J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 94
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In this procedure, the basic idea is to modify what corresponds to the function F(q>) in (12.2.14) in such a way that themodified problem (which is arranged to contain one or more parameters) can always be solved. Afterwards, conditions are written downto ensure that the modified problem is identical with the originalproblem; these conditions arc called the bifurcation conditions. Sucha process could have been used here in conjunction with an iterationscheme, as a substitute for the process of fixing the parameters e n atstage of the iteration procedure in the manner indicated above.Basically the method of solution of our nonlinear problem just out-t-cichWeturnlined requires the solution of a sequence of linear problems.next, therefore, in sec. 12.2c to the derivation of the solution of theselinear problems,and afterwards,in sec.
12.2dof solutions of the linearweprove that anproblems converges toshallappropriate sequencethe desired solution of the nonlinear problem.12.2c.TheThesolution of a class of linear problemslinear problemswe haveinmindto solve, in accordance with=the above discussion, are problems for co(<p, \p]0(<p, y) -f- ir((p,function.asaThat is,iswhen F(q>) in (12.2.14)givenregardedshould satisfyall\p)u>of the conditions formulated above, except for theon we shall begin our iteration processwith a function o^ which has the period 2n/A in <p; F(q>), all subsequent iterates, and the solution itself, will have the same period.Since we expect the waves to be symmetrical about a crest or troughfree surface condition.
LaterWATER WAVES530(indeed, our existence proof will yield only waves with this property),that the origin is taken at such a point, and hence that atwe supposeany stage of theiteration process0)6(<p,= 6(<p) would be an odd func-while r(<p, 0)r((p) would be an even function of 9?, and9?,both would have the period 2n/L Thus F(q>) in (12.2.14) as definedby (12.2.12) should be taken for our purposes as an odd function of ywith the real period 2n/LIf we were to work at the outset with Fourier series, it follows that6 would be represented as a sine series, and F(<p) also. However, wewish later on to carry out an iteration process in which only continuous, and not necessarily differentiable, functions of q> are employed,and in which the existence of a certain continuous periodic functiontion ofof period 2;rc/A is first proved; this function will furnish theboundary values of the solution a)(q> ip).
(Afterwards, the question offt)(9?)9the existence of the normal derivative6^(99,0) in (12.2.14),and ofother derivatives, will be dealt with separately. ) In doing so, we shallhave occasion to approximate such continuous periodic functions byfinite Fourier series, or Fourier polynomials, a process justified by theWeierstrass approximation theorem which states that such a polynomial can always be constructed to yield a uniform approximationfor all values of<pand any arbitrary degree of approximation.Suppose, therefore, that F(<p) in (12.2.14) had been approximatedat some stage in the iteration procedure by a function g((p] in theform of the following(12.2.15)finite sine series:g(<p)= Jbv=lvsinand we seek the bounded harmonic function6(99, y) which satisfiesthe boundary condition (12.2.14) withg.
For this purpose wewrite the solution 6(<p, y>)Ste 00(99, \p) also as a finite Fourier sum:F=== J a e ** sin vhp.v=lv(12.2.16)0(q> 9 \p)vF=sum in (12.2.14), withg, leads to the followingfortheofdeterminationthecoefficients a vequationsInsertion of this:and thus to the conditionsvv7>-l)a,= 6,,.....LEVI-CIVITA'Sa t arbitrary,[(12.2.17)=aTHEORY531fcjfcvery important for the following to observe that the Fourier sinemust lack the first order term: otherwise, as we haveremarked above, our problem would have no solution: a term of theform &! sin T^p in F(q>) is a "resonance" term, the presence of whichItisseries for F(<p)would preclude the existence of the solution of the nonhomogeneousproblem. The unique solution forn= ^~v=2^( v0(<p.
\p)y)ise^ sin vty +ft-(12.2.18)0(<p,i^sinA<p,1)is assumed to be an oddis prescribed (cf. Chapter 3) andfunction of (p. The harmonic conjugate r(cp y>) of 0(<p, \p) is obtained byr^, with the resultintegrating the Cauchy-Riemann equation 6 Vonce a x9n(12.2.19)T(Q?,= V -w)fv2b-~ve^ cos vhy +a^cos Ay.1)A(v(A possible additive integration constant is set equal to /ero sincewhen ip ->oo.) Thus we would have for co(y, \p) under theassumed circumstances the expressionr ->= 2 ~~2 X(ve^-^V(12.2.20)co(o>,In other words,series of sines*V)ifisF(<p)which lacks1)given as in (12.2.15) by a finite Fourierits lowest order term, i.e. is such that2n/(12.2.21*F((p ) sinI*)jothen, aswefunctiona)(<p)(12.2.22)seeXydy=0,from (12.2.20) and the discussion precedingco((p,a>(<p)0)6((p)- 2i+ ii((p)-Ait,thegiven by-- e-<"**+ ia^""boundary values of an analytic function co(<p, ip) whichwould satisfy the boundary condition (12.2.14).
Evidently, o) wouldyields thealso satisfy all of the conditions a) to e) formulated above, if theamplitude a x of thefirstorder term of the Fourier seriesischosenWATER WAVES582small enough, except that the boundary condition b)a linear condition.Itisisreplacedbyclear that the insertion of a function a)(<p) as given by (12.2.22)determine a new function F(cp) in order to continue thein (12.2.12) toiteration process would not yield in general a function representableas a finite Fourier sum, but rather to one representable only by aseries.
However, we have already stated that we wish toour iteration scheme in the nonlinear problem within theoutcarryclass of continuous functions, which need not possess convergentFourier series. Nevertheless, the general scheme outlined above forFourierdetermining the successive iterations can still be used once it has beenextended in an appropriate way to the wider class of functions. Forthis purpose, and later purposes as well, it is convenient to introducethe terminology of functional analysis. Thus we speak of the linearvector space of elements which are complex-valued functionscontinuous for all <p and of period 2rc/A, such thatifi(<p)g(q>)<x.((p)a is an odd function and ft an even function of <p.
The scalars are thereal numbers. This space is made into a normed linear space by introducing as the distance from the origin to the "point" g the following+norm, written||g9||g=||||:||a+ift||- maxA/a 2+2]8- maxg||,9andbetween two elements or points g l5 g 2 the norm ofi.e.glg 2 ||. This space, which we shall call theas the distancetheir difference,,||complete, i.e. it has the property that every Cauchy seinthespace converges to an element in the space.* By a Cauquencechy sequence g n we mean a sequence such that || gmg n II -*spaceJB, iswhen m, n-> oo.
Since thenormisthemaximumfollows that a sequence g n whichof the absolutea Cauchy sequence isandhencehasacontinuousfunction as a limit.uniformly convergentWe remark also that the notion of distance thus introduced in ourspace has the usual properties required for the distance function in avalue ofg, itmetric space,i.e.,||and theg||the distance^ 0,and||positive definite:isg||is=implies g= 0,triangle inequalityll*i+foil^11gill+11foilholds.* We remark that acomplete linear normed space is called a Banach spacehowever, such properties of these spaces as are needed will be developed here.;THEORYLEVI-CIVITA'SWeintroduce next the subspaeeBlconsists of all real functions g(<p) givenlacking the term of first order:(12.2.23)g(<p)= J6v=2WithSrespect to this setas follows:(12.2.24)SinceipSg(<p)^we haveFrom Cauchy'sobtainedBvBanach space B whichby finite Fourier sums of sinesof ourb v real.sinvA<p,weof functionsl= Ji--define a transformation#** e- i9**,-norm*for the533of-oo<\p^isthen0.Sg the boundinequality the following inequality for||Sg\\:the last step resulting from (12.2.23) because of the orthogonalityof the functions sin vA,q>.
SinceitKfollows that a constant(though not ofA),(12.2.25)Thus S\\Sg\\iswhatisexistswhichisindependent of g and nsuch that<Kcalled atransforms each element ofB\\&\\,torgCB,.in B since itB with a norm bound-bounded transformationlinto an element ofled by a constant times the norm of the original element. Clearly, Stransforms a certain class of boundary data given in terms of the realan analytic function defined in the lower half plane.nextto extend the domain of definition of the transproceedformation S in such a way that it applies to a certain set of realfunctiong(<p) intoWe*itsBy the norm of a function of two variablesabsolute value.we meanthe least upperbound ofWATER WAVES584B which contains the set B i.e., to the set JB of continuous real functions g in B with vanishing first Fourier coefficients, that*= 0; this subspace B 2 isto functions such thatg((p) sin h<pdyfunctions in219is,jojalso complete, with the same norm.
The extension of the definition ofS is made in the following rather natural way: Take any function ginB 2 an d let g n be a sequence of functions inB^(i.e.,a set of realtri-order terms) which approximategonometric polynomials lackingis known from the WeierexistsThatsuchag uniformly.sequenceformthe sequence Sg n whichthenstrass approximation theorem. Wefirstispossible sinceSbecauseisSisapplicable to these functionsand observe thatobviously a linear transformation, and hence\\Sgm-Sg n^K\\gm\\~ gj|from (12.2.25) since gmg n is an clement of J5 X Thus the sequence->isabecause the functionsCauchy sequence, for gmSg ngng n are assumed to furnish uniform approximations to g; hence the sequence Sg n has a unique continuous limit function which we defineas Sg.
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