METHODS OF POTENTIAL THEORY.I. Agoshkov and P.B. Dubovsk Insttute of Numecal Mathematcs, Russan Academy of Scences, Moscow, Russa Keywods: Potental, volume potental, Newton s potental, smple laye potental, double laye potental, logathmc potental, Fedholm equaton, Schwatz method, cylndcal coodnates, sphecal coodnates, Posson equaton, Dchlet poblem, Neumann poblem, Geen s functon, sweepng-out method, Helmholtz equaton, etaded potental, heat conductvty equaton, telegaph equaton. Contents. Intoducton 2. Fundamentals of the Potental Theoy 2.. Some Elements fom Calculus 2... Basc Othogonal Coodnates 2..2. Basc Dffeental Opeatons on a ecto Feld 2... Fomulae fom the Feld Theoy 2..4. Basc Popetes of Hamonc Functons 2.2. olume Mass o Chage potental 2.2.. Newton s (Coulomb s) potental 2.2.2. Popetes of Newton s Potental 2.2.. Potental of a Homogenous Sphee 2.2.4. Popetes of the Potental of olume-dstbuted Masses 2.. Logathmc Potentals 2... Defnton of the Logathmc Potental 2..2. Popetes of the Logathmc Potental 2... Logathmc Potental of a Dsk of Constant Densty 2.4. Smple Laye Potental 2.4.. Smple Laye Potental n the D space 2.4.2. Popetes of the Smple Laye Potental 2.4.. Potental of a Homogenous Sphee 2.4.4. Smple Laye Potental on the Plane 2.5. Double Laye Potental 2.5.. Dpole Potental 2.5.2. Double Laye Potental n the Space and ts Popetes 2.5.. Double Laye Logathmc Potental and ts Popetes. Applcaton of the Potental Theoy to the Classcal Poblems of Mathematcal Physcs.. Soluton of the Laplace and Posson Equatons... Fomulaton of Bounday alue Poblems fo the Laplace Equaton..2. The Dchlet Poblem n the D Space... The Dchlet Poblem on the Plane..4. The Neumann Poblem..5. The Thd Bounday alue Poblem fo the Laplace Equaton..6. The Bounday alue Poblem fo the Posson Equaton
.2. Geen s Functon of the Laplace Opeato.2.. The Posson Equaton.2.2. Geen s Functon.2.. The Dchlet Poblem n a Smple Doman.. The Laplace Equaton n a Complex-Shaped Doman... The Schwaz Method..2. Sweepng-out Method 4. Othe Applcatons of the Potental Method 4.. Applcaton of the Potental Method to the Helmholtz Equaton 4... Basc facts 4..2. Bounday alue Poblems fo the Helmholtz Equaton 4... Geen s Functons 4..4. The Equaton Δυ - λ υ = 0 4.2. Non-statonay Potentals 4.2.. Potentals fo the D Heat Conductvty Equaton 4.2.2. Heat Souces n a Mult-dmensonal Case 4.2.. Bounday alue Poblem fo the Telegaph Equaton Glossay Bblogaphy Bogaphcal Sketches Summay 2 The Laplace equaton Δ u = 0 o u = 0 s one of the basc classcal equatons of mathematcal physcs. Its soluton s epesented as the ntegal of the poduct of some functon (potental densty) and the fundamental soluton of the Laplace equaton. An ntegal of ths knd s sad to be a potental ntegal. In the D case the fundamental soluton of the Laplace equaton s the functon and n the 2D case- the functon ln, whee s the dstance between ponts. If we look fo a soluton of a bounday value poblem n the fom of a potental, then fo potental densty we obtan the Fedholm ntegal equaton, whee the ntegaton s pefomed ove the bounday of a gven doman. The potental theoy can be natually extended to moe complcated ellptc equatons and othe equatons of mathematcal physcs.. Intoducton The noton of Newton s potental was fst ntoduced at the end of the 8 th centuy by P.Laplace and J.Lagange and then by L. Eule fo poblems of hydodynamcs. The noton of a potental beng consdeed as a functon whose gadent s a vecto feld s due to Gauss. The popetes of the smple laye potental wee fst studed by Coulomb and Posson, the geat contbuton to the development of the potental theoy was made by Geen. Nowadays the potental theoy s an actvely developed tool fo studyng and solvng poblems n dffeent felds of mathematcal physcs. Let F = = Fe be a gven vecto feld, whee F = F( x, y, z) ae the component of the = vecto F appled at the pont ( x, yz,, ) e ae the basc vectos of the othogonal
coodnate system; let uxyz (,, ) be a scala functon (scala feld). A scala feld uxyz (,, ) whose gadent equals F: gad u= u= ( u x, u y, u z) = F, s called the potental of a vecto feld F. So knowledge of a potental functon (potental) allows one to calculate actng foces. Many poblems of electomagnetsm, hydodynamcs, acoustcs, heat conductvty and dffuson ae educed to bounday value poblems fo ellptc equatons. The smplest and mpotant examples of such equatons ae the Laplace equaton Δ u = 0 and the Posson equaton Δ u= f. Hee Δ s the Laplace opeato equal to = 2 2 Δ u= u x (4 ). The fundamental solutons of the Laplace equaton beng π n the D case and to (2 π ) ln( ) n the 2D case play key ole n the methods of the potental theoy. On the bass of these solutons a potental s constucted as the ntegal of the poduct of some functon (potental densty) and a fundamental soluton (o ts devatve). Dependng on an ntegaton doman and on the use of a fundamental soluton o ts nomal devatve, the volume potentals and the smple and double laye potentals ae dstngushed. If we look fo a potental (a soluton of the coespondng ellptc equaton) n the fom of the ntegal of densty, then we obtan an ntegal equaton fo the unknown densty. Snce the soluton can be expessed n tems of dffeent potentals, the pefeed choce of a potental s that whch yelds the smplest ntegal equaton. Thus, to obtan the Fedholm equaton of the second knd, the Dchlet poblem should be solved wth the help of the double laye potental and the Neumann poblem should be solved wth the smple laye potental. Below we consde the potentals fo the Laplace and Helmholtz equatons and the wave and heat conductvty equatons beng the basc types of equatons of mathematcal physcs that ase n enegetcs, ecology, the theoy of electcty, atmosphee and ocean. 2. Fundamentals of the Potental Theoy 2.. Some Elements fom Calculus 2... Basc Othogonal Coodnates Gven a system of thee sngle-valued functons of thee vaables: x = ϕ( u, u2, u), x2 = ϕ2( u, u2, u), x ϕ( u, u2, u) =. () Suppose that to each set of values u, u2, u thee coesponds a cetan pont M n the space wth Catesan coodnates x, x 2, x. The quanttes u, u 2, u can be consdeed as cuvlnea coodnates of the pont M. They defne a coodnate system whch s sad to be cuvlnea. A system s called othogonal f at each pont the coodnate lnes passng though ths pont mutually ntesect at ght angles. Let us consde two basc examples of cuvlnea othogonal coodnates.
Cylndcal coodnates: [ ] x = cos ϕ, u = sn ϕ, z = z ( ϕ 0, 2 π, > 0). Hee nstead of x, x 2, x we have x, yz, and nstead of u, u 2, u, ϕ, z. In the 2D case beng ndependent of z cylndcal coodnates ae called pola coodnates. 2 Sphecal coodnates [ ] [ ] x = snθ cos ϕ, y = snθ sn ϕ, z = cos θ ( θ 0, π, ϕ 0,2 π, > 0). 2..2. Basc Dffeental Opeatons on a ecto Feld = be a scala feld, F = F ( u, u2, u ) be a vecto feld, F= Fe. In Let ϕ ϕ( u, u2, u) the Catesan ectangle coodnates the followng opeatons ae defned. Gadent: gad ϕ = ϕ = ϕ e ; Dvegence: ( F) = dv F=, = F ; Roto (votcty): [,F] = e e e 2 ot F = = ; 2 F F F 2 The Laplace opeato (Laplacan) = 2 Δ ϕ = dv gad ϕ = ϕ, = 2 2 2 whee the desgnatons = / u, = / u, = (, 2, ) ae ntoduced fo the sake of convenence. In cylndcal coodnates the Laplace opeato has the fom 2 2 υ υ υ Δ= + + (2) 2 2 2 ϕ z and n sphecal coodnates
2 2 2 2 2 2 2 υ υ sn υ Δ υ = + θ +. snθ θ sn θ ϕ () 2... Fomulae fom the Feld Theoy Let uandυ be two abtay functons wth contnuous patal devatves up to the second ode nclusve. Instead of u = u( x, y, z), we wte u = u( A) whee a pont A has coodnates ( x, yz)., The dstance between a pont A( xyz,, ) and a pont P( ξ, η, ζ ) s defned by AP ( ξ) ( η) ( ζ ) 2 2 2. = x + y + z The symbols of dffeental opeatos on functons of A and P wll be equpped wth the sub-scpts A o P dependng on whethe the dffeentaton s pefomed wth espect to xyz,, o ξ, η, ζ. Fo, example, 2 2 2 Δ Au = u + u + u 2 2 2, gad u u u Pu = + j + k. x y z ξ η ζ The symbol ( u n) P denotes the devatve n the decton of the nomal n to a suface at a pont P: u u u u = cosα + cos β + cos γ, n ξ η ζ P whee cos α, cos β, cosγ ae the decton cosnes of the oute nomal n. Recall the Ostogadsk-Gauss fomula P Q R + + d = ( P cosα + Q cos β + R cos γ ) ds, x y z S whee the cosnes ae the decton cosnes of an oute nomal n. Settng P = u υ, Q = u υ, R = u υ, we ave at Geen s fst fomula 2 υ gad u,gad d + uδ υd = u ds. n ( υ) (4) S Let us ntechange u and v n the fomula (4) and subtact the obtaned equalty fom (4). Ths yelds Geen s second fomula: { uδυ υδ u} d = u υ ds. S υ n u n (5)
If A, then we can not substtute υ = AP at once nto (5). Constuct a small sphee wth the cente at A. Applyng Geen s second fomula (5) to the functons u and υ outsde the sphee and assumng that the adus of the sphee tends to zeo, we obtan Geen s man ntegal fomula υ ( AP ) ΔuP ( ) P P (6) Ω u( A) = u( P) ds d. AP n n AP S Dependng on the locaton of the pont A, the coeffcent Ω takes the values Ω= 4 π, A, Ω= 2 π, A, Ω= 0, A. Smlaly, n the 2D case we denote some doman n the plane (x, y), bounded by a smooth closed cuve L (o by seveal cuves), by D. Then fo abtay functons u and υ, whch have contnuous patal devatves up to the second ode nclusve, the followng expessons take place: u υ u υ υ + ds + uδ υds = u dl, ξ ξ η η n (7) D D L D υ n u n { uδυ υδ u} ds = u υ dl, (8) ( ln ( ) ) u ua ( ) = ln up ( ) dl uln ds, 2π Δ n n 2π (9) L D whee n s the dffeentaton opeato n the decton of oute nomal to L, 2 2 2 ξ η 2, AP Δ= + = s the dstance between the pont A and a vaable pont P. 2..4. Basc Popetes of Hamonc Functons Functons, satsfyng the Laplace equaton Δ u = 0 n a doman, ae called hamonc functons. Fo a hamonc functon U the followng popetes hold. U ds = 0, n S.e., the ntegal of the nomal devatve of a hamonc functon ove the bounday of a doman s equal to zeo. 2 The value of a hamonc functon at any nteo pont of a doman s expessed n tems of the values of ths functon and ts nomal devatve at the bounday of the doman by the fomula
( ) U U( A) = U ds. 4π n n S. The value of a hamonc functon at the cente A of a sphee S R of adus R s equal to the athmetc mean of the values of ths functon at the suface of the sphee,.e., to the ntegal of the functon ove the suface of the sphee, dvded by the aea of ths suface: U( A) = U ds. 2 4π R S R 4 Fom the maxmum pncple follows: a functon, hamonc nsde a doman and contnuous up to ts bounday, takes ts maxmum and mnmum values at the bounday of the doman. 2.2. olume Mass o Chage potental 2.2.. Newton s (Coulomb s) potental Let be some fnte doman n R, bounded by a pecewse smooth closed suface S. Let ρ ( P) be a contnuous bounded functon n. Then ( ) u A ρ( P) = d (0) s called the nfnte mass potental o Newton s mass potental dstbuted ove volume wth densty ρ. The functon ua ( ) can also be consdeed as Coulomb s potental of volume-dstbuted chages. 2.2.2. Popetes of Newton s Potental At any pont A outsde the functon ua ( ) fom (0) s contnuous and dffeentable wth espect to x, yz, unde the ntegal sgn as much tmes as desed. In patcula, gad ua ( ) = ρ( P)gad( d ) = ρ( P) d, () whee s a adus-vecto, = AP = ( x ξ) + ( y η) j+ ( z ζ ) k, A= A( x, y, z), P = P( ξ, η, ζ ). Δ = 0, A, P, we have Snce ( ) AP
( ) Δ ua ( ) = ρ( P) Δ d= 0, A. AP Thus, the potental ua ( ) of masses o chages dstbuted ove volume satsfes the Laplace equaton at all ponts outsde. Away fom the ogn o, whch s the same, fom the doman we have the appoxmate equalty M P d, = (2) ( ) ρ ( ) u A whee M= ρd s the total mass. In othe wods, at nfnty the potental of volume dstbuted masses (o chages) behaves lke the potental of a mass pont (o of a pont chage) located at the ogn such that ts mass o chage s equal to the total mass (o to the total chage) dstbuted ove volume. In patcula, u( A) 0 as. Fo patal devatves of the potental of volume dstbuted masses we have the estmate u C u C u C <, <, <, x 2 y 2 z 2 whee C s some constant. 2.2.. Potental of a Homogenous Sphee Assume that a sphee of adus R wth the cente at the ogn has constant densty ρ = const. Passng to sphecal coodnates, ϕ, θ whee ξ = snθ cos ϕ, η = sn ϕ, ζ = cosθ, we obtan the potental of a homogenous sphee at a pont : u () M, f > R, M, f < R. 2R R = 2 () It s easy to see that u ( ) and ts fst-ode devatve u ( ) ae contnuous fo all 0 u becomes dscontnuous at the pont = R., but the second-ode devatve ( ) At all exteo ponts the potental of a homogenous sphee s equal to the potental of the mass pont of the same mass, placed at ts cente, and satsfes the Laplace equaton. At all nteo ponts of the sphee the potental satsfes the Posson equaton Δ u = 4 πρ.
- - - TO ACCESS ALL THE 46 PAGES OF THIS CHAPTER, st: http://www.eolss.net/eolss-sampleallchapte.aspx Bblogaphy Belo M. (964). Basc Elements of Classcal Potental Theoy, 25 p. Moscow: Physmathlyt. [Abstact genealzatons of the potental theoy ae pesented.] Gunte N.M. (95). The Potental Theoy and ts Applcaton to Basc Poblems of Mathematcal Physcs, 46 p. Moscow: Gostekhzdat. [A fundamental book on the potental theoy. Popetes of potentals ae poved and methods fo solvng the Neumann and Dchlet poblems ae descbed.] Kupadze.D. (950). Bounday alue Poblems n the Oscllaton Theoy and Integal Equatons, 280 p. Moscow-Lenngad: Gostekhzdat. [Bounday value poblems n the electomagnetc oscllaton theoy ae consdeed. The theoy of ntegal equatons wth sngula kenels s developed. The antenna potentals and othe non-statonay potentals of the oscllaton theoy ae constucted.] Ladyzhenskaya O.A. (970). Mathematcal Poblems n scous Incompessble Flud Dynamcs, 288 p. Moscow: Nauka. [In one chaptes of ths book the theoy of hydodynamc potentals s pesented.] Landkof N.S. (966). Basc Elements of Moden Potental Theoy, 56 p. Moscow: Nauka. [The fundamentals of the abstact potental theoy ae systematcally pesented. In patcula, potental densty s genealzed to the case of an abtay Boel measue. The book s ntended fo an advanced eade.] Levn.I. and Gosbeg Yu.I. (95). Dffeental Equatons of Mathematcal Physcs, 576 p. Moscow- Lenngad: Gostekhzdat. [In the second chapte of ths evew book the basc elements of the potental theoy ae pesented.] Smnov.I. (98). Textbook of Hghe Mathematcs.. I, pat 2, 55 p. Moscow: Nauka. [In ths fundamental textbook of n hghe mathematcs the potental theoy s pesented ncludng methods of etaded potentals fo paabolc equatons and the wave equaton.] Sobolev S.L. (966). Equatons of Mathematcal Physcs, 444 p. Moscow: Nauka. [In seveal sectons of ths textbook the basc elements of the potental theoy wth detaled poofs ae pesented. The book s ntended fo students.] Sologub.S. (975). Development of the Theoy of Ellptc Equatons n the 8 th and 9 th Centues, 280 p. Kev: Naukova Dumka. [The hstoy of the development of the mathematcal theoy of potental, statng wth the woks of Laplace and Lagange, s pesented as well as the development of the theoy of bounday value poblems fo the Laplace equaton and fo equatons nvolvng the Laplace opeato. Intal fomulatons of theoems and appoaches ae gven and the gadual specfcaton and development ae shown.] Setensk L.N. (946). Newton s Potental Theoy, 550 p. Moscow-Lenngad: Gostekhzdat. [A fundamental book on the potental theoy wth detaled poofs.] Tkhonov A.N. and Samask A.A. (966). Equatons of Mathematcal Physcs, 724 p. Moscow: Nauka. [In seveal sectons of ths textbook the fundamentals of the potental theoy ae pesented wth detaled poofs. Retaded potentals ae descbed. The book s ntended fo students.]
ladmov.s. (988). Equatons of Mathematcal Physcs, 52 p. Moscow: Nauka. [The basc felds of moden mathematcal physcs ae pesented. The noton of a genealzed soluton s wdely used.] Weme J. (980). Potental Theoy, 4 p. Moscow: M. [The book s an ntoducton to the moden potental theoy. The abstact potental theoy as well as ts applcatons s pesented. The poblems ae consdeed n spaces of dmenson n.] Bogaphcal Sketches Agoshkov aley Ivanovch s a Docto of Physcal and Mathematcal Scences, pofesso of Insttute of Numecal Mathematcs of Russan Academy of Scences (Moscow). He s the expet n the feld of computatonal and appled mathematcs, the theoy of bounday poblems fo the patal dffeental equatons and tanspot equaton, the theoy of the conjugate opeatos and the applcatons. He s also the autho of moe than 60 eseach woks, ncludng 9 monogaphs. Hs basc eseach woks ae devoted to: the development of the effectve methods of numecal mathematcs; the theoy of Poncae-Steklov opeatos and methods of the doman decomposton; the development of methods of the optmal contol theoy and the theoy of conjugate equatons and the applcatons n the nvese poblems of mathematcal physcs; the development and justfcaton of new teatve algothms of the nvese poblems soluton; the development of the theoy of functonal spaces used n the theoy of bounday poblems fo the tanspot equaton; the detemnaton of new qualtatve popetes of the conjugate equatons soluton. Dubovsk Pavel Bosovch s a Docto of Physcal and Mathematcal Scences, docent of Insttute of Numecal Mathematcs of Russan Academy of Scences (Moscow). He s the expet n the feld of dffeental and ntegal equatons, the theoy of Smolukhovsky equatons, mathematcal modelng. He s the autho of moe than 40 eseach woks, ncludng one monogaph. Hs basc woks ae devoted to the development of the mathematcal theoy of coagulaton and cushng knetcs, ncludng the evelaton of new knetc models and tanston to a hydodynamc lmt, the development of the theoy of ntegal equatons and nonlnea equatons n patal devatves, and the eseach of some poblems of hydodynamcs.