A Note on the Axisymmetric Diffusion equation
aa r X i v : . [ m a t h . C A ] J un A NOTE ON THE AXISYMMETRIC DIFFUSION EQUATION
ALEXANDER E PATKOWSKI
Abstract.
We consider the explicit solution to the axisymmetric diffusionequation. We recast the solution in the form of a Mellin inversion formula, andoutline a method to compute a formula for u ( r, t ) as a series using the Cauchyresidue theorem. As a consequence, we are able to represent the solution tothe axisymmetric diffusion equation as rapidly converging series. Keywords:
Axisymmetric Diffusion equation; Bessel functions; Mellin transforms Introduction and Main results
The axisymmetric diffusion equation is [2, pg.61](1.1) κ ∇ u ≡ κ r ∂∂r (cid:18) r ∂u∂r (cid:19) = κ (cid:18) u rr + 1 r u r (cid:19) = ∂u∂t , where t > , r ∈ (0 , ∞ ) , u ( r,
0) = g ( r ) , and positive diffusitivity constant κ. TheHankel transform of a function f ( x ) is defined as [2, pg.58, eq.(1.10.1)] H ( f ( y ))( x ) := Z ∞ yJ ( xy ) f ( y ) dy. We may temporarily drop the integrating variable in denoting integral transformsaccording to when the context is appropriate throughout. The known explicitsolution is obtained by taking Hankel transform of (1.1), which gives(1.2) ∂∂t H ( u ( r, t ))( x ) + x κ H ( u ( r, t ))( x ) = 0 , with initial condition H ( u ( r, x ) = H ( g ( r )) . Applying the inverse Hankel trans-form H − to (1.2) gives the explicit solution [2, pg.62, eq.(1.10.25)](1.3) u ( r, t ) = 12 κt e − r / (4 κt ) Z ∞ yg ( y ) I ( yr κt ) e − y / (4 κt ) dy, where the modified Bessel function of the first kind is given by I v ( x ) = X n ≥ n !Γ( v + n + 1) (cid:16) x (cid:17) n + v . Some simple examples include the bell-shaped temperature profile g ( r ) = e − cr , orthe uniform temperature profile g ( r ) = 1 on (0 , . In both these instances it is asimple task to appeal to the tables.The purpose of this note is to provide further analysis on (1.3) by means of Mellin in-version. In applying methods from [4] we can better understand u ( r, t ) by providinga method to obtain an infinite series representation involving Laguerre polynomialsor a hypergeometric function. For a general overview applying Mellin transformsto evaluating integrals involving Bessel functions see [6, pg.196].Recall [4] the Mellin transform is given by M ( g )( s ) := Z ∞ y s − g ( y ) dy. Parseval’s identity is [4, pg.83, eq.(3.1.11)](1.4) Z ∞ k ( y ) g ( y ) dy = 12 πi Z ( c ) M ( k )( s ) M ( g )(1 − s ) ds Recall from [3, pg.709, eq.(6.643), x → x and µ = s )that(1.5) Z ∞ y s e − αy I v (2 βy ) dy = Γ( s + v + ) e β / (2 α ) v + 1) β α − s/ M − s/ ,v ( β α ) , valid for ℜ ( s + v + ) > . Here M µ,v ( x ) is the Whittaker hypergeometric function [3, pg.1024](1.6) M µ,v ( x ) = x v + e − x/ F ( v − µ + 12 ; 2 v + 1; x ) , and F ( a ; b ; x ) is the confluent hypergeometric function. Theorem 1.1. If M ( g )(1 − s ) is analytic in a subset S of the region { s ∈ C : ℜ ( s ) > − } , then u ( x, t ) = 1 r e − ( r − r ) / (4 κt ) πi Z ( c ) Γ( s κt ) s/ M − s/ , ( r κt ) M ( g )(1 − s ) ds,c ∈ S ∩ { s ∈ C : ℜ ( s ) > − } . Proof.
We choose the k ( y ) to be the integrand in (1.5) with v = 0 , α = κt , β = r κt , and apply (1.4). (cid:3) NOTE ON THE AXISYMMETRIC DIFFUSION EQUATION 3
Some relevant notes are in order to apply Theorem 1.1. First, Theorem 1.1 requiresthat M ( g )( s ) is analytic in the region { s ∈ C : ℜ ( s ) < } . It is known that M µ,v ( x )only has simple poles for fixed µ and x, at v = − ( k + 1) , k ∈ N . By [3, pg.1026,9.228], M µ,v ( x ) ∼ √ π Γ(2 v + 1) µ − v − x / cos(2 √ µx − vπ − π , as | µ | → ∞ , and further we have the functional relationship [3, pg.1026, eq.(9.231, x − − v M µ,v ( x ) = ( − x ) − − v M − µ,v ( − x ) . In the model with u ( r,
0) = J ( ar ) , the Bessel function of the first kind, we mayproceed in the following way. Note that for − v < ℜ ( s ) < , [4, pg. 407](1.7) M ( J v ( ay ))( s ) = 2 s − Γ( v + s )Γ(1 + v − s ) a − s . We set v = 0 and insert (1.7) into Theorem 1.1 to obtain for − < ℜ ( s ) = c < , (1.8) u ( r, t ) = 1 r e − ( r − r ) / (4 κt ) πi Z ( c ) (4 κt ) s/ M − s/ , ( r κt )2 − s Γ( 1 − s a s − ds. It is known that(1.9) F ( a, x ) = e x L a − ( − x ) , where L n ( x ) is the Laguerre polynomial [3]. This can be seen by using [3, pg.1001] L a ( x ) = F ( − a ; 1; x ) , together with Kummer’s [1, pg.509] F ( a ; b ; x ) = e x F (1 − a ; b ; − x ) with b = 1 . Now (1.6) with (1.8) leads to(1.10) u ( r, t ) = e − r / (4 κt ) √ κt πi Z ( c ) (4 κt ) s/ F ( s r κt )2 − s Γ( 1 − s a s − ds = e − r / (4 κt ) √ κt πi Z (1 − c ) (4 κt ) (1 − s ) / F (1 − s r κt )2 s − Γ( s a − s ds. Here we made the change of variable s → − s. This integral has simple poles at s = 0 , and the negative even integers s = − n. Computing these residues and using(1.9) gives u ( r, t ) = X n ≥ L n ( − r κt ) n ! (cid:0) − a κt (cid:1) n = e − a κt J ( ar ) . Here we have applied the α = 0 case of [5, pg.102, Theorem 5.1, eq.(5.1.16)] X n ≥ L ( α ) n ( x )Γ( n + α + 1) w n = e w ( xw ) − α/ J α (2 √ xw ) . ALEXANDER E PATKOWSKI
Next we consider an example of Theorem 1.1 with a function for which it is difficultto evaluate (1.3), and is apparently new.
Theorem 1.2.
The solution of (1.1) with u ( r,
0) = J ( ar ) , is given by u ( r, t ) = 12 X n ≥ (2 n )!( n !) L n ( − r κt )( − a κt ) n . Proof.
First we write down [4, pg.407](1.11) M ( J v ( ay ))( s ) = 2 s − Γ( s + v )Γ(1 − s )Γ (1 − s )Γ(1 + v − s ) a − s . valid for −ℜ ( v ) < ℜ ( s ) < . We set v = 0 in (1.11), and insert it into Theorem 1.1to find for 0 < c < ,u ( r, t ) = 1 r e − r / (4 κt ) πi Z ( c ) (4 κt ) s/ M − s/ , ( r κt ) 2 − s Γ( − s )Γ( s )Γ ( + s ) a s − ds = e − r / (4 κt ) √ κt πi Z (1 − c ) (4 κt ) (1 − s ) / F (1 − s r κt ) 2 s − Γ( s )Γ(1 − s )Γ (1 − s ) a − s ds. The resulting integral has simple poles s = − n for each integer n ≥ . Therefore,computing the residues at these poles gives, by Cauchy’s residue theorem and (1.9), u ( r, t ) = 12 X n ≥ (2 n )!( n !) L n ( − r κt )( − a κt ) n . (cid:3) It is interesting to note that taking the limit r → r → X n ≥ (2 n )!( n !) L n ( − r κt )( − a κt ) n = e − a κt I ( a κt ) , by means of [3, pg.1024, eq.(9.215), , p = 0 , z = ix ]. Next we consider an initialcondition involving the modified Bessel function of the second kind K v ( x ) , whichhas the general relationship [3] K v ( x ) = π ( I − v ( x ) − I v ( x ))2 sin( πv ) . Theorem 1.3.
The solution to (1.1), with u ( r,
0) = I v ( ar ) K v ( ar ) , is given by u ( r, t ) = e − r / (4 κt ) (4 κta ) v √ π X n ≥ F (1+ v + n ; 1; r κt ) Γ(1 + v + n )Γ( − v − n )Γ( + n + v ) n !Γ(2 v + 1 + n ) ( − a κt ) n + e − r / (4 κt ) √ π X n ≥ F (1 + n ; 1; r κt ) Γ( v − n )Γ( + n )Γ( v + 1 + n ) ( − a κt ) n . NOTE ON THE AXISYMMETRIC DIFFUSION EQUATION 5 provided that v is not an integer or equal to . Proof.
From [6, pg.199, eq.(7.10.8)] with 0 < ℜ ( s ) < , (1.12) M ( I v ( ay ) K v ( ay ))( s ) = Γ( s + v )Γ( − s )Γ( s )4 √ π Γ( v + 1 − s ) a − s . Setting v = 0 in (1.12) and applying Theorem 1.1, we have that u ( r, t ) is equal to1 r √ π e − r / (4 κt ) πi Z ( c ) (4 κt ) s/ Γ( 12 + s M − s/ , ( r κt ) Γ( (1 − s ) + v )Γ( − s )Γ( s )Γ( v + + s ) a s − ds = e − r / (4 κt ) √ π √ κt πi Z (1 − c ) (4 κt ) (1 − s ) / F (1 − s r κt ) Γ(1 − s )Γ( s + v )Γ( s )Γ( − s )Γ( v + 1 − s ) a − s ds. Now we see that if v = 0 then the gamma functions would have a pole of order twoat the negative even integers s = − n, which we want to avoid due to the lengthyresulting formula. Hence we restrict v to be a non-integer and v = 0 , and the polesat s = − n − v, and s = − n are simple. For the poles at s = − n − v, we havethe residue e − r / (4 κt ) (4 κta ) v √ π X n ≥ F (1+ v + n ; 1; r κt ) Γ(1 + v + n )Γ( − v − n )Γ( + n + v ) n !Γ(2 v + 1 + n ) ( − a κt ) n , and for the poles at s = − n, we have the residue e − r / (4 κt ) √ π X n ≥ F (1 + n ; 1; r κt ) Γ( v − n )Γ( + n )Γ( v + 1 + n ) ( − a κt ) n . (cid:3) A nice consequence of our series representations of u ( r, t ) is that they are rapidlyconverging, and so should be of great interest for numerical calculations. From [pg.1003, eq.(8.978), , α = 0], we have the asymptotic expansion for the Laguerrepolynomial(1.13) L n ( x ) = e x/ √ π ( xn ) − / cos(2 √ nx − π O ( n − / ) , as n → ∞ , uniformly in x > . In conjunction with our series involving Laguerrepolynomials, (1.13) may be used to obtain approximations to u ( r, t ) . ALEXANDER E PATKOWSKI Some related observations
We mention a method of evaluating (1.3) when g ( y ) = h ( y ) log( y ) for a suitablefunction h ( y ) . It is known [3, pg.919, eq.(8.447] that(2.1) I ( x ) log( x − K ( x ) + X n ≥ x n n ( n !) ψ ( n + 1) , where ψ ( x ) is the digamma function [3]. The formula (2.1) appears to providean effective way of computing special cases of (1.3). We provide an outline of amethod. Theorem 2.1.
Let h ( y ) be a suitable function chosen so the series converges. Thesolution to (1.1) with initial condition u ( r,
0) = h ( r ) log( r ) , satisfies u ( r, t ) = 12 κt e − r / (4 κt ) (cid:18) log( 4 κtr ) Z ( h ) − Z ∞ yh ( y ) e − y / (4 κt ) K ( yr κt ) dy + X n ≥ ψ ( n + 1)2 n ( n !) (cid:16) r κt (cid:17) n Z n +1 ( h ) (cid:19) , where Z s ( h ) := M ( yh ( y ) e − y / (4 κt ) )( s ) = Z ∞ h ( y ) y s e − y / (4 κt ) dy. Proof.
Note that (2.1) implies I ( yr κt ) log( y ) = log( 4 κtr ) − K ( yr κt ) + X n ≥ ψ ( k + 1)2 k ( k !) (cid:16) yr κt (cid:17) k . Hence, Z ∞ yh ( y ) I ( yr κt ) log( y ) e − y / (4 κt ) dy = log( 4 κtr ) Z ∞ yh ( y ) e − y / (4 κt ) dy − Z ∞ yh ( y ) e − y / (4 κt ) K ( yr κt ) dy + X n ≥ ψ ( k + 1)2 k ( k !) (cid:16) r κt (cid:17) k Z k ( h ( y )) , provided yh ( y ) log( y ) satisfies certain growth conditions. In particular by [9, pg.920] K ( t ) = O ( e − t / √ t ) , when t → ∞ in | arg( t ) | < π , and so we require the very mildnecessary (but not sufficient) condition that for a positive constant c , and any t > , (cid:12)(cid:12) yh ( y ) (cid:12)(cid:12) < c e y / (4 κt ) , by the first integral on the right side. (cid:3) In closing we mention it is possible to utilize many of our results outside of theirrange of convergence as asymptotic formulas.
NOTE ON THE AXISYMMETRIC DIFFUSION EQUATION 7
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