModelling Bsic concepts to understnd n ODE. Description nd properties
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
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 informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
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 informationMethod of Localisation and Controlled Ejection of Swarms of Likely Charged Particles
Method of Loclistion nd Controlled Ejection of Swrms of Likely Chrged Prticles I. N. Tukev July 3, 17 Astrct This work considers Coulom forces cting on chrged point prticle locted etween the two coxil,
More informationCalculus of Variations: The Direct Approach
Clculus of Vritions: The Direct Approch Lecture by Andrejs Treibergs, Notes by Bryn Wilson June 7, 2010 The originl lecture slides re vilble online t: http://www.mth.uth.edu/~treiberg/directmethodslides.pdf
More informationII. Integration and Cauchy s Theorem
MTH6111 Complex Anlysis 200910 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationProperties of the Riemann Integral
Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationThe presentation of a new type of quantum calculus
DOI.55/tmj2722 The presenttion of new type of quntum clculus Abdolli Nemty nd Mehdi Tourni b Deprtment of Mthemtics, University of Mzndrn, Bbolsr, Irn Emil: nmty@umz.c.ir, mehdi.tourni@gmil.com b Abstrct
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 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 HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) YnBin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
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 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 informationCandidates must show on each answer book the type of calculator used.
UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor
More informationProblem set 1: Solutions Math 207B, Winter 2016
Problem set 1: Solutions Mth 27B, Winter 216 1. Define f : R 2 R by f(,) = nd f(x,y) = xy3 x 2 +y 6 if (x,y) (,). ()Show tht thedirectionl derivtives of f t (,)exist inevery direction. Wht is its Gâteux
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationOn the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 2, 28, no. 2, 5762 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, ShhreRey
More informationMath Fall 2006 Sample problems for the final exam: Solutions
Mth 425 Fll 26 Smple problems for the finl exm: Solutions Any problem my be ltered or replced by different one! Some possibly useful informtion Prsevl s equlity for the complex form of the Fourier series
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationINDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012
Lecture 6: Line Integrls INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Anlysis Autumn 2012 August 8, 2012 Lecture 6: Line Integrls Lecture 6: Line Integrls Lecture 6: Line Integrls Integrls of complex
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, BbeşBolyi University Str. Koglnicenu
More informationThinPlate Splines. Contents
ThinPlte Splines Dvid Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Cretive Commons Attribution 4.0 Interntionl License. To view copy of this
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L13. 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 informationApproximation of functions belonging to the class L p (ω) β by linear operators
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 3, 9, Approximtion of functions belonging to the clss L p ω) β by liner opertors W lodzimierz Lenski nd Bogdn Szl Abstrct. We prove
More information