Solution of 1d/2d Advection-Diffusion Equation Using the Method of Inverse Differential Operators (MIDO) Robert Kragler Weingarten University of Applied Sciences P.O.Box 1261 D-88241 Weingarten, Germany
[email protected] http://portal.hs-weingarten.de/web/kragler/mathematica The Method of Inverse Differential Operators (MIDO) is applied to the linear advectiondiffusion equation which is a 2nd order PDEs with homogeneous Dirichlet boundary conditions (BC) and initial value problem (IVP). By means of the transformation u t, x, ... v t, x, ... 1 t 2 x ... applied to the PDE the gradient term of the advectiondiffusion equation vanishes and the problem reduces to the known heat equation.
Advection-Diffusion Equation The advection-diffusion equation [1] is a combination of a diffusion equation and advection/convection equation; it describes phenomena where physical quantities are transferred inside a systemdue to two processes : diffusion and advection/diffusion. Depending on the context, the same equation can be either called advection-diffusion equation, drift-diffusion equation or scalar transport equation. The general equation in 3 dimensions is of the following form t u Dv u c u R where u is the (dependent) variable of interest (i.e. temperature for heat transfer or specific concentration for mass transfer). Dv is the diffusion coefficient (i.e. thermal diffusivity for heat transport or mass diffusivity for particle motion). c is the (average) velocity of the moving quantity. R describes sources/sinks of the quantity u, e.g. for heat transport, R > 0 is a source for thermal energy. is the gradient and · is the divergence. Hence, the term Dv u describes diffusion; if u denotes temperature distribution and has a local maximum, then the temperature local maximum will diffuse out. The net diffusion is proportional to the Laplacian of u. The second term c u describes advection (or convection). If u is a concentration then concentration at a given location can change due to the flow. The third term R describes the creation or destruction of the quantity u. In general, R may be a function of u and other parameters. Common simplifications are that the diffusion coefficient Dv = const , there are no sources/sinks, i.e. R=0, and the velocity field c describes, for example, an incomprssible flow, thus has zero divergence. Then, the advection-diffusion equation simplifies to t u Dv 2 u c u . In this form, the advection-diffusion equation combines both parabolic and hyperbolic PDEs. Subsequently, the advection-diffusion equation is even further simplified to the 1d or 2d case so that it will assume the form t u t, x
Dv x,x u t, x c0 x u t, x
or
182
t u t, x, y
Dv x,x y,y u t, x, y c0 x y u t, x, y .
According to the paper of A. Mojtabi & M.O. Deville [2] a time-dependent 1d linear advectiondiffusion equation with homogeneous Dirichlet boundary conditions and an initial sine function is solved analytically by separation of variables in the framework of MIDO too. In [1] and [2] the linear advection-diffusion equation t u c0 x u Dv x,x u (-1 < x < 1, t ]0,T] ) with homogeneous Dirichlet boundary conditions is treated. As shown therein for high Reynolds numbers Re
c0 L Dv
2 Dv
c0 1, L 2
for
(hence Dv 2 Re ) advection is
dominating diffusion but the boundary conditions complicates the solution because boundary layers develop and influence the flow dynamics. No-slip wall conditions impede the use of periodic Fourier representation. Although the advection-diffusion equation under investigation is linear it is difficult to find a closed form analytical solution in the literature; the paper of Guérrero et al. [3] uses a 'change of (dependent) variable' (i.e. introducing t x… 1 2 u t, x, … v t, x, … ) to reduce the problem posed to the well-known heat equation for v(t, x, …) which is solved. The final solution u(t,x, …) wanted must only be multiplied by the exponential factor 1 t2 x… . Below we follow the treatment in the paper [2] which obtains a closed form analytical solution of the linear advection-diffusion problem with homogeneous boundary conditions (BVC) and a smooth initial value condition (IVC) like a sine function. However, our approach generalizes this procedure; the main function (which is included in the package DESolve0.m) transformAdvDiff2HeatEqn is able to transform the advectiondiffusion equation and the corresponding function u(t, x, …) with up to three arbitrary spatial coordinates into the (homogeneous) heat equations, see 2 t c0 x Dv x 2 with u(t, x)
2 trans t Dv 2x with v(t,x) and 1 t 2 x
c2 0
t
4 Dv
c0
x
2 Dv
3 t c2 x c3 y D2 x 2 D3 y 2 with u(t, x, y) 3 trans
t D2 2x D3
2y
1 t 2 x3 y
with v(t,x, y) and
c2 D2 c2 D3 3 2
t
4 D2 D3
c2
x
2 D2
c3
y
2 D3
4 t c0 Dv 2 2 2 with u(t, , , )
4 trans t Dv 2 2 2 with v(t, , , ) and 1 t 2 3 4
3 c2 0
t
4 Dv
c0
2 Dv
c0
2 Dv
c0
2 Dv
The 1d or 2d cases are already covered by the procedure BIVProblem (with parameter BC=1 or BC=7) but specific procedures homogeneousBVC1a and homogeneousBVC2a will be used which are tailored for the advection-diffusion problem.
183
Procedures used for the Advection-Diffusion Equation The main procedures being used for the evaluation of the advection-diffusion equation are : ? advectionDiffusionEqnSolution
advectionDiffusionEqnSolution ,BC,xT,xL,F,,onoff,opts_:Identity calculates the solution u t,x,… 1 t2 x… v t,x,… of the homogeneous advectiondiffusion equation t c0 x Dv 2x … with given Dirichlet boundary conditions BC1 or 7 by transforming this PDE into the homogeneous heat equation t Dv 2x … for 1 or 2 spatial dimensions which could instead of x,y
be , etc. . The parameters xT T0 ,T1 and xL L0 ,L1 Tx ,Ty and xyL L0 ,L1 ,L2
for BC1 or xyT
for BC7 describe the initial temperatures
at the boundary in x respectively y direction. F f x or f x,y defines the initial temperature distribution u t0,x f x or u t0,x,y f x,y . Parameter specifies the order of the double Fourier expansion series for the approximate solution uk t,x,… . With the parameter onoff On Off additional printed information can be switched on or off; opts with default Identity is an optional parameter default is Identity which allows an additional simplification Simplify, FullSimplify of the final solution uk . Further simplification can be achieved with the global rule $rule see e.g. exp2hyp .
and the procedure ? transformAdvDiff2HeatEqn
transformAdvDiff2HeatEqn 0,onoff transforms the advectiondiffusion equation in 1,2 or 3 spatial dimensions by means of u t,x,… 1 t2 x… v t,x,… into the homogeneous heat equation in 1,2 or 3 spatial dimensions . The parameters 1 ,2 ,… are determined such that the gradient terms c2 x c3 y … i.e. first order derivatives with respect to coordinates x,y,… are eliminated so that the heat equation for v t,x,… results. Parameter ' 0' is the differential polynomial t c0 x Dv 2x … for
the advectiondiffusion equation; by means of the parameter 'onoff' On Off additional printed information can be switched on or off. A list Trans,expFactTrans,vars is returned where ' Trans' is the transformed differential polynomial, 'expFactTrans' the transformed exponential factor 1 t2 x… . 1 1 ,2 2 ,… and 'vars' the variables t,x,… extracted from 0.
which transforms the advection-diffusion equation into the well-known heat equation; an exponential factor 1 t 2 x … compensates the elimination of the gradient term(s) c0 x etc. The coefficients i are determined by the parameters in the differential polynomial AdvDiff . Examples for 1, 2 and 3 spatial dimensions are shown below :
184
2 t c0 x Dv x 2 ; 1d case : variable x t,expF,var transformAdvDiff2HeatEqn 2,"On"
0 t c0 x Dv 2x
t Dv 2x ,
c2 t
c x
v
v
40D 20D
trans t Dv 2x
with 1 t 2 x …
c2 0t 4 Dv
c0 x 2 Dv
, t, x
3 t c0 x y
Dv 2x 2y ;
2d case : variables x, y
t,expF,var transformAdvDiff2HeatEqn 3,"On"
31 t c2 x D2 x 2 c3 y D3 y 2 ; t,expF,var transformAdvDiff2HeatEqn 31,"Off"
4 t c0 Dv
3d case : variables ,,
2
2 2
;
t,expF,var transformAdvDiff2HeatEqn 4,"Off"
41 t c1 D1 2 c2 D2 2 c3 D3 2 ; t,expF,var transformAdvDiff2HeatEqn 41,"Off"
t
D1 2
D2 2
D3 2 ,
c23 D1 D2 c22 D1 D3 c21 D2 D3 t 4 D1 D2 D3
c
c
c
1
2
3
21D 22D 23D
, t, , ,
The procedure for the (homogeneous) heat equation ? homogeneousHeatEqnSolution1
provides solution for 1 or 2 spatial dimensions as shown below : These solutions have to fulfill the Dirichlet boundary conditions (at the domain boundaries) ? homogeneousBVC1 homogeneousBVC2
by means of which the eigenvalues are determined. The procedures homogeneousBVC1a and homogeneousBVC2a, however, used in the context of the advection-diffusion equation have an additional option simplifyRule (with default value {11} ) which enables further simplification of the series terms. ? determineEigenvalue2
Additonal auxiliary procedures used for the treatment of the advection-diffusion equation are
185
? diffSequence vars2arg determineEigenvalue1
Applying advectionDiffusionEqnSolution to problems with 1 and 2 spatial dimensions Below we follow the treatment of the advection-diffusion problem delineated in the paper of Mojtabi and Deville [2] which provides the analytical solution of the linear advection-diffusion problem with (homogeneous) Dirichlet boundary conditions (BC) and a smooth initial value condition (IV) like a sine function.
Solution of 1d advection-diffusion equation (using homogeneousBVC1a) Starting from the PDE for the advection-diffusion problem (or its representation in terms of the differential polynomial AdvDiff ) together with the (homogeneous) Dirichlet boundary conditions (case BC=1) and a smooth initial value condition (IV) like a sine function. Approximate solution (finite Fourier series up to finite order =1000) : $Rule is a global substitution rule which is used inside advectionDiffusionEqnSolution for simplification of the results.
$Rule
1c
c
2 2Sinh
c
, 1c
2 Sec n 1
n
,
1
2n
c
c
2 2Cosh
, 2
1 ;
AdvDiff t c 2 ; BC 1; . ; xT0,xT1 xT 0,0 ; xL 1,1 ; L0, L1 F Sin ; 200 sec u t, advectionDiffusionEqnSolution AdvDiff, BC1,xT,xL,F,,"On",sf
0 t c 2
trans t 2
with 1 t 2 x …
1dheat diffusion equation : u0,2 t,
u1,0 t,
separates into 2 ODEs :
with u t, T t X T t 2 T t 0, 2 X X 0
186
0
c2 t 4
c 2
returned from homogeneousHeatEqnSolution : uh t, T t X t
1
Tn t 4
n2 2 t
2
0 c1 Cos c2 Sin
n
eigenvalues n
2 n
Wn c1 Cos
n
c2 Sin
2 for boundary values
2 W 1 0, W 1 0
vp t,
1
4
12 p2 2 t
B 1 2 p Cos
1
1 2 p p
2
2 t
A 2 p Sin p
2
p0
A 2p
rule for A2 p : 32 1
c 2c
54p 1p
2
c4 8 c2 1 p2 2 2 16 1 p2
2
B 1 2 p 16 1
rule for B2 p1 : 4
c
c p 2 3 Sinh
p
p
c 1 2 p 2 3 Cosh
34p 1p 2
2
2
2
4 4
c 2
4
4
v t,
16 1
c 2 3
p
p0
1 4
2 p 2 t
1 2 p Cos
p Cosh
2 2 p 34p 1p
2
4
2
2 t
c
c4 2 c2 5 4 p 1 p
2
p Sin p Sinh
4
2 2
c 2
c4 8 c2 1 p2 2 2 16 1 p2
2
4 4
approximate solution u t,,
c2 t 4
c
16 1
2
p
c 2 3
1 4
2 p 2 t
1 2 p Cos
2
p0
c 2c 4
2 p
p Cosh
2
2
2 t
54p 1p
c 2
34p 1p c p Sin p Sinh c4 8 c2 1 p2 2 2 16 1 p2 2 2
2
2
187
4
4
2
4 4
c2 t 4
c
1
16 1
2
p
c 2 3
4
2 p 2 t
1 2 p Cos
2
p0
p Cosh
c 2
c4 2 c2 5 4 p 1 p 2 p
2
2 t
2 2 3 4 p 1 p 2 4 4 c p Sin p Sinh c4 8 c2 1 p2 2 2 16 1 p2 2
2
4 4
The result returned from advectionDiffusionEqnSolution after a lengthy calculation (~ 200 sec) (achieved without any user interaction) is an infinite Fourier series u[t,][]. For practical calculations the exact analytical solution is approximated by a finite sum of terms up to order =1000. u t, _ u t, ; 1000 t_,_ u t, 1000 ;
It turns out that this solution is identical with the result given in paper [2] from Mojtabi & Deville, see eqn. (18). The numerical results calculated up to = 1000 terms agree. 1000 t_, _ u t,
1000 1.6, .5
.
1000 ;
c 1.,
1 10
0.197242
As was pointed out in [2] the behavior of the solution depends sensitively on the exponential pref
c2 t
c
actor 4 2 which blows up (irrespective of the Fourier series of order 1000) as approaches 1 because the numerical values of competing series terms become huge. However, the exponential prefactor is uninteresting because the series terms will become very large and hence compensate each other so that values of order 1 are finally obtained. Visualization of uk t, x For the kinematic viscosity 0 the evaluation of the series terms requires high precision with 1000 or more digits. In order to achieve this goal the calculation must be performed with exact number which means integer and rational numbers are used throughout and only at the very end the numerical approximation is applied. Below, a sequence of curves with increasing values of the ‘parameter’ t = {~0, 0.4, 0.8, 1.2, 1.6, 2.0} (corresponding to the colors red, yellow, green, light blue, blue, dark blue) is shown for different values of the kinematic viscosity 103 , 102 , 1 10 , 1 2 , 1 , 2 3, 1 . It turns out that the numerical calculations are very subtle with respect to small values of t. To display the sequence of (discrete) points xi , u tk , xi the procedure showTimeSteps2 is used (instead of showTimeSteps which is used for functions). As to the first two values of = 103 and 102 the curves are calculated pointwise and the data sets {x, 3 ti , x } are displayed by showTimeSteps1.
188
In order to set up the structure of xuValues a dummy array is pre-defined for the data sets of six curves (with colors red, yellow, green, light blue, blue, dark blue) corresponding to the parameters tk 0, 0.4, 0.8, 1.2, 1.6, 2.0 . For the first two examples given for = 103 and = 102 for the first (red) sinusoidal curve values t = 101 and 102 are chosen instead of the exact value t = 0. è Parameter c 1, = 1/1000, t [0,2] , x [-1,1] For = 1/1000 the value for $MaxExtraPrecision should be increased to > 500 at least, and the number of terms in the Fourier series expansion taken into account must include more than 1000 terms otherwise the values for the first curve 3 t, x
. t 102 will blow up to
10400 . Numerical calculations show that particularly the first graph for t~0 (i.e. choosing = 103 or 102 but not 0 for the first (red) curve) ) depends in a very sensitive way on the number of series terms and the extra precision. xuValues. ; xuValues Table
,
, i,6
;
Due to the high precision required the calculation of the (discrete) data points xuValues is very time consuming. 3 t_,_ u t,
1000
. c 1, 103 ;
600 sec
$MaxExtraPrecision 2000; 4
1
xVal Union Table x, x,1, , 5 9
Table x, x, 10
,1,
4
,Table x, x, , 10 5
1 ,
,
5 10 50
; 1000
corresponds to t 2 4 6 8 , , , , ,2 ; 10 5 5 5 5 Table xuValues i N ,5 &
9
1
0,.4,.8,1.2,1.6,2.0
where t 103 is critical
tVal
Table
x,3 t,x
. t tVal i , x,xVal , i,1,6
;
Structure of the data sets xk , uk i for 6 curves with t-values {0, .4, .8, 1.2, 1.6, 2.} is given in xuValues 1
The following plot depicts the analytical solution for = 1/1000 corresponding to a Reynolds number of Re=2000. The boundary layer at x=1 behaves like a sharp exponential layer which is clearly shown for values of t = {1.2, 1.6, 2.0 }. showTimeSteps2 xuValues,"1 1000"
189
1 1000 1.0
0.5
u t,x
0.0 0.5 1.0 1.0
0.5
0.0
0.5
1.0
x
A comparison shows that this plot is identical with Fig. 4 in [2]. è Parameter c 1, = 1/100,
t [0,2] , x [-1,1]
For larger values of high digital precision is no longer essential. The sequence of (continuous) curves for subsequent -values are displayed with showTimeSteps1 and showGraph1. è Parameter c 1, = 1/(10), t [0,2] , x [-1,1] For t = { 104 , .4, .8, 1.2, 1.6, 2. } a series of curves is shown for the Reynolds number Re =
cL
2
with values for the kinetic viscosity = 1/(10), the advection velocity c set to 1
and interval length L L1 L0 2 . 1 t_,_ u t,
showTimeSteps1
. c 1.,
1000
1.
;
104 ;
10
1 t,x , t,,2,.4 , x,1,1 ,6,"1 10" 1 10
1.0
0.5 u t,x
0.0 0.5 1.0 1.0
0.5
0.0
0.5
1.0
x
The color function used is col2 = Hue
4.85.2 1 6.5
& where the lowest temperatures are colored
in blue, the highest in red. The function Rasterize returns a rasterized graphic.
190
raster Rasterize ,ImageResolution 210 &; showGraph1 1 t,x , t,,2.2 , x,1,1 ,col2,ViewPoint .89,.85,.62 , " t","x"," u t,x " raster
è Parameter c 1, = 1/(2), t [0,2] , x [-1,1] è Parameter c 1, = 1/,
t [0,2] , x [-1,1]
è Parameter c 1, = 2/3,
t [0,1] , x [-1,1]
è Parameter c 1, = 1,
t [0,.4] , x [-1,1]
5 t_,_ u t,
showTimeSteps1
. c 1., 1.
1000
;
5 t,x , t,0,.4,.1 , x,1,1 ,6,"1" 1
1.0
0.5 u t,x
0.0 0.5 1.0 1.0
0.5
0.0
0.5
1.0
x
showGraph1 5 t,x , t,0,.42 , x,1,1 ,col2,ViewPoint .89,.85,.62 , " t","x"," u t,x "
191
raster
Solution of 2d advection-diffusion equation (using homogeneousBVC2a) The advection-diffusion problem investigated is extended to the case with 2 spatial dimensions for a rectangular domain = { -1 < < 1, -1 < < 1 } together with (homogeneous) Dirichlet boundary conditions (case BC=7) ; initial value condition (IV) for u(t=0, , ) is a smooth function F(, ) = sin(4)·sin(2) of two sine functions. Approximate solution (finite double Fourier series up to finite order ) : c
$Rule
1c 2 2Sinh
c 2
Sec n_ 1 AdvDiff t c
n
,
1
c
c
1c 2 2Cosh
,
, 2
2n_
1 ;
2 2
Expand;
BC 7; . ; T0,T1,T2 xyT 0,0,0 ; xyL 1,1,1 ; L0,L1,L2 F Sin 4 Sin 2 ; 3160 sec 53 min u t,, advectionDiffusionEqnSolution AdvDiff,BC7,xyT,xyL, F,,"On",sf 0 t c c 2 2 t 2 2
trans
with 1 t 2 x …
c2 t 2
c 2
192
c 2
2dheat diffusion equation : u0,0,2 t, , u0,2,0 t, ,
u1,0,0 t, ,
0 separates into 3 ODEs :
with u t,, T t X Y T t 2 T t 0, 12 X X 0, 22 Y Y 0 with 2 1 2 2 2
returned from homogeneousHeatEqnSolution1 : uh t,, T t X Y t
2
0 c1 Cos 1 c2 Sin 1
c3 Cos 2 c4 Sin 2
returned from determineEigenvalue2 m n eigenvalues: 1 m for m 1; 2 n for n 1 2 2
1
Tm,n t 4
m2 n2 2 t
eigenvalues m,n 2
1
m2 n2 2
4 Xm c1 Cos Yn c3 Cos domain :
m
c2 Sin
2 n
m 2 n
c4 Sin 2 2 L0 ,L1 1, 1 ; L0 ,L2 1, 1
vp,q t,, p,q0
1 4
2 12 p2 12 q2 t
2
p2 q2 t
B1 1 2 p B2 1 2 q Cos
2
p Cos
q
2
A1 2 p A2 2 q Sin p Sin q
1 p 2 3 rule for A21 p ,B2 p1 : A12 p 128 1 c p Sinh
B11 2 p 64 1p c 1 2 p 2 3 Cosh
c 2
c 2
2
c4 8 c2 16 p2 2 2 16 16 p2 2 4 4 ,
c2 7 2 p2 2 2 c2 9 2 p2 2 2
2 q 2 3 rule for A22 q ,B2 q1 : A22 q 64 1 c q Sinh
B21 2 q 32 1q c 2 1 2 q 3 Cosh
c
c 2
c4 8 c2 2 4 q2 2 16 4 4 q2 2 4 ,
c4 2 c2 2 17 4 q 1 q 2 4 15 4 q 4 q2 2 4
193
v t,, p,q0
2048 1
pq
c2
1 4
2 12 p2 12 q2 t
1 2 p 4 1 2 q 6 Cos
1
12p
2 1 Cos
1 2 q Cosh
2
c
8192 1
pq
c2
4
c2 7 2 p
2
c4 2 c2 2 17 4 q 1 q 1
2
2
2 4 15 4 q 4 q2
2 4 p2 4 q2 t
2 2 2
c2 9 2 p
4
2
4 4
2 2
c
p 4 q 6 Sin p Sin q Sinh
c4 8 c2 16 p2 2 2 16 16 p2
2
2
2
c4 8 c2 2 4 q2 2 16 4 4 q2
2
4
approximate solution u t,,,
c2 t 2
c 2
c 2
2048 1
pq
c2
1 4
2 12 p2 12 q2 t
1 2 p 4 1 2 q 6 Cos
1
12p
2
p0 q0
1 Cos
1 2 q Cosh
2
c
8192 1
pq
c2
4
c2 7 2 p
2
c4 2 c2 2 17 4 q 1 q 1
2
2 4 p2 4 q2 t
2
2 2
2 4 15 4 q 4 q2
2
4
c2 9 2 p
2
c4 8 c2 2 4 q2 2 16 4 4 q2
2 2
p 4 q 6 Sin p Sin q Sinh
c4 8 c2 16 p2 2 2 16 16 p2
2
c
2
2
4 4 2
4
The solution u[t,,][] obtained after a lengthy evaluation (about 52 min on a QuadCore i7 computer) is used for further numerical calculations u t,,
_ u t,,
20 t_,_,_ u t,,
testing finite double sum with 2 202 term
20 ;
A numerical value of 20 t, , is in the reasonable order of magnitude. 20 1,.5,.5
.
c .1, .2
2.03367 106
Visualization of uk t, x, y The function u t, x, y is given as an animation using Manipulate which displays the result as a 3d contour plot together with a 2d contour plot for different time steps attached to a slider. The corresponding graphics function is showAnimation1 which is a modification of the procedure showAnimationed (used to display the solutions of the
194
which is a modification of the procedure (used to display the solutions of the 2d heat equation for different boundary conditions in Part 4) with respect to a specific color function col2 and some other improvements of options. 4.85.2 col2
Hue
&
preferred ColorFunction
;
6.5
è Parameter c .1, = 0.2, t [0,2] , [-1,1], [-1,1] g1 20 t,, . c .1, .2 ; l0,l1,l2 1,1,1 ; showAnimation1 g1, t,0.6,2 , ,l0,l1 , ,l0,l2 ,col2,PlotRangeAll raster
è Parameter c .1, = 1. , t [0,2.] , x [-1,1], x [-1,1]
195
g2 20 t,, . c .1, 1. ; l0,l1,l2 1,1,1 ; showAnimation1 g2, t,0.08,.2 , ,l0,l1 , ,l0,l2 ,col2,PlotRangeAll raster
196
Summary Explicit analytical solutions for the advection-diffusion equation were obtained. Due to a transformation [3] u t, x, … 1 t 2 x… v(t,x,…), named as ‘change of the dependent variable’ the gradient term c0 x etc. in the advection-diffusion equation is eliminated so that the PDE is reduced to the well-known heat equation. The type of equations which had been investigated are (i) the 1d- advection-diffusion equation for (homogeneous) Dirichlet boundary conditions (ii) the 2d advection-diffusion equation for (homogeneous) Dirichlet boundary conditions. The exact solutions are given in terms of (infinite) Fourier sine/cosine series expansions and are rendered as 3d plots and animated contour plots.
Acknowledgement The author would like to thank A. Mojtabi & M.O. Deville for the inspiration to investigate in the framework of MIDO the problem of how to obtain the analytical solution of the linear advectiondiffusion equation given in their publication [2] and the discussion of sophisticated technical details. Special thanks to Vladimir Gerdt from JINR, Dubna/Russia for his continuous interest and to Michael Trott from WRI for his readiness to discuss Mathematica related technical details of the implementation of the algorithm and make valuable suggestions and improvements.
References [1] Convection-diffusion equation, seehttp://en.wikipedia.org/wiki/Convection–diffusion_equation [2] Abdelkader Mojtabi, Michel O. Deville “One-dimensional advection-diffusion equation : Analytical and finite element solution” Computers & Fluids 107 (2015), pp. 189-195. [3] J. S. Pérez Guérrero, L.C.G. Pimentel, T. H. Skaggs, M. Th. van Genuchten “Analytical solution of the one-dimensional advection-diffusion transport equation using a change-of-variable and integral transform technique” Int. J. Heat and Mass Transfer 52 (2009), pp. 3297-3304 [4] P. K. Kythe, P. Puri and M. R. Schäferkotter, "Partial Differential Equations and Boundary Value Problems with Mathematica", 2nd Ed. , Chapman & Hall / CRC, 2002, Chapter 3.2-3.4, pp. 75-81. [5] R. Kragler "Method of Inverse Differential Operators for the Solution of PDEs," in Computer Algebra Systems in Teaching and Research, 6th International. Workshop, vol. Differential Equations, Dynamical Systems and Celestial Mechanics (CASTR 2011), Siedlce ( L. Gadomski et al. eds.), Siedlce : Wydawnictwo Collegium Mazovia, 2011 pp. 79-95. [6] R. Kragler "Method of Inverse Differential Operators applied to certain classes of non-homogeneous PDEs and ODEs", in Proceedings of "The Third International Conference : Mathematical Modelling and Differential Equations", Brest State University, Brest/Belarus ( Sept. 2012), Publishing Center BSU, Minsk, 2012 pp. 290-307, ISBN 978-985-553-054-2 [7] R. Kragler "Method of Inverse Differential Operators; Analytic Solutions of 2nd Order PDEs with Initial Value and Boundary Condition Problems", in Proceedings of "12th International Mathematica Symposium (IMS 2015), Eds. Zdenĕk Buk & Miroslav Čepek, Publ. by České vysoké technické v Praze, ISBN 978-80-01-05623-3 [8] Martha L Abell, James P. Braselton “Differential Equations with Mathematica”, 2nd Ed. , Academic Press , 1997, Chapter 11.2, pp. 723-737. [9] Selwyn Hollis “ A Mathematica Companion for Differential Equations” 2nd Ed., Prentice Hall/Pearson Education , 2003, Chap. 11.1, pp. 227-233.
197
