TMA4329 Intro til vitensk. beregn. V2017

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1 Norges eknisk naurvienskapelige universie Insiu for Maemaiske Fag TMA439 Inro il viensk. beregn. V7 ving 6 [S]=T. Sauer, Numerical Analsis, Second Inernaional Ediion, Pearson, 4 Teorioppgaver Oppgave 6..3, (e), s. 9, [S] : Separaion of variables leads o e equaion d/ = d, or log() = 3 /3 + C were C is an arbirar inegraion consan. Tus () = ep( 3 /3 + C), were C = from e equaion () =. (e): Here we ge d = d, or 3 /3 = + C, () = (3 + 3C) /3. Finall () = and erefore 3C =. Oppgave 6..4,, s. 9, [S] : Te general soluion o e omogeneous ssem = is () = C ep(), were C is an arbirar consan. Furer, a paricular soluion o e ssem = + can be searc in e form () = a + b, from wic i follows a a = b =. Tus we can pu () = () + (). Te consan C is en deermined from e equaion () = and us () = + ep(). : Similarl o, () = C ep( ), () =, and as a resul () = + ep( ). 3 Oppgave 6..9, s. 9, [S] : Here f(, ) = - independen from and us f is uniforml Lipsciz coninuous wi respec o on [a, b] (, + ) wi e consan L =. Teorem 6. en guaranees e eisence and uniqueness of soluions on an arbirar inerval., (c): Here f(, ) = ±, wic is a linear funcion wi e slope ±. In eier case f is uniforml Lipsciz coninuous wi respec o wi consan L = on 5. mars 7 Side av 6

2 ving 6 [a, b] (, + ), regardless of a, b. Teorem 6. en guaranees e eisence and uniqueness of soluions on an arbirar inerval. (d): Here f(, ) = 3. Tis funcion is independen from and is coninuousl differeniable wi respec o. Tus i is uniforml Lipsciz coninuous on an finie square S = [a, b] [α, β] wi e Lipsciz consan L = ma [α,β] df/d = ma [α,β] 3. Noe a e Lipsciz consan depends and grows wi e inerval [α, β]. Tus Teorem 6. guaranees e eisence and uniqueness of soluions on some sub-inerval [a, c], a < c b. As a resul we canno guaranee eisence of soluions on e wole inerval [, ], onl on a sub-inerval. 4 Oppgave 6.., s. 9, [S] (c) (d) 5 Oppgave 6.., s. 9, [S] : () = / + (). Teorem 6.3 is clearl verified wi L =. (b,c): () = () ep(±). Teorem 6.3 is verified wi L =. In fac, in (c) an non-negaive L or even L is sufficien, bu is is of course is no a valid value for e Lipsciz consan. 5. mars 7 Side av 6

3 ving 6 (d): If () = we can use e soluion (). For () = we can use separaion of variables o find a () = ( + C) /, were = () = C /. Noe a e difference beween e wo soluions decreases wi ime, and erefore e esimae of Teorem 6.3 olds wi an L in is case. 6 Oppgave 6.., s. 3, [S] : _i _i.e+.e+.5e 3.5e 5.e.466e 7.5e 3.533e.e e err =.34 : _i _i.e+.e+.5e 3.5e 5.e.356e 7.5e.6837e.e e err =.46 7 Oppgave 6.3.3, s. 3, [S] We inroduce a new variable z = so a z =. : ( ) ( ) z = z : ( ) ( ) z = z z 5. mars 7 Side 3 av 6

4 ving 6 (c): ( ) ( ) z = z z + 8 Consider e iniial value problem () = λ(), > () =, were λ C. Is soluion is () = ep(λ). Suppose a we use a numerical meod (suc as e.g. forward Euler or eplici rapezoid) o solve is problem saring from a poin w =. Te sabili region for e meod is a se of poins z = λ in e comple plane, suc a e numerical soluion (w, w,... ) sas bounded (i.e., C > : i, w i C). Find e sabili region for forward Euler meod; eplici Trapezoid meod. : In is case w k = w k + f( k, w k ) = ( + λ)w k = ( + λ) k w. Tus w k sas bounded iff + λ. Ta is, e sabili region for e forward Euler meod is a circle in e comple plane of radius around e poin. Noe: is implies, in paricular, a if λ = iω is purel imaginar en ere is no > suc a λ is in e sabili region. Tus wereas ep(iω) = cos(ω) + i sin(ω) is oscillaor (bounded) in is case, e Euler s meod will resul in an unbounded soluion (long erm beaviour) regardless of ow small we cose. : Now we ave w k = w k +/[f( k, w k )+f( k +, w k +f( k, w k ))]+ w k +/[λw k +λ(w k +λw k )][+λ+.5(λ) ]w k = [+λ+.5(λ) ] k w. Tus z = λ is in e sabili region of e eplici rapezoid meod if and onl if + z +.5z. For an arbirar purel imaginar number λ = iω and an > we ave p(iω) =.5ω + iω = [(.5ω ) + ω ] / = [ + ω 4 4 /4] / > as in e case of forward Euler, wi e same implicaions. However, for small > we can use a firs order Talor series epansion o ge [ + ω 4 4 /4] / + ω 4 4 /8 wic is muc closer o an + iω = ( + ω ) / + ω / (forward Euler). Here is e plo of sabili regions: imag imag real real forward Euler eplici Trapezoid 5. mars 7 Side 4 av 6

5 ving 6 Compueroppgaver 9 Oppgave 6..5, s. 93, [S] See oppgave_6 5.m Repea e previous eercise, bu use e eplici rapezoid meod insead. See oppgave_.m Oppgave 6.3., s. 34, [S]. Use e iniial condiions specified in e book bu differen masses: m = m 3 =.3, m =.3. Use e eplici Trapezoid meod. See ree_bod_problem.m 5. mars 7 Side 5 av 6

6 ving 6.5 m m m m () =. () =. Oppgave 6.3., s. 34, [S]. Use e eplici Trapezoid meod. See ree_bod_problem.m 4 m m m m mars 7 Side 6 av 6