8 Separation of Variables in Other Coordinate Systems

Transcription

1 8 Sepaation of Vaiables in Othe Coodinate Systems Fo the method of sepaation of vaiables to succeed you need to be able to expess the poblem at hand in a coodinate system in which the physical boundaies of you poblem ae given by setting appopiate coodinates to constant values. Up to now we have consideed only poblems whee this was the case fo ectangula coodinates, but ectangula coodinates ae not always up to the job. 8. Steady-State Tempeatue in a Cicula Plate Let the disk D = (x, y) :x + y a ª model a thin heat conducting plate that is lateally insulated. To bette undestand the opening paagaph of Sec. 8, lets find the steady state tempeatue fo u t = α (u xx + u yy ) fo (x, y) inside D, t >, (HE) IBVP u (x, y, t) =f (x, y) fo (x, y) on D, t >, (BC) u (x, y, ) = g (x, y) fo (x, y) in D. (IC) Unde these conditions we expect the solution (tempeatue) u (x, y, t) u (x, y), a steady state tempeatue, as t. The steady state tempeatue u (x, y) is a solution to the bounday value poblem BVP uxx + u yy = fo (x, y) inside D, (LE) u (x, y) =f (x, y) fo (x, y) on D. (BC) That is, we need to solve Laplace s equation (LE), u =, fo a disk with Diichlet bounday conditions. If we seek sepaated solutions of the fom u (x, y) = X (x) Y (y) to this poblem disaste stikes. Ty it and see why! It should be appaent that, disaste o not, we should not be tying to solve this poblem in ectangula coodinates. We should expess all the equations in pola coodinates. This is easy in pinciple. To convet Laplace s equation to pola coodinates, you just use the elations x = cos θ and y = sin θ and the chain ule to tade in x-deivatives and y-deivatives fo -deivatives and θ-deivatives. You get (check it) in pola coodinates, and u = u xx + u yy = u + u + u θθ u = u xx + u yy + u zz = u + u + u θθ + u zz 39

2 in cylindical coodinates. In pola coodinates the bounday of the disk, D, has equation = a. So the steady state solution u = u (, θ), satisfies BVP u + u + u θθ = fo <a, θ π (LE) u (a, θ) =f (θ) fo θ π, (BC) whee f (θ) =f (a cos θ, a sin θ). Now, we seek sepaated solutions to Laplace s equation of the fom u (, θ) = R () Θ (θ) and ae lead to R + R λ R = fo <a, R () bounded, and Θ + λ Θ =, Θ () = Θ (π), Θ () = Θ (π), whee the sepaation constant is λ. Solving the Θ-EVPandthentheR-poblem leads to λ = λ n = n Θ = Θ n (θ) =c n cos nθ + d n sin nθ, c n and d n constants R = R n () = n fo n =,,,.... Thus, sepaated solutions to LE ae u n (, θ) = n (c n cos nθ + d n sin nθ) fo n =,,,.... Fo n =the nontivial sepaated solutions ae the nonzeo multiples of u (, θ) =. In the now standad way, we ty to satisfy the emaining inhomogeneous condition BC by supeposing the solutions to the homogeneous condition LE: u (, θ) = c + X (c n n cos nθ + d n n sin nθ), whee the coefficient of u (, θ) =is expessed as c / in anticipation of a Fouie seies expansion. The constants c n and d n must be chosen to satisfy u (a, θ) = a + X (c n a n cos nθ + d n a n sin nθ) =f (θ). Hence, c n a n and d n a n must be the Fouie coefficients of f (θ) : c n a n = a n = π d n a n = b n = π Z π Z π 4 f (θ)cosnθ dθ, f (θ)sinnθ dθ,

3 fo n =,,, 3,.... Thus, the solution to BVP is u (, θ) = a + X ³ ³ n ³ a n cos nθ + bn sin nθ. a a n 8. Steady-State Tempeatue in a Semiing This time we seek the steady-state tempeatue u = u (, θ) in a lateally insulated, themally homogeneous, heat conducting plate in the shape of a semiing. The plate is modeled as the egion in the fistquadantoutsidethediskwith adius and inside the disk with adius. Thesideoftheplatealongthey-axis is insulated, the cicula acs ae kept at tempeatue C, and the side of the plate along the x-axisisheldattempeatue C. y x The coesponding BVP fo u is u + u + u θθ = fo <<, <θ<π/ (LE) BVP u (,θ)=,u(a, θ) = <θ<π/, fo (BC) u θ (, π/) =, u(, ) =. Now, we seek sepaated solutions u (, θ) =R () Θ (θ) to Laplace s equation and the homogeneous bounday conditions. This leads to R-EVP R + R + λ R =, R () =, R() =, and Θ λ Θ =, Θ (π/) =, whee the sepaation constant is λ. The R-EVP has eigenvalue, eigenfunction pais λ = λ n and R = R n () =sin(λ n ln ) whee λ n = nπ ln. 4

4 With these choices fo λ, the Θ-poblem has solutions the nonzeo multiples of ³ π Θ = Θ n (θ) =cosh ³λ θ n. So nontivial sepaated solutions to the homogeneous equations in BVP ae ³ π u n (, θ) =R n () Θ n (θ) =sin(λ n ln )cosh ³λ θ n and thei nonzeo multiples. We now ty to satisfy the full BVP with u (, θ) = ³ π c n sin (λ n ln )cosh ³λ θ n () whee λ n = nπ/ ln. The constants c n must be chosen to satisfy u (, ) = µ λn π c n sin (λ n ln )cosh =. () So now what? We ae sunk unless we ecognize that the R-EVP is a SLEVP. To see this expess the R-EVP as R-EVP (R ) + λ R =, R () =, R() =. By the SLEVP facts we see that the eigenvalues λ n > as we aleady discoveed diectly. What is new is that the eigenfunctions R n () =sin(λ n ln ) ae othogonal on with weight function /. Consequently, Z sin (λ n ln )sin(λ m ln ) d =fo n 6= m and by diect calculation Z sin (λ n ln ) Z ln ³ d = sin nπ u du = ln. ln = ln Use these othogonality elations in () to find µ ln c n cosh λn π = c n = Z = sin (λ n ln ) d = ln ³ nπ u ln nπ cos = ln ³ 4 ( ) n+ + nπ cosh (λ n π/). Z ln ln nπ ³ nπ u sin ln du ³ ( ) n+ +, 4

5 Finally we can expess the solution () as ³ 4 ( ) n+ + ³ π u (, θ) = nπ cosh (λ n π/) cos (λ n ln )cosh ³λ θ n = 8 π ³ π (n ) cosh (λ n π/) cos (λ n ln )cosh ³λ θ n. 8.3 Radially Symmetic Tempeatue in a Cicula Plate We continue with the lateally insulated, homogeneous, heat-conducting cicula plate D of Sec. 8.. If the initial conditions and bounday conditions ae adially symmetic functions, then it is natual to expect that the tempeatue in the plate also will be adially symmetic. We seek such adially symmetic tempeatue distibutions hee and use pola coodinates fom the stat. So we seek the solution u = u (, t) to IBVP u t = α u + u fo inside D, t >, (HE) u (a, t) =T a fo on D, t >, (BC) u (, ) = g () fo in D (IC) whee T a is the given constant tempeatue on = a. This IBVP has an obvious steady state solution, the constant T a. Consequently, u solves IBVP iff v = u T a solves IBVP-v v t = α v + v fo inside D, t >, (HE) v (a, t) = fo on D, t >, (BC) v (, ) = h () fo in D (IC) whee h () =g () T a is known. We can solve IBVP-v by sepaation of vaiables. Nontivial sepaated solutions v = T (t) R () to the homogeneous equations HE and BC must satisfy T + α λ T = and R R-EVP + R + λ R =, R () bounded, R (a) =. At this point, the T -poblem has solutions the nonzeo multiples of T (t) =exp α λ t. The R-diffeential equation is Bessel s equation of ode with paamete λ. (Stay tuned.) A Vey Shot Couse in Bessel Functions 43

6 The Bessel s equation of ode n is x y (x)+xy (x)+ x n y (x) =, whee n =,,, 3,... (See B & D Sec. 5.8.) Fact. BE has two linealy independent solutions named J n (x) and Y n (x); the fist solution is smooth and behaves oughly like a damped cosine while the second has a logaithmic singulaity at x =;hence, Y n (x) is unbounded as x. Fact. J n (x) satisfies J n (x) = k= ( ) k ³ x n+k = k!(n + k)! π Z π cos (nθ x sin θ) dθ, and, fo lage x J n (x) ³x πx cos nπ π. 4 Fact 3. J n (x) is an even function when n isevenandanoddfunctionwhen n is odd. In eithe case, it has an infinite numbe of positive zeos; say, x n <x n <x n3 <. Its othe eal zeos ae the negatives of its positive zeos. (Why?) Fo some numeical data see The Bessel s equation of ode n with paamete λ is whee n =,,, 3,.... x y (x)+xy (x)+ λ x n y (x) =, Fact 4. Two linealy independent solutions of BE of ode n with paamete λ ae J n (λx) and Y n (λx). Fact 5. Fix n and let λ np = x np /a. Then J n (λ np x) and J n (λ nq x) with p 6= q ae othogonal on x a with weight function x. Fact 6. Z a (J n (λ np x)) xdx= a (J n+ (x np )) 44

7 Retun to the R-EVP. Multiply the DE by to see it is Bessel s equation of ode with paamete λ. So the DE has geneal solution R = AJ (λx)+by (λx). Since R () must be bounded, we must choose B =. Then R = AJ (λx). To satisfy R (a) =and get a nontivial solution we must choose λ so that J (λa) =. By Fact 3 thee ae an infinite sequence of eal positive solutions λ = λ n = x n a whee x <x <x 3 < ae the positive zeos of J (x). These λ n ae the eigenvalues of the R-EVP and the coesponding eigenfunctions ae the nonzeo multiples of R n () =J (λ n ). Consequently, HE and BC in IBVP-v have nontivial sepaated solutions v n (, t) =T n (t) R n () =exp λ nα t J (λ n ) fo n =,, 3,.... We seek to satisfy the full IBVP-v by supeposition with v (, t) = c n exp λ nα t J (λ n ). This seies will solve IBVP-v if we can choose the c n so that v (, ) = UseofFacts5and6foJ yields c n = c n J (λ n )=h (). a (J (x n )) Z a h () J (λ n ) d. With these choices fo c n the oiginal IBVP has solution u (, t) =T + c n exp λ nα t J (λ n ). 8.4 Vibations of a Dum As befoe D = (x, y) :x + y a ª 45

8 is the disk with adius a. This time we use it to model a thin membane a dum head. The small tansvese vibations u (a function of space and time) of themembanecanbemodeledby IBVP u tt = c u inside D, t >, (WE) u = on D, and fo t>, (BC) u = f in D at t =, (IC) u t = g whee f and g ae given functions of position. To save a little effot we assume the membane is initially at est so that g =. Of couse, with this geomety, we should locate positions by thei pola coodinates and expess ou poblem as u t = c u + u + u θθ fo (, θ) inside D, t >, (WE) IBVP u (a, θ, t) = fo all θ and t>, (BC) u (, θ, ) = f (, θ) u t (, θ, ) = fo (, θ) in D. (IC) Nontivial sepaated solutions u = T (t) R () Θ (θ) to the homogeneous equations, WE, BC, and IC must satisfy T + λ c T =, T () =, Θ + ν Θ =, Θ π-peiodic, and R + R + λ ν R =, R () bounded, R (a) =. At this point, the T -poblem has solutions the nonzeo multiples of The Θ-poblem has T (t) =cosλct. eigenvalues ν = ν n = n with coesponding linealy independent eigenfunctions cos nθ and sin nθ, fo n =,,, 3,....(Whenn =, sin nθ is omitted.) Thus, T = T n (t) =cosnct and the R-poblem educes to R + R + λ n R =, R () bounded, R (a) =, 46

9 which is Bessel s equation of ode n with paamete λ. Its geneal solution is R = AJ n (λ)+by n (λ). The fist bounday condition equies B =;so, R = AJ n (λ) and the second bounday bounday condition equies J n (λa) =. That is, λa must be a zeo of Bessel s function of ode n; we conclude that λ = λ nm = x n a, x n a, x n3 a,..., whee x n <x n <x n3 < ae the zeos of J n (x). Thus, the R-poblem has nontivial solutions R = R nm () =J n (λ nm ). Summay: WE, BC, and IC in the dum IBVP have sepaated solutions u nm (, θ) =cos(λ nm ct)(a nm cos nθ + b nm sin nθ) J n (λ nm ) fo n =,,, 3,... and m =,, 3,.... Finally we ty to satisfy IC by supeposing the sepaated solutions. Evidently, u (, θ, t) = will satisfy IC if u (, θ, ) = n=,m= n=,m= cos (λ nm ct)(a nm cos nθ + b nm sin nθ) J n (λ nm ) (a nm cos nθ + b nm sin nθ) J n (λ nm )=g (, θ). To avoid a silly mistake we split of the tems with n =and wite a m J (λ m )+ (a nm cos nθ + b nm sin nθ) J n (λ nm )=g (, θ), m= n,m= Ã! Ã! X a m J (λ m ) + a nm J n (λ nm ) cos nθ + m= m= Ã X! b nm J n (λ nm ) sin nθ = g (, θ). m= The expessions in ound paentheses must be the Fouie coefficients of g (, θ) 47

10 egaded as a function of θ with teated as a paamete. So λ m a m J (λ m ) = Z π g (, θ) dθ, π m= λ nm ca nm J n (λ nm ) = π m= λ nm cb nm J n (λ nm ) = π m= Z π Z π g (, θ)cosnθ dθ, g (, θ)sinnθ dθ. Multiply the fist equation by J (λ p ), the second and thid by J n (λ np ), integate fom to a, and use the othogonality popeties of the Bessel functions to get a a p (J (x np )) = π a a np (J n+ (x np )) = π a b np (J n+ (x np )) = π Z a Z π Z a Z π Z a Z π J (λ p ) g (, θ) dθd, J n (λ np ) g (, θ)cosnθ dθ, J n (λ np ) g (, θ)sinnθ dθ. These elations detemine the coefficients in the solution to the full IBVP: u (, θ, t) = n=,m= cos (λ nm ct)(a nm cos nθ + b nm sin nθ) J n (λ nm ). 48