Amarnath An Elementary Course in Partial Di64256erential Equa tions Part A Uniqueness of solution for one dimensional wave equation with 64257nite length Theorem The solution of the following problem if it exists is unique tt xx xt 0 1 x 0 l x 0 l ID: 27962
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Hencevx0;vt08t�0;0xl:Thisispossibleonlyofv(x;t)=constant,sincev(x;0)=0;v0:Hencethetheorem.PartB:UniquenessofsolutionforonedimensionalheatequationwithnitelengthTheorem:Thesolutionofthefollowingproblem,ifitexists,isunique.ut�uxx=F(x;t);0xl;t&#x-277;0(3)u(x;0)=f(x);0xl;u(0;t)=u(l;t)=0;t0ProofTheaboveuniquenessresultforIBPofheatequationisequivalenttoshowingthatthefollowingIBPhasonlytrivialsolution,vt=vxx;0xl;t&#x-277;0(4)v(x;0)=0;0xl;v(0;t)=v(l;t)=0;t0Letv(x;t)beasolutionofproblem(4).Nowconsider,E(t)=1 2Z10v2(x;t)dx:ObservethatE(t)isadi erentiablefunctionoft,sincev(x;t)istwicedi eren-tiable.ThereforedE dt=1 Z10vvtdx;=Z10vvxxdx=vvxjl0�Zl0v2xdxSincev(0;t)=v(l;t)=0;wehavedE dt=�Zl0v2xdx0;i.e.Eisadecreasingfunctionoft.Also,bydenition,E(t)isnonnegativeandfromtheconditionv((x;0)=0wehaveE(0)=0:ThereforeE(t)08t�0)v(x;t)0in0xl;t0:Hencethetheorem.2