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COO-3077-160MF-94

Courant Institute ofMathematical SciencesMagneto-Fluid Dynamics Division

Similarity Solutions of Systemsof Partial DifferentialEquations Using MACSYMAp. Rosenau and J. L. Schwarzmeier

Air Force Office ofScientific Research ReportandU. S. Department of Energy Report

Plasma PhysicsOctober 1979

New York University


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UNCLASSIFIEDNew York UniversityCourant Institute of Mathematical SciencesMagneto-Fluid Dynamics Division

MF-94 COO-3077-160

SIMILARITY SOLUTIONS OF SYSTEMS OFPARTIAL DIFFERENTIAL EQUATIONS

USING MACSYMA

P. Rosenau and J. L. Schwarzmeier

October 1979

Air Force Office of Scientific ResearchContract No. AFOSR-76-2881and

U. S. Department of EnergyContract No. EY-76-C-02-3077*000

UNCLASSIFIED


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ABSTRACT

A code has been written to use the algebraic computersystem MACSYMA to generate systematically the in-finitesimal similarity groups corresponding to systemsof quasi-linear partial differential equations. Theinfinitesimal similarity groups can be used to findexact solutions of the partial differential equations.In an example from fluid mechanics the similaritymethod using the computer code reproduces immediately asolution obtained from dimensional analysis.

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1. IntroductionIn recent years Lie infinitesimal contact transformation

group theory has become a widely used tool for systematic ex-plorations of similarity solutions of partial differentialequations [1] . Though the implementation of this method in-volves mostly standard manipulations (algebra and chain deri-vatives) , in practice these become extremely involved for allbut the simplest of equations.

Here we introduce the use of the algebraic computing

system MACSYMA to facilitate these calculations. Specifically,MACSYMA is used to calculate systematically the generators ofthe infinitesimal group under which the considered equationsare invariant. As far as the operational side of the similaritymethod is concerned, this is the most involved and tedious step.Once the general similarity group is found, a (hopefully non-trivial) subgroup is found which leaves invariant boundarycurves and boundary conditions. Finally, once this subgroupis found, its invariants and consequently the similarity formsof the solutions of the partial differential equations can befound.

A primary advantage of the similarity method is that itis one of the few general techniques for obtaining exact (non-linear) solutions of partial differential equations. A primarydisadvantage of the similarity method is that the solution foundmay satisfy only a very restricted set of initial and boundaryconditions.

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In general, the existence of a one parameter group oftransformations leads to a reduction by one in the number ofindependent variables. Thus, in the most encountered case oftwo independent variables, the reduction leaves us with anordinary differential equation.

We remark here that the similarity method contains as aspecial case those solutions which can be obtained via dimen-sional analysis. That is, if an equation has a solution whichcan be obtained from dimensional analysis, then the presentmethod will reproduce immediately that special similaritysolution, and possibly more general similarity solutions aswell.

2 . Sketch of the Similarity MethodIn this section we present a brief summary of the similar-

ity method. The similarity transformation solver (STS) usesMACSYMA to construct the similarity transformations which areadmitted by a set of partial differential equations of theform ( Slims over j,k, and 1)

a. . =- U . + b . . yr-rr U . + C . ., -r U .ID 3t^ ^ ID 9t ] i:k 9Xj^ J

^ijkl 93^^33^;; ^j = ^i ' (^)

where u. = u.(x,t) are solutions, and a,b,c,d, and e arearbitrary, specified functions of (x,t,u) . The one parameter

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infinitesimal transformations considered are

x' = X + eX(x,t,u) (2a)t' = t + eT(x,t,u) (2b)u' = u + eU(x,t,u) . (2c)

MACSYMA is used to determine the differential equations,called the determining equations, that are satisfied bythe infinitesimal groups X,T,U. Technically, this is achievedby requiring that the partial differential equations (1) be

2 -*-*invariant to order e in terms of the new variables (x',t',u')of Eqs. (2). Thus the annuling of 0(e) terms becomes an in-variance condition.

Let M[u] =0 be a system of partial differentialequations in the old variables, and let M'[u'] =0 be thesame system in the new variables, where M'[u'] is obtainedfrom M[u] by replacing everywhere (x,t,u) -* (x',t',u')The application of the transformations (2) to M'[u'] showsthat M'[u'] is of the form M'[u'] = M[u] + eH + O(e^),where H=H(x,t,u,X,T,U) . We make the system invariant tothe infinitesimal transformations (2) by requiring that H 5 0.Suppose we denote by {S } , for an appropriate set of values m,all possible first or second derivatives of u . with respectto X, and/or t . Then the expression H is a sum of termsinvolving the S^^'s, with coefficients functions of (X,T,U)and their derivatives with respect to (x, ,t,u.) . After theoriginal system of partial differential equations is used to

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guarantee that all the ^^' that appear in H are independent,H is made to vanish by requiring that all the coefficients ofthe S 's vanish. This, as explained in [1], is a sufficientbut not a necessary condition. The coefficients of the S 'smare differential equations satisfied by the infinitesimalgenerators (X,T,U); these are the determining equations mentionedearlier. In practice the determining equations usually can besolved easily, as they are an overdetermined set of linearequations. A knowledge of the infinitesimal transformationsenables one to construct a continuous transformation or in turnto find the invariants of the group that serve as the new de-pendent and independent variable. it is in these new variablesthat Eqs. (1) exhibit their symruetry and result in a re-duction by one in the number of independent variables.

To illustrate the desirability of using MACSYMA to calcu-late the determining equations, suppose a partial differentialequation contains a derivative t (assume scalar u and x)o XIn terms of the new variables, the transformed partial differential

3uequation contains a term r But from Eq. (2c)9x3u' a8x' 8x'

[u(x,t) + eU(x,t,u)]

9u dx _ _ 8u dt _^^ ^ ax- ^^ ax-

au a X ^ au at^ ^ ax' ^^ ax'

au au ax _ _ au at^^ ax' ^^ ax' (3)

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Now for 9x_3x

and at9x'

use Eqs. {2a-b) to obtain

9x_9x'

= 1 - e 8X 3u8u 9x + O(e^) (4)

and3t9x'

= - e 3T _ _ 3T 9u9x 9u 9x + O(e^) (5)

By substituting Eqs. (4) and (5) into Eq. (3) andsimplifying, we obtain

9u'9x'

9u9x + e

r9U _^ f 9U _ 9X'j_9x [9u ~ 9x^

9u _ 9T 9u _ 9X f 9ul9x ~ 9x 9t ~ 9u 9x

9T 9u 9x0]9u 9x 9tJ + O(e^) (6)

Correspondingly, any second derivatives such as 92u'9x .2 becomevery involved. The system of partial differential equations(1) is written in a completely vectorized form. SinceMACSYMA knows the chain rule and can simplify and factor, itis quite feasible for MACSYMA to reliably transform compli-cated systems of partial differential equations in several in-dependent variables.

3. Programming ConsiderationsIn this section we describe the implemention of the STS

on the MACSYMA system. The basic mode of operation is that a

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reference version of the STS is modified as required with theTECO editor to solve the problem at hand, and then the modifiedversion is executed while in MACSYM using the BATCH command.Once program execution is initiated with the BATCH command,no further user interaction is required.

Input procedures for the STS are described as follows.Once a user obtains a reference copy of the STS program in hisuser space, a listing of the program should be made while inTECO. In the program listing the portions of the code whichmust be changed from one problem to another are preceded by asingle asterisk and a blank line and are followed by a blankline and a double asterisk. There are three places in theSTS program where the user must modify the text to solve anew problem. First, a title should be specified which identifiesthe system of partial differential equations being considered.Second, the user must specify the number of equations (i.e.,the number of dependent variables u.), NEQ, the spatialdimension of the problem, NX (NX ^ 1) / and whether or notthere are time derivatives (NT = , no time derivatives;NT = 1, there are time derivatives) . Finally, the user mustspecify the nonzero coefficients a. ., b. ., c. ., , d. ., ,, ande. of Eq. (1). Before assigning the subscript i to thecoefficients, the equations should be ranked in ascendingorder according to the number of terms in the equations . Forexample, the equation with the fewest terms goes first, etc.After these changes are made in the reference STS program, the

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updated version is stored with TECO as a new file, to beexcuted later with BATCH. In this way a user might build alibrary of STS programs appropriate to different systemsof equations.

The output of the STS is in two stages. In the firststage the BATCH command causes listing and execution of thecode. During this stage some of the intermediate calculationsare printed out, so that the progress of the calculation canbe inspected as it proceeds. This portion of the computingtime takes about 10 minutes, nearly independent of the systemof equations considered. The first intermediate printout isa listing of the functional dependencies of the functions in-volved in the infinitesimal transformations (2) . The STSprogram is vectorized, so the variables (x, r t,u . ,Xj^,T,U . ) arereplaced with the program variables (X[K] ,T,U [j] ,FX[K] ,FT,FU[J]), respectively. The second intermediate printout is alisting of the system of equations to be solved. This servesas a check that the equations were ordered properly and thatthe coefficients were entered correctly. Next the linearized-in-epsilon versions of the new coefficients of the transformedpartial differential equations are displayed. Recall that theoriginal system of equations must be used to guarantee thatthe S 's that appear in H are independent. The fourthintermediate printout lists NEQ of the S 's which the^ mSTS has chosen to eliminate from H by using the originalsystem of equations. The solutions of the original equations

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for the NEQ S^'s is then listed in the last intermediateprintout.

The second stage of output in the STS is the displayingof the components of H, from which the determining equationsare derived. Each component of H is factored convenientlywith respect to all the S 's; the determining equations canbe read off immediately as the coefficients of the independentterms involving the S ' s . The amount of time spent computingH depends significantly on the system of equations solved, aswell as on the current load on the MACSYMA system. A veryrough estimate of the computing time to calculate H is

T (min) = NEQ (NX+1) (NX/2 + 2) (7)

4 . An ExampleAs an example of the use of the STS, consider the classical

Blasius problem from fluid mechanics; solve the problem of two-dimensional, steady state, incompressible, viscous, laminar flowin a boundary layer. The equations to be solved are

^ V + I- v =0 (8a)9x X 3y yV -r V + V -r V = V 55 , (8b)X 3x X y 3y X g 2

where v = (v ,v ,0) is the fluid velocity, and v is theX ycoefficient of viscosity. A semi-infinite plate exists in theX - z plane for x > , and far from the plate, y-*- , v

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assumes a constant value, u . The viscous boundaryconditions are:

V (y = 00) = u (9a)X -^ o

V (y = 0) = V (y = 0) = . (9b)X y

Using arguments of dimensional analysis, one concludes that(Landau and Lifshitz [2]) the new independent similarityvariable is

^ = y /uTx (10a)oand that v and v are of the formsX y

v^ = u^ f^(;) (10b)

V =y yj f2 (C) . (10c)

Let us now obtain these same results as well as the gen-eral infinitesimal similarity group by means of the STS. Ina program listing of the STS we first insert in the appropriateplace a title for problem being solved. Next we insert onseparate lines in the STS:

NEQ:2$NX:2$ (11)NT:0$.

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Finally we insert the nonzero coefficients:

C[l,l,l]:l$C[l,2,2]:l$

C[2,l,l] :U[1]$ (12)C[2,1,2]:U[2]$D[2,l,2,2] : - NU$ ,

where NU is the coefficient of viscosity, v . After thisversion of the STS is stored in TECO and executed in MACSYMwith BATCH, the printouts begin.

The components of H are printed out, and the determiningequations can be read-off by inspection (see the Appendix)They are (letting U[l] ^ u, U[2] ^ v, FU[1] ^ U, and FU[2] ^ V)

X=X=X=X =Y =YV u y w w V= U =u=Y=0VV V uV -Y =V +U =0 (13)u X y XV -V -X +U =U-uX + 2uY =y V X u X yV (-U + 2Y ) + 2vY = - vU + vU + uU =0uu uy u yy y XV (Y -2U ) + vY - uY + V = .yy uy y x

This set of differential equations for X,Y,U, and V asfunctions of x,y,u and v is overdetermined.

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The integrations involved is solving Eqs. (13) are straight-forward. The result is that the general infinitesimal similaritygroup corresponding to Eqs. (8) is

X{x,y,u,v) = (a + 2b)x + cY(x,y,u,v) = by + g (x) (14)U(x,y,u,v) = auV(x,y,u,v) = ug^ - bv

where a,b, and c are arbitrary constants, and g(x) isan arbitrary function of x alone.

Now we find the subgroup of (12) that leaves invariantthe boundary curves (9)

Invariance of y = 0=*y' =0=*Y (x,y=0,u,v) =0= g(x) = . (15a)

From the transformations (2a) , the boundary curves y = 0=>X (x> 0,y,u,v) >=> 2b + a > . (15c)

Equations (15) will restrict the group (14) so that theboundary curves are left invariant. Now we further restrict

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the group (14) so that the boundary conditions are invariant:

u(y = 0) = ^ u' (y = 0) = => U(x,y = 0, u=0,v) ==> no restriction on a . (16a)

Similarly, v(y = 0) = is trivially satisfied=* no restriction on b . (16b)

u (y = ) = u =* U(x,y = 00, u = u ,v) =

=* a = (16c)

By combining Eqs . (15-16) the final subgroup of Eqs(7-8) that leaves invariant the boundary data is

X(x,y,u,v) = 2bx

Y(x,y,u,v) = by (17)

U(x,y,u,v) =

V(x,y,u,v) = - bv.

The first invariant surface condition is

dx _ dy _ du ^^ dx _ dy _ du .,q.jr-y-u-' ''' 2b^-b7~~- ^^^'

Integration of the first equality in (18) is

C = ^/ /x = constant. (19a)

Integration of the second equality in (18)

u = Fj^(C) . (19b)-13-


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The second invariant surface condition is

dx _ dy _ dv ^^ dx dy dv ,-^,

Integration of the second equality in (2 0) gives

V = F^U) /y . (19c)Equations (19) are the same results as Eqs . (10) obtained

by Landau and Lifshitz. Furthermore, the infinitesimal group(14) is the general infinitesimal group corresponding to Eqs.(8) . Thus the similarity solution corresponding to any otherboundary conditions can be found by analyzing the group (14)

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Acknowledgements

The MACSYMA computer system is being developed by theMathlab Group of the MIT Laboratory for ComputerScience. This work is supported, in part by the De-partment of Energy under Contract Number E(ll-])-3070and by the National Science Foundation under GrantNSG 1323.

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References1. G. W. Bluman and J. D. Cole, Similarity Methods for

Differential Equations, (Springer-Verlag, New York, 1974)Part II.

2. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley Publishing Co., Reading, Massachusetts, 1959)pp. 145-150.

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Appendix

In this appendix we illustrate how the determiningequations are obtained from a printout of the functions H .As an example we consider the system of partial differentialequations (8), so there are two components H.,i= 1 and 2.

The STS factors each H. to produce a polynomial in theS 's, where the factoring is performed first with respect toS- , then with respect to S^, and so on. That is, H. firstis factored to produce a polynomial in S^ . Then the coef-ficient of each power of S, is factored to produce a poly-nomial in S_, etc. In this way, once a particular productof powers of the S ' s appears in H. , that same combinationof the S 's never appears again. Consequently the coef-ficients of the independent terms of H. (i.e., the deter-mining equations) can be identified by inspection.

The printout of H corresponding to the partial differentialequations (8) is

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COMPONENT NUMBER 1 OF THE EXPRESSION H FROM WHICH THEDETERMINING EQUATIONS rtRE DETERMINED IS:

d d 2 d d d dZEROP =


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2 2 2d d d d

+ ((- 2*( FX ) + 2( FX ) + 2( FU ))*NU - U ( FX )dU dX 2 dU dX 1 dU dU 1 2 dU 22 2 12 12 2d d d

+ 2*U *( FX ))*S - U ( FX )*S + ( FX )NU*S1 dU 2 7 1 dU 1 13 dU 1 14

1 2 22 2d d d d

+ ( FX + 2*< FU ))*NU + 2*U < FX ) - U *< FX )2 1 dU dX 1 1 dX 2 2 dX 1dX 2 2 2 22

2d d d 3

- U *< FX ) - U *( FU ) + FU )*S + ( FX )*NUSIdXl 2dUl 14 22 712 dU 12 2d d d 2+ ((2*( FX ) FU )*NU + 2*U *( FX ))SdU dX 2 2 1 2 dU 2 7

1 2 dU 11

2 2d d d d+ *NU*S + ( FX - 2*( FU ))*NU

1 dU 2 13 dU 2 14 2 2 dU dX 12 2 dX 122d d d d

+ U *( FX ) - U *( FX ) + FU )*S + U *( FU ) S - ( FU )*NUS2 dX 2 1 dX 2 2 7 1 dU 1 13 dU 1 14

2 1 2 22

d d d- ( FU )*NU + U *< FU ) + U *< FU ))*EPSLON2 1 2 dX 1 1 dX 1dX 2 1

(088) BATCH DONE(C89)

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The functions H. can be simplified considerably bymaking use of the following procedure: if all the independentcoefficients of the S ' s that contain only a single termmare set to zero, these conditions on the groups can be usedimmediately to discard many of the remaining terms of H .For instance, in H we note that

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and

1x7 ^ 2 ^ 1x7 ^ 1 =

Similar reductions occur in the size of H_ by applyingEqs. (A. 1)

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This report was prepared as an account ofGovernment sponsored work. Neither theUnited States, nor the Administration,nor any person acting on behalf of theAdministrationA. Makes any warranty or representation,express or implied, with respect to theaccuracy, completeness, or usefulness ofthe information contained in this report,or that the use of any information,apparatus, method, or process disclosedin this report may not infringe privately

owned rights ; orB. Assumes any liabilities with respect tothe use of, or for damages resulting fromthe use of any information, apparatus,method, or process disclosed in thisreport.As used in the above, person acting on behalfof the Administration includes any employeeor contractor of the Administration, oremployee of such contractor, to the extentthat such employee or contractor of theAdministration, or employee of such contractorprepares, disseminates, or provides access to,any information pursuant to his employment orcontract with the Administration, or hisemployment with such contractor.

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.Similarity solutions ofsystf iR of p.d. e. ...

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