ON THE PARTIAL DIFFERENCE EQUATIONS OP MATHEMATICAL PHYSICS by R. Courant, K. Friedrichs, and H. Lewy Mathematische Annalen, volume 100 (1928).

Transcription

1 ON THE PARTIAL DIFFERENCE EQUATIONS OP MATHEMATICAL PHYSICS b R Courant, K Friedrics, and H Lew Matematisce Annalen, volume 00 (98) TABLE OF CONTENTS Introduction Part I Te Elliptic Case Preliminar Remarks ) Definitions ) Difference Epressions and reen's Function Boundar Value and Eigenvalue Problems ) Te Teor of Boundar Value Problems ) Relation to te Minimum Problem 3) reen's Function ) Eigenvalue Problems 3 Connections wit te Problem of te Random Walk Te Solution of te Differential Equation as a Limiting Form of te Solution of te Difference Equation ) Te Boundar Value Problem of Potential Teor ) Proof of te Lemmas 3) Te Boundar Condition ) Applicabilit of te Metod to Oter Problems 5) Te Boundar Value Problem u = 0 Part II Te Hperbolic Case Te Equation of te Vibrating String On te Influence of te Coice of Mes Te Domains of Dependence of te Difference and Differential Equations 3 Limiting Values for Arbitrar Rectangular rids Te Wave Equation in Tree Variables Appendi - Supplements and eneralizations 5 Eample of a Differential Equation of First Order 6 Te Equation of Heat Conduction 7 Te eneral Homogeneous Linear Equation of Second Order in te Plane 8 Te Initial Value Problem for an Arbitrar Linear Hperbolic Differential Equation of Second Order

2 Introduction Problems involving te classical linear partial differential equations of matematical psics can be reduced to algebraic ones of a ver muc simpler structure b replacing te differentials b difference quotients on some (sa rectilinear) mes Tis paper will undertake an elementar discussion of tese algebraic problems, in particular of te beavior of te solution as te mes widt tends to zero For present purposes we limit ourselves mainl to simple but tpical cases, and treat tem in suc a wa tat te applicabilit of te metod to more general difference equations and to tose wit arbitraril man independent variables is made clear Corresponding to te correctl posed problems for partial differential equations we will treat boundar value and eigenvalue problems for elliptic difference equations, and initial value problems for te perbolic or parabolic cases We will sow b tpical eamples tat te passage to te limit is Indeed possible, ie, tat te solution of te difference equation converges to te solution of te corresponding differential equation; in fact we will find tat for elliptic equation in general a difference quotient of arbitraril ig order tends to te corresponding derivative Nowere do we assume te eistence of te solution to te differential equation problem on te contrar, we obtain a simple eistence proof b using te limiting process For te case of elliptic equations convergence is obtained independentl of te coice of mes, but we will find tat for te case of te initial value problem for perbolic equations, convergence is obtained onl if te ratio of te mes widts in different directions satisfies certain inequalities wic in turn depend on te position of te caracteristics relative to te mes We take as a tpical case te boundar value problem of potential teor Its solution and its relation to te solution of te corresponding difference equation as been etensivel treated during te past few ears However in Our metod of proof ma be etended witout difficult to cover boundar value and eigenvalue problems for arbitrar linear elliptic differential equations and initial value problems for arbitrar linear perbolic differential equations J lerou, "Sur le problem de Diriclet", Journ de matem pur et appl (6) 0 (9), p 89 R D Ricardson, "A new metod in boundar problems for differential equations", Trans of te Am Mat Soc 8 (97), p 89ff H B Pilips and N Wiener, "Nets and te Diriclet Problem", Publ of M I T (95) Unfortunatel tese papers were not known b te first of te tree autors wen e prepared is note "On te teor of partial difference equations", ött Nacr 3, X, 95, on wic te present work depends See also L Lusternik, "On an application of te direct metod in variation calculus", Recuell dela Société Matém de Moscou, 96 Bouligand, "Sur le problème de Diriclet," Ann de la soc polon de matém, Cracow 96 On te meaning of te difference epressions and on furter applications of tem, see R Courant, "Uber dlrekte Metoden in der Variationsrecnung", Mat Annalen 97, p 7 > and te references given terein

3 contrast to te present paper, te previous work as involved te use of quite special caracteristics of te potential equation so tat te applicabilit of te metod used tere to oter problems as not been immediatel evident In addition to te main part of te paper, we append an elementar algebraic discussion of te connection of te boundar value problem of elliptic equations wit te random walk problem arising in statistics Preliminar Remarks Definitions Part I Te Elliptic Case Consider a rectangular arra of points in te (, ) -plane, suc tat for mes widt > 0 te points of te lattice are given b = n = m m, n = 0, ±, ±, Let be a region of te plane bounded b a continuous closed curve wic as no double points Ten te corresponding mes region, wic is uniquel determined for sufficientl small mes widt consists of all tose mes points ling in wic can be connected to an oter given point in b a connected cain of mes points B a connected cain of mes points we mean a sequence of points suc tat eac point follows in te sequence one of its four neigboring points We denote as a boundar point of All oter points of a point wose four neigboring points do not all belong to we call interior points We sall consider functions u, v, of position on te grid, ie, functions wic are defined onl for grid points, but we sall denote tem as u(,), v(,), For teir forward and backward difference quotients we emplo te following notation, u u = [ u( +, ) u(, )], u = [ u(, + ) u(, )], = [ u(, ) u(, )], u = [ u(, ) u(, )] In te same wa te difference quotients of iger order are formed, eg, ( u ) = u = u = [ u( +, ) u(, ) + u(, )}, etc Difference Epressions and reen's Function In order to stud linear difference epressions of second order, we form (using as a model te teor of partial differential equations), a bilinear epression from te forward difference quotients of two functions, u and v, 3

4 B( u, v) = au v + bu v + cu v + du v + αu v+ βu v+ γuv + δuv + guv were are functions on te mes a = a(, ),, α = α(, ),, g = g(, ) From te bilinear epression of first order we derive a difference epression of second order in te following wa: we form te sum over all points of a region B( u, v) In te mes, were in B(u, v) te difference quotients between boundar points and points not belonging to te mes are to be set equal to zero We now transform te sum troug partial summation, ie, we arrange te sum according to v, and split it up into a sum over te set of interior points, we obtain: ' and a sum over te set of boundar points, () B( u, v) = vl( vr( ' Γ Γ Tus L( is a linear difference epression of second order defined for all interior points of C n : L( = ( au ) αu + ( bu ) βu + ( γ + ( cu ) + ( δ + ( du ) gu R( is, for ever boundar point, a linear difference epression wose eact form will not be given ere If we arrange B ( u, v) according to u, we find () B( u, v) = um ( v) us( v) ' M(v) is called te adjoint difference epression of L( and as te form M ( v) = ( av ) + ( bv ) + ( α v) + ( βv) + ( cv ) γv δv Γ ' + ( dv ) gv wile S(v) is a difference epression corresponding to R( for te boundar Formulas () and () give (3) ( vl( um ( v)) + ( vr( us( v)) = 0 ' Γ ' Formulas (), () and (3) are called reen s Formula Te simplest and most important case results if te bilinear form is smmetric, ie, if te relations b = c, α = γ,, β = δ old In tis case L( is identical wit M( te self-adjoint case and it can be derived from te quadratic epression

5 + bu u + du + u u+ u u+ B( u, = au α β gu In te following we sall limit ourselves mainl to epressions L( wic are self-adjoint Te caracter of te difference epression L( depends principall on te nature of tose terms in te quadratic form B(u, v) wic are quadratic in te first difference quotients We call tis part of B(u, te caracteristic form: ( u, au + bu u du P = + We call te corresponding difference epression L( elliptic or perbolic depending on weter te function P(u, of te difference quotients is (positive) definite or indefinite Te difference epression u = u + u wit wic we sall concern ourselves in te following paragrap is elliptic, ie, it comes from te quadratic epression ( u, u u B = + or u u + Te corresponding reen s formulas are 3 () ( u + u ) = u v ur(, (5) ( v u u v) + ( vr( ur( v)) = 0 ' ' ' Te difference epression, u = u + u, is obviousl te analogue of te differential epression u u + for a function u(, ) of te continuous variables and Written out eplicitl te difference epression is u = { u( +, ) + u(, + ) + u(, ) + u(, ) u(, ) } Terefore u is te ecess of te aritmetic mean of te functional values at te four neigborood points over te functional value at te point in question Completel similar considerations lead to linear difference epressions of te fourt order and corresponding reen's formula, provided one starts 3 Te boundar epression R( ma be written as follows: Let u 0, u, K, uν be values of te function at a boundar point and at its ν neigboring points ( ν 3 ), ten R( = ( u uν νu0 ) + L + 5

6 from bilinear difference epressions wic are formed from te difference quotients of second order Consider for eample te difference epression u= u ++ u u Tis corresponds to te quadratic epression B( u, = ( u + u ) = (, provided one orders te sum ' u v according to v, or equivalentl replaces u b notice owever tat in te epression u in equation (5) One must u, te functional value at a point is connected wit te values at its neigboring points and at teir neigboring points, and accordingl is defined onl for suc points of te region are also interior points of te region as ' ' (See Section 5) Te entiret of suc points we designate as '' We obtain ten reen's formula (6) u v= v u+ v R(, D ' D' Γ+ Γ ' were R is a definable linear difference epression for eac point of te boundar strip ' Γ + Γ Γ ' indicates te set of boundar points of ' Boundar Value and Eigenvalue Problems Te teor of boundar value problems Te boundar value problem for linear elliptic omogeneous difference equations of second order, wic corresponds to te classical boundar value problem for partial differential equations, can be formulated in te following wa Let tere be given a self-adjoint elliptic linear difference epression of second order, L(, in a mes region, L( results from a quadratic epression B(u, v) wic is positive definite in te sense tat it cannot vanis if u and u do not temselves vanis A function, u, is to be determined satisfing in equation L( = 0, and assuming prescribed values at te boundar points te difference Under tese requirements tere will be eactl as man linear equations as tere are interior points of te mes at wic te function u is to be determined Some of tese equations wic involve onl mes points wose If te matri of te linear sstem of equations corresponding to an arbitrar difference 6

7 neigbors also lie in te interior of te region are omogeneous; oters wic involve boundar points of te mes region are inomogeneous If te rigt-and side of te inomogeneous equations is set equal to zero, tat is if u = 0 on te boundar, ten it follows from reen's formula (), b setting u = v tat B(u, vanises, and furter from te definiteness of B(u, tat u and u vanis, and ence tat u itself vanises Tus te difference equation for zero boundar value as te solution u = 0, or in oter words te solution is uniquel determined since te difference of two solutions wit te same boundar value must vanis Furter, if a linear sstem of equations wit as man unknowns as equations is suc tat for vanising rigt-and side te unknowns must vanis, ten te fundamental teorem of te teor of equations asserts tat for an arbitrar rigt-and side eactl one solution must eist In our case tere follows at once te eistence of a solution of te boundar value problem Terefore we see tat for elliptic difference equations te uniqueness and eistence of te solution of te boundar value problem are related to eac oter troug te fundamental teorem of te teor of linear equations, wereas for partial differential equations bot facts must be proved b quite different metods Te basis for tis difficult in te latter case is to be found in te fact tat te differential equations are no longer equivalent to a finite number of equations, and so one can no longer depend on te equalit of te number of unknowns and equations Since te difference epression u = 0 can be derived from te positive definite quadratic epression ( + u u ), te boundar value problem for te difference epression is uniquel solvable Te teor for difference equations of iger order is developed in eactl te same wa as tat for difference equations of second order; for eample one can treat te fourt order difference equation u = 0 In tis case te values of te function must be prescribed on te boundar strip Γ + Γ ' Evidentl ere also te difference equation ields just as man linear equations as tere are unknown functional values at te points of equation of second order, L( = 0, is transposed, ten te transposed set of equations corresponds to te adjoint difference equation M(v) = 0 Tus te above self-adjoint sstem gives rise to a set of linear equations wit smmetric coefficients 7

8 '' In order to demonstrate te eistence and uniqueness of te solution one needs onl to sow tat a solution wic as te value zero in te boundar strip Γ + Γ ' necessaril vanises identicall To tis end we note tat te sum over te corresponding quadratic epression (7) ( '' for suc a function vanises, as can be seen b transforming te sum according to reen's formula (6) Te vanising of te sum (7) owever implies tat u vanises at all points of ', and according to te above proof tis can onl appen for vanising boundar values if te function u assumes te value zero trougout te region Tus our assertion is proved, and bot te eistence and uniqueness of te solution to te boundar value problem for te difference equations are guaranteed 5 Relation to te minimum problem Te above boundar value problem is related to te following minimum problem: among all functions ϕ (, ) defined in te mes region and assuming given values at te boundar points, tat function ϕ = u(, ) is to be found for wic te sum B( ϕ, ϕ) over te mes region assumes te least possible value We assume tat te quadratic difference epression of first order, B(u, is positive definite in te sense described in Section, part One can sow tat te difference equation L ( ϕ) = 0 results from tis minimum requirement on te solution ϕ = u(, ), were L (ϕ ) is te difference epression of second order derived previousl from B ( ϕ, ϕ) In fact tis can be seen eiter b appling te rules of differential calculus to te sums B( ϕ, ϕ) as a function of a finite number of values of ϕ at te grid points, or b emploing te usual metods from te calculus of variations B wa of eample, solving te boundar value problem of finding te solution to te equation ϕ = 0 wic assumes given boundar values, is equivalent to minimizing te sum ( + ϕ ϕ ) over all functions 5 For te actual process of carring troug te solution of te boundar value problem b an iterative metod, see among oters te treatment: "Uber Randwertanjgaen bei partieller Differenzenglertunger" b R Courant, Zeitscr f angew, Matematik u Mecanik 6 (95), pp 3-35 Also tere is a report b H Henk, in Zeitscr f angew Mat u Mec (9), p 58 ffrr 8

9 assuming given values on te boundar strip Γ ' Besides tis sum tere are et oter quadratic epressions in te second derivatives wic give rise to te equation u = 0 under te process of being minimized For eample tis is true in for te sum u + u + ' ( u ) Tat te minimum problem posed above alwas as a solution follows from te teorem tat a continuous function of a finite number of variables (te functional values of ϕ at te grid points) alwas as a minimum if it is bounded from below and if it tends to infinit as soon as an of te independent variables goes to infinit 6 3 reen's function It is possible to treat te boundar value problem for te inomogeneous equation, L( = -f, in muc te same wa as te omogeneous case, L( = 0 For te inomogeneous case it is sufficient to consider onl te case of zero boundar conditions, since different boundar conditions can be taken care of b adding a suitable solution of te omogeneous equation To solve te linear sstem of equations representing te boundar value problem, L( = -f, we first coose as te function f(, ) a function wic assumes te value at te point = ξ, = η of te mes If K (, ; ξ, η) is te solution (vanising on te boundar) of tis difference equation wic depends on te parametric point ( ξ, η), ten te solution for an arbitrar boundar condition can be represented b te sum u(, ) = ( ξ, η) K(, ; ξ, η) f ( ξ, η) Te function K (, ; ξ, η) wic depends on te points (,) and ( ξ, η) is called te reen's function of te differential epressions L( If we call te reen's function for te adjoint epression M(v), K (, ; ξ, η), ten te equivalence K ( ξ, η ; ξ, η) = K ( ξ, η; ξ, η ) olds, as can be seen to follow immediatel from reen's formula (3) wen u = K(, ; ξ, η) and v= K (, ; ξ, η ) For a self-adjoint difference epression te above relation gives te smmetric epression: K ( ξ, η ; ξ, η) = K( ξ, η; ξ, η ) 6 It can easil be verified tat te poteses for te application of tis teorem are satisfied 9

10 Eigenvalue problems Self-adjoint difference epressions, L(, give rise to eigenvalue problems of te following tpe: find te value of a parameter λ, te eigenvalue, suc tat in equation, a solution, te eigenfunction can be found for te difference were u is to be zero on te boundar, L( + λ u = 0, Γ Te eigenvalue problem is equivalent to finding te principle aes of te quadratic form B(u, Eactl as man eigenvalues and corresponding eigenfunctions eist as tere are interior mes points of te region Te sstem of eigenfunctions and eigenvalues (and a proof of teir eistence) is given b te minimum problem: Among all functions, ϕ (, ), vanising on te boundar, and satisfing te ortogonalit relation and normalized suc tat te function, ϕ = ( ν ) ϕ u = 0, ( ν =, K, m ) ϕ =, u, is to be found for wic te sum B( ϕ, ϕ) assumes its minimum value Te value of tis minimum is te m-t eigenvalue, and te function for wic it is assumed is te m-t eigenfunction 7 3 Connections wit te Problem of te Random Walk 8 Te teme of te following is related to a question from te teor of 7 From te ortogonalit condition on te eigenfunctions, ( ν ) ( µ ) u u = 0, ( ν µ ) it follows tat eac function, g(,), wic vanises on te boundar of te mes can be epanded in terms of te eigenfunctions in te form of g = N ν= c 0 ( ν ) ( ν ), were te coefficients are determined from te equations ( ν ) ( ν ) c = gu u In tis wa in particular te following representation for te reen's functions ma be derived 8 Section 3 is not prerequisite to Section N ν= ( ν ) ( ν ) u (, ) u ( ξ, η) K(, ; ξ, η) = ( ν ) λ

11 probabilit, namel te problem of te random walk in a bounded region 9 We consider te lines of a mes region as pats along wic a particle can move from one grid point to a neigboring one In tis net of streets te particle can wander aimlessl, and it can coose at random one of te four directions leading from eac intersection of pats of te net - all four directions being equall probable Te walk ends as soon as a boundar point of is reaced because ere te particle must be absorbed We ask: ) Wat is te probabilit w(p;r) tat a random walk starting from a point P reaces a particle point R of te boundar? ) Wat is te matematical epectation v(p; tat a random walk starting from P reaces a point Q, of witout toucing te boundar? Tis probabilit or matematical epectation, respectivel will be defined more precisel b te following process Assume tat at te point P tere is a unit concentration of matter Let tis matter diffuse into te mes wit constant velocit, traveling sa a mes widt in unit time At eac mespoint let eactl one fourt of te matter at te point diffuse outwards in eac of te four possible directions Te matter wic reaces a boundar point is to remain at tat point If te point of origin P is itself a boundar point, ten te matter never leaves tat point We define te probabilit w(p, R) tat a random walk starting from P reaces te boundar point R (witout previousl attaining te boundar), as te amount of matter wic accumulates at tis boundar point over an infinite amount of time We define te probabilit E n ( P; tat te walk starting from te point P reaces te point Q in eactl n steps witout toucing te boundar b te amount of matter wic accumulate in n units of time provided P and Q are bot interior points If eiter P or Q are boundar points ten set equal to zero Te value E n ( P; is eactl equal to n E n is / times te number of pats of n steps leading from P to Q witout toucing te boundar Tus En ( P; = En ( Q; P) We define te matematical epectation V(P; tat one of te pats considered above leads from P to te point Q, b te infinite sum of all of tese possibilities, 0 9 Te present treatment is essentiall different from te familiar treatments wic can be carried troug, sa for eample in te case of Brownian motion for molecules Te difference lies precisel in te wa in wic te boundar of te region enters 0 Te convergence will be sown below

12 = V ( P; Eν ( P;, =0 ν ie, for te interior points P and Q te sum of all te concentrations wic ave passed troug te point Q at different times Tis will be assigned te epected value for a concentration originating at Q If te amount arriving at te boundar point R in eactl n steps is designated as ( P; R), ten te probabilit w(p;r) is given b te series F n = w ( P; R) Fν ( P; R) =0 ν All te terms of tis series are positive and te partial sum is bounded b one (since te concentration reacing te boundar can be made up of onl part of te initial concentration), and terefore te convergence of te series is assured Now it is eas to see tat te probabilit ( P;, tat is, te concentration reacing te point Q in eactl n steps tends to zero as n increases For if at an point Q, from wic te boundar point R can be reaced in m steps, we ave ( P; > a> 0, ten at least E n E n a a / of te concentration at Q will reac te point R after m steps However, since te sum ν=0 F ( P; R) converges, te concentration reacing R goes to zero wit ν increasing time, were te value of E n ( P; must itself vanis as time increases; tat is te probabilit of an infinitel long walk remaining in te interior of te region is zero From tis it follows tat te entire concentration eventuall reaces te boundar; or in oter words tat te sum R w ( P; R) over all te boundar points R is equal to one Te convergence of te infinite series for te matematical epectation v(p; remains to be sown = v ( P; E =0 ν ν ( P; To tis end we remark tat te quantit E n ( P; satisfies te following relations En + ( P; = { En ( P; Q ) + En ( P; Q ) + En ( P; Q3 ) + En ( P; Q) }, ( n ), were Q troug Q are te four neigboring points of te interior point Q Tat is, te concentration at te point Q at te (n+)-st step consists of /

13 times te sum of te concentrations at te four neigboring points at te n-t step If one of te neigbors of Q, for eample Q = R, is a boundar point ten it follows tat no concentration flows from tis boundar point to Q since te epression E n ( P; is zero in tis case Furtermore, for an interior point, E ( P; P) and of course E ( P; 0 0 = = From tese relationsips we find for te partial sum te equation v ( P; = n n ν= 0 E ( P; ν for v P; = { v ( P; Q ) + v ( P; Q ) + v ( P; Q ) v ( P; )} n + ( n n n 3 + n Q, P Q For P = Q, v P; P) = + { v ( P; P ) + v ( P; P ) + v ( P; P ) v ( P; )} n + ( n n n 3 + n P, tat is, te epectation tat a point goes back into itself consists of te epectation tat a non-vanising pat leads from P back again to itself namel { v ( P; P ) v ( P; P ) + v ( P; P ) v ( P; )} +, n n n 3 + n P togeter wit te epectation unit tat epresses te initial position of te concentration itself at tis point Te quantit v n ( P; tus satisfies te following difference equation vn ( P; = E ( P; n, for P Q, vn ( P; = ( E ( P; ) n, for P = Q v n ( P; is equal to zero wen Q is a boundar point Te solution of tis boundar value problem for arbitrar rigt-and side is uniquel determined as we ave eplained earlier (Section, Part ), and depends continuousl on te rigt-and side Since te variables ( P; tend to zero, te solution v n ( P; converges to te solution v ( P; of E n Tis defines te -operation for te variable point Q Tis equation can be interpreted as an equation of te eat conduction tpe Tat is if te function v n ( P; is considered, not as a function of te inde n as in our presentation above, but rater as a function of time, t, wic is proportional to n, so tat t = nτ and v n ( P; = v( P; Q; t) = v( t), ten te above equations can be written in te following form: t v( t+ τ ) v( t) v( t) =, for P Q, τ t v( t+ τ ) v( t) v ( t) =, for P = Q τ τ For a similar difference equation wic as a parabolic differential equation as its limiting form, see Section 6 of te second alf of te paper 3

14 te difference equation v( P; = 0 for P Q, v( P; = for P = Q, wit boundar values v ( P; R) = 0 Tus we see tat te matematical epectation v(p; eists and is non oter tan te reen's function for te difference equation u = 0, ecept for a factor of Te smmetr of te reen's function is an immediate consequence of te smmetr of te quantit E n ( P; wic was used to define it Te probabilit w(p;r) satisfies, wit respect to P, te relation w ( P; P) = { w( P ; R) + w( P ; R) + w( P3 ; R) + w( P ; R) }, and tus te difference equation w=0 Tat is, if P, P, P3, P are te four neigboring points of te interior point P, ten eac pat from P to R must pass troug one of tese four directions, and eac of te four is equall likel Furtermore, te probabilit of going from one boundar point R to anoter R' is w(r;r') = 0 unless te two points R and R' coincide, in wic case w(r;r) = Tus w(p;r) is tat solution of te boundar value problem w= 0 wic assumes te value at te boundar point R and te value 0 at all oter points of te boundar Terefore te solution of te boundar value problem for an arbitrar boundar value u(r) as te simple form u ( P) w( P; R) u( R), were te = R sum is to be etended over all te boundar points If te function u is substituted for u in tis epression, ten we ceck te relation = w ( P; R) R Te interpretation given above for reen's function as an epectation ields immediatel furter properties We mention onl te fact tat te reen's function decreases if one goes from te region to a subregion ling witin ; tat is te number of possible pats for steps on te mes leading from one point P to anoter Q (witout toucing te boundar), decreases as te region decreases Of course corresponding relationsips old for more tan two Moreover it is eas to sow tat te probabilit w(p;r) of reacing te boundar is te boundar epression ~ R ( K( P, ), constructed from te reen's function K(P; in terms of Q, were u(,) is to be identified wit w(p,, and v(,) wit v(p, in reen's formula (5)

15 independent variables We note onl tat oter elliptic difference equations admit a similar probabilit interpretation If te limit for vanising mes widt is considered b metods given in te following section, ten te reen's function on te mes goes over to te reen's function of te potential equation ecept for a numerical factor; a similar relationsip olds between te epression w ( P; R) and te normal derivative of te reen's function at te boundar of te region In tis wa for eample te reen's function for te potential equation could be interpreted as te specific matematical epectation of going from one point to anoter, 3 witout reacing te boundar In going over to te limit of a continuum from te mes, te influence of te direction in te mes prescribed for te random walk vanises Tis fact suggests tat it would be of some interest to consider limiting cases of more general random walks for wic te limitations on te direction of te walk are relaed Tis lies outside of te scope of tis presentation owever and we can onl ope to renew te question at some oter opportunit Te Solution of te Differential Equation as a Limiting Form of te Solution of te Difference Equation Te boundar value problem of potential teor In letting te solution of te difference equation tend to te solution of te corresponding differential equation we sall relinquis te greatest possible degree of generalit wit regard to te boundar and boundar values in order to demonstrate more clearl te caracter of our metod Terefore we assume tat we are given a simpl connected region wit a boundar formed of a finite number of arcs wit continuousl turning tangents Let f(,) be a given function wic is continuous and as continuous partial derivatives of first and second order in a region containing If is te mes region wit mes widt belonging to te region, ten let te boundar value problem for te difference equation u = 0 be solved for te same boundar values wic te function f(,) assumes on te boundar; let u (, ) be te solution We sall prove tat as 0 te function u (, ) converges to a function u(,) wic satisfies in te partial 3 Here te a priori epectation of reacing a certain area element is understood to be equal to te area of te element We mention owever tat carring troug our metod for more general boundaries and boundar values in no wa causes an fundamental difficult 5

16 u u differential equation + = 0 and takes te value of f(, ) at eac of te points of te boundar We sall sow furter tat for an region ling entirel witin te difference quotients of u of arbitrar order tend uniforml towards te corresponding partial derivatives of u(, ) In te convergence proof it is convenient to replace te boundar condition u = f b te weaker requirement tat were r S r ( u f ) dd 0 as r 0, S r is tat strip of wose points are at a distance from te boundar smaller tan r 5 Te convergence proof depends on te fact tat for an subregion * ling entirel witin, te function u (, ) and eac of its difference quotients is bounded and uniforml continuous as 0 in te following sense: For eac of tese functions, sa w (, ), tere eists a δ (ε ) depending onl on te subregion and not on suc tat w ( P) w ( P ) <ε provided te mes points P and P lie in te same subregion of are separated from eac oter b a distance leas tan δ (ε ) and Once uniform continuit in tis sense (equi-continuit) as been establised we can in te usual wa select from te functions u a subsequence of functions wic tend uniforml in an subregion * to a limit function u(,), wile te difference quotients of u tend uniforml towards te partial derivatives of u Te limit function ten possesses derivatives of arbitraril ig order in an proper subregion * of and satisfies u = 0 in tis region If we can sow also tat u satisfies te boundar condition we can regard it as te solution of our boundar value problem for te region Since tis solution is uniquel determined, it is clear tat not onl a partial sequence of te functions of functions itself possesses te required convergence properties u but tis sequence 5 Te weaker boundar value requirement does in fact provide te unique caracterization of te solution, as can be seen from te easil proved teorem: If te boundar condition above is satisfied for f(,) = 0 for a function satisfing te equation u u + = 0 in te interior of and if u u eists, ten u(, ) is identicall zero (See + dd Courant, "Uber die Losung der Differentialgleicungen der Psik", Mat Ann 85 ep p 96 ff ) In te case of two independent variables te boundar values are actuall attained, as follows from te weaker requirement; but in te case of more variables te corresponding result cannot in general be epected since tere can eist eceptional points on te boundar at wic te boundar value is not taken on even toug a solution eists under te weaker requirement 6

17 Te uniform continuit (equi-continuit) of our quantities ma be establised b proving te following lemmas ) As 0 te sums over te mes region remain bounded 6 u and ( + u u ) ) If w = w satisfies te difference equation w= 0 at a mes point of, and if, as 0 te sum w * etended over a mes region * associated wit a subregion * of remains bounded, ten for an fied subregion ** ling entirel witin * te sum over te mes region as 0 ( + ** w w ) ** associated wit ** likewise remains bounded Using () tere follows from tis, since all of te difference quotients w of te function sums is bounded u again satisf te difference equation w= 0, tat eac of te * 3) From te boundedness of tese sums tere follows finall te boundedness and uniform continuit of all te difference quotients temselves w Proof of te lemmas Te proof of ) follows from te fact tat te functional values u are temselves bounded For te greatest (or least) value of te function is assumed on te boundar 7 and so is bounded b a prescribed finite value Te boundedness of te sum ( + u u ) is an immediate consequence of te minimum propert of our mes function formulated in Part of Section wic gives in particular ( u + u ) ( f + f ), but as 0 te sum on te rigt tends to te integral 6 Here and in te following we drop te inde from te grid functions 7 We note owever wit a view to carring over te metod to oter differential equations, tat we can rela tese conditions To tis end we need onl to bring into pla te inequalit (5) or to use te reasoning of te alternative (see Part, Section ) 7

18 wic, b potesis, eists f f + dd To prove ) we consider te quadratic sum ( w + w + w + int Q 8 w ), were te summation etends over all te interior points of a square Q, (see Fig ) We call te values of te function on te boundar S of te square Q, w, and tose on te boundar S 0, of Q 0, w 0 Ten reen s formula gives (8) ( w + w + w + w ) = w w w, int Q were S and S 0 are respectivel te boundar of Q and te square boundar of te lattice points Q 0 ling witin S, wile C consists of te four corner points of te boundar of Q We now consider a sequence of concentric squares boundaries S, S,, S S S0 C Q, Q,, Q 0 K n wit 0 K n, were eac boundar is separated from te net b a mes widt Appling te formula to eac of tese squares and observing tat we ave alwas we obtain ( w + w ) ( w + w + Q0 Q k w ), ( k ) ( w + w ) w w w, ( k < n ) Q0 Sk Sk Ck We strengten te inequalit b neglecting te last non positive term on te rigt and we ten add te n inequalities to obtain n ( w + w ) w w w Q0 Sn S0 S n Summing tis inequalit from n = to n = N we get N w + w ) Q0 ( w Diminising te mes widt we can make te squares Q 0 and converge towards two fied squares ling witin and aving corresponding sides separated b a distance a In tis process independent of te mes widt (9) ( w + w ) w a Q0 Q N Q N Q n N a and we ave,

19 For sufficientl small tis inequalit olds of course not onl for two squares Q 0 and Q N but wit a cange in te constant, a, for an two subregions of suc tat one is contained entirel witin te oter Tus lemma () is proved 8 In order to prove te tird result, tat quotients u and all its partial difference w remain bounded and uniforml continuous as 0, we consider a rectangle R wit corners PQ of lengt a parallel to te -ais We start wit te relation and te inequalit P 0, Q0, P, Q and wit sides P 0 Q 0 and = w( Q0 ) w( P0 ) w w, () + w ( Q ) w( P0 ) w w wic is a consequence of it PQ 0, PQ R We ten let te side PQ of te rectangle var between an initial line P Q, a distance b from P 0 Q 0 and a final line P Q a distance b from P 0 Q 0, R b and we sum te corresponding + inequalities () We obtain te estimate + + w ( Q0 ) w( P0 ) w w, b were te summations etend over te entire rectangle, R = P0 Q0P Q R From Scwarz's inequalit it ten follows tat, () w + R w ( Q0 ) w( P0 ) ab ab w b R R Since, b potesis, te sums wic occur ere multiplied b remain bounded, it follows tat as a 0 te difference 0 0 w ( P ) w( Q ) 0 independentl of te mes-widt, since for eac subregion * of te quantit b can be eld fied Consequentl te uniform continuit (equi-continuit) of w= w is proved for te -direction Similarl it olds for te -direction and so also for an subregion * of Te boundedness of te function w in * finall follows from its uniform 8 If we do not assume tat w= 0, ten in place of te inequalit (9) we find (0) + + ( w w ) c w c ( w) ** * for suitable constants c and c independent of, were ** lies entirel witin * and * in turn is contained in te interior of 9 *

20 continuit (equi-continuit) and te boundedness of * w B tis proof we establis te eistence of a subsequence of functions u wic converge towards a limit function u(, ) and wic do so uniforml togeter wit all teir difference quotients, in te sense discussed above for ever interior subregion of Tis limit function u(, ) as trougout continuous partial derivatives of arbitrar order, and satisfies tere te potential equation: u u + = 0 3 Te boundar condition In order to prove tat te solution satisfies te boundar condition formulated above we sall first of all establis te inequalit (3) v Ar ( v + v ) + Brv, were boundar strip Sr, Sr, S r, is tat part of te mes region S r Tis boundar strip Γ wic lies inside a S r consists of all points of wose distance from te boundar is less tan r; it is bounded b Γ and anoter curve Γ r Te constants A and B depend onl on te region and not on te function v nor te mes widt In order to prove te above inequalit, we divide te boundar, Γ, of into a finite number of pieces for wic te angle of te tangent wit one of te or -aes is greater tan some positive value (sa 30 ) Let γ, for instance, be a piece of Γ wic is sufficientl steep (in te above sense) relative to te -ais, (See Fig ) Lines parallel to te -ais intersecting γ will cut a section togeter wit γ and smbol γ r from te neigboring curve γ r a piece s r, to denote te portion of associated portion of te boundar s r of te boundar strip Γ b contained in γ Γ r, and will define S r We use te s r and denote te We now imagine a parallel to te -ais to be drawn troug a mes point P of s r, Let it meet te boundar line wic lies in s r, we call p r, γ in a point P Te portion of tis Its lengt is certainl smaller tan cr, were te constant c depends onl on te smallest angle of inclination of a tangent γ to te -ais Between te values of v at P and P we ave te relation v ( P ) = r( P ) ± v P P 0

21 Squaring bot sides and appling Scwarz's inequalit, we obtain Summing wit respect to v( P ) v( P ) + cr v p r, P in te -direction, we get v crv( P ) + c r v pr p r Summing again in te -direction we obtain te relation () v crv( P ) + c r v Sr, Γ Writing down te inequalities associated wit te oter portions of Γ and adding all te inequalities togeter we obtain te desired inequalit (3) 9 We net set v Sr, = u f so tat v = 0 on Γ Ten since ( + v v ) remains bounded as 0, we obtain from (3) ~ (6) v Hr r were H ~, S r, is a constant wic does not depend on te function v or te mes S, Sρ, size Etending te sum on te left to te difference ν of two boundar strips, te inequalit (6) stills olds wit te same constant H ~ and we can pass to te limit 0 From te inequalit (6) we ten get ~ r S r v dd Hr, v = u f S ρ Now letting te narrower boundar strip obtain te inequalit ~ v dd ( u f ) dd Hr r = r, S r S r S ρ approac te boundar we wic states tat te limit function u satisfies te prescribed boundar condition Applicabilit of te Metod to Oter Problems Our metod is based essentiall on te inequalities arising from te lemmas stated previousl since te principal points of te proofs follow from 9 B similar reasoning we can also establis te inequalit (5) v c v + c ( v + v ) Γ

22 te two last lemmas in part of Section 0 No use is made of special fundamental solutions or special properties of te difference epression, and terefore te metod can be carried over directl to te case of arbitraril man independent variables as well as to te eigenvalue problem, u u + + λ u= 0 Te same sort of convergence relations will obtain in tis case as above Also te metod applies to linear partial differential equations of oter tpes, in particular its application to equations wit variable coefficients requires onl some minor modifications Te essential difference in tis case lies onl in proving te boundedness of u wic of course does not alwas old for an arbitrar linear problem However in case tis sum is not bounded it can be sown tat te general boundar value problem for te differential equation in question also possesses effectivel no solutions, but tat in tis case tere eist non-vanising solutions of te corresponding omogeneous problem, ie, eigenfunctions 5 Te boundar value problem u= 0 In order to sow tat te metod can be carried over to te case of differential equations of iger order, we will treat briefl te boundar value problem of te differential equation: u u + u + = 0 We seek, in, a solution of tis equation for wic te values of u and its first derivative are given on te boundar, being specified tere b some function f(, ) To tis end we assume as previousl tat f(, ) togeter wit its first and second derivatives is continuous in tat region of te plane containing te region We replace our differential equation problem b te new problem of solving te difference equation u= 0 in te mes region subject to te condition tat at te points of te boundar strip Γ + Γ te function u in wic te constants c and c depend onl on te region and not on te mes division 0 For an application of corresponding integral inequalities see K Friedrics, "Die Rand-und Eigen Wert problems aus der Teorle der elastiscen Platten", Mat Ann 98, p Similarl one proves at te same time tat ever solution of suc a differential equation problem as derivatives of ever order See Courant-Hilbert, Metoden der matematiscen Psik, Cap III, Section 3, were te teor of integral equations is andled wit te elp of te corresponding principle of te alternative See also te Dissertation (ottingen) of W v Koppenfels wic will appear soon

23 takes on te values f(, ) From Section we know tat tis boundar value problem as a unique solution We will sow tat as te mes size decreases tis solution, in eac interior subregion of, converges to te solution of te differential equation, and tat all of its difference quotients converge to te corresponding partial derivatives We note first tat for te solution u = u, te sum ( u + u + u ) remains bounded as 0 Tat is, b appling te minimum requirements on te solution (part, Section ) one finds tat tis sum is not larger tan te corresponding sum ( f + f + f ) and tis converges as 0 to wic eists, b potesis f From te boundedness of te sum f f + + dd ( u + u + u ) follows immediatel te boundedness of ( u + u ) and ( and also tat of Tat is, for arbitrar w te following inequalit olds (see footnote 9), (5) w c( w + w ) + c w Ten if one substitutes te first difference quotients of w for w itself in tis inequalit and applies te epression over te subregion of te are defined, ten one finds te furter inequalit, ( w + w ) c ( w + w + w u ) + c Γ Γ + Γ for wic ( w + w ) were again te constant c is independent of te function and of te mes size We appl tis inequalit to te sum over w= u and tus find te boundedness of Γ + Γ on te rigt-and side; (b definition tese boundar sums converge to te corresponding integral containing f(, ) ) Tus from te boundedness of 3

24 ( u + u + u ) follows te boundedness of ( + u u ) and tence tat of u u, For te tird step we substitute one after te oter te epressions u, u,, for w in te inequalit ( w + w ) c w + c ( w) u, (see part, Section ) were is a subregion of containing in its interior Te epressions introduced all satisf te equation w= 0 follows ten tat for eac epression in turn and for all subregions tat te sums, ( w + w ) It of, tat is, ( u + u ), ( u + u ),, are bounded togeter wit te sums: u, ( + wic are alread known to be bounded u u ), and ( Finall we substitute into te inequalit (0), in place of w, te sequence of functions u, u, u, u,, for wic ( w), ie, ( u ), are bounded as sown above We ten find tat for all subregions te sums remain bounded ( u + u ), ( u + u ), From tese facts we can now conclude as previousl tat from our sequence of mes functions a subsequence can be cosen wic in eac interior subregion of converges (togeter wit all its difference quotients) uniforml to a limit function (or respectivel its derivatives) wic is continuous in te interior of We ave et to sow tat tis limit function wic obviousl satisfies te differential equation u= 0 also takes on te prescribed boundar conditions For tis purpose we sa ere onl tat, analogous to te foregoing, te epressions

25 ( u f ) dd cr, S r S r u f u f + dd cr old 3 3 Tat te limit function fulfills tese conditions ma be seen b carring over te treatment in Part 3, Section to te function u and its first difference quotient From te uniqueness of our boundar value problem we see furtermore tat not onl a selected subsequence, but also te original sequence of functions u possesses te asserted convergence properties (End of Part I) 3 Tat te boundar values for te function and its derivatives actuall are assumed is not difficult to prove See for instance te corresponding treatment of K Friedrics, loc cit 5