u ρ ω x u ρ λ x = 0, ρ ω x + ρ λ y + u σ

Transcription

1 > restart: > with(oremodules): > with(oremorphisms); > with(linalg): We onsider the approximation of the steady two dimensional rotational isentropi flow studied in page 436 of R. Courant, D. Hilbert, Methods of Mathematial Physis, Wiley Classis Library, Wiley, 989, ω x + σ 2 x =, λ x + σ 2 y =, () ρ ω x + ρ λ y + u σ x =, where u denotes the onstant veloity parallel to the x-axis, ρ the onstant density and the speed of sound. Let us introdue the ring A = Q(u, ρ, )d x, d y of differential operators in d x and d y with oeffiients in the field Q(u, ρ, ) > A:=DefineOreAlgebra(diff=dx,x,diff=dy,y,polynom=x,y, > omm=u,rho,): and the system matrix R A 3 3 of () defined by: > R:=matrix(3,3,u*rho*dx,^2*dx,,,^2*dy,u*rho*dx, > rho*dx,u*dx,rho*dy); d x 2 d x R := 2 d y d x ρ d x u d x ρ d y We denote by M = A 3 /(A 3 R) the A-module finitely presented by the matrix R. Let us study the endomorphism ring E = end A (M) of M: > Endo:=MorphismsConstCoeff(R,R,A): Endo;, 2, 2 u 2 ρ 2 ( u 2) Hene, we obtain that E is finitely generated by three endomorphisms f = id M, f 2 and f 3 defined by f i (π(λ)) = π(λ P i ), where π : A 3 M denotes the projetion onto M, λ A 3 and P i is one of the three previous matries. The generators f i s of E satisfy the relations Endo2 (f f 2 f 3 ) T =, where Endo2 is the following matrix: > Endo2; u 2 ρ d x + u 3 d x ρ d y 2 d y d x 2 d x + u 2 d x d y Let us study the idempotents of the endomorphism ring E defined by means of onstant matries, i.e., matries defined by with zero-order differential operators: > Idem:=IdempotentsConstCoeff(R,Endo,A,,alpha);

2 Idem := 2 α α α ( 2 +u 2 ) 2,,,, 2 2 α α α ( 2 +u 2 ) Ore algebra, diff, diff, x, y, d x, d y, x, y, u, ρ,, α,,, 4 α α 2 u 2, x, y,,, diff = d x, x, diff = d y, y We obtain the two trivial idempotents and id M of E respetively defined by the matries or I 3, two nontrivial idempotents respetively defined by the matries Idem, 3 and Idem, 4 whose entries belong to A and two non-trivial idempotents of end B (B A M), where B = Q(u, ρ, )α /(4 (u 2 2 ) α 2 )d x, d y, respetively defined by the matries Idem, and Idem, 6. Let us onsider the matrix P = Idem, 3 and Q satisfying R P = Q R: > P:=Idem,3; Q:=Fatorize(Mult(R,P,A),R,A); 2 P := Q := u We an hek that we have P 2 = P and Q 2 = Q : > VERIF:=simplify(evalm(Mult(P,P,A)-P)); > VERIF2:=simplify(evalm(Mult(Q,Q,A)-Q)); VERIF := Using the fat that matries P and Q are idempotents of A 3, we obtain that the A-modules ker A (.P ), im A (.P ) = ker A (.(I 3 P )), ker A (.Q ) and im A (.Q ) = ker A (.(I 3 Q )) are projetive, and thus, free by the Quillen-Suslin theorem. As the entries of P and Q only belong to the field Q(u, ρ, ), using linear algebrai tehniques, we an easily ompute bases of the orresponding Q(u, ρ, )-vetor spaes, and thus, bases over the ring A: > U:=SyzygyModule(P,A): U2:=SyzygyModule(evalm(-P),A): > U:=stakmatrix(U,U2); > V:=SyzygyModule(Q,A): V2:=SyzygyModule(evalm(-Q),A): > V:=stakmatrix(V,V2); U := 2 V := u The matries U GL 3 (A) and V GL 3 (A) are suh that the matries U P U and V Q V are two blok-diagonal matries formed by the diagonal matries 2 and : > VERIF:=Mult(U,P,LeftInverse(U,A),A); > VERIF2:=Mult(V,Q,LeftInverse(V,A),A); VERIF :=, 2

3 Then, the matrix R is equivalent to the blok-diagonal matrix V R U defined by: > R_de:=Mult(V,R,LeftInverse(U,A),A); ( d x 2 + u 2) d y R de := 2 d y d x d x This last result an be obtained using the ommand HeuristiDeomposition: > HeuristiDeomposition(R,P,A); ( d x 2 + u 2) d y 2 d y d x d x Let us now onsider the first 2 2 blok-diagonal matrix S of R de defined by: > S:=submatrix(R_de,..2,..2); ( dx 2 + u 2) d y S := 2 d y d x Let us try hek whether or not the matrix S is equivalent to a blok-diagonal matrix. To do that, we introdue the A-module L = A 2 /(A 2 S) finitely presented by the matrix S and ompute the endomorphism ring F = end A (L) of L: > Endo:=MorphismsConstCoeff(S,S,A): Endo; Endo2; u 2 ρ 2 dy ( u 2 ρ d x + u 3 d x ρ u 2), d x 2 d y Let us hek whether or not we an find idempotents of F defined by means of onstant matries: > Idem:=IdempotentsConstCoeff(S,Endo,A,,alpha); α Idem :=,, α ( 2 +u 2 ), Ore algebra, diff, diff, x, y, d x, d y, x, y, u, ρ,, α,,, 4 α α 2 u 2, x, y,,, diff = d x, x, diff = d y, y We obtain the two trivial idempotents and id F of F and an idempotent of end B (B A L), where B = Q(u, ρ, )α /(4 (u 2 2 ) α 2 )d x, d y, defined by the following matrix: > B:=Idem2: P2:=Idem,; Q2:=Fatorize(Mult(S,P2,B),S,B); α α 2 α u 2 P 2 := Q α ( 2 +u 2 ) 2 := α We an hek that the matries P 2 and Q 2 satisfy P 2 2 = P 2 and Q 2 2 = Q 2 : > VERIF:=simplify(subs(alpha^2=/4/(u^2-^2),alpha^4=/6/(u^2-^2)^2, > simplify(evalm(mult(p2,p2,b)-p2)))); > VERIF2:=simplify(subs(alpha^2=/4/(u^2-^2),alpha^4=/6/(u^2-^2)^2, > simplify(evalm(mult(q2,q2,b)-q2)))); VERIF := 3

4 As the matries P 2 and Q 2 are idempotents of B 2 2, we know that the B-modules ker B (.P 2 ), im B (.P 2 ) = ker B (.(I 2 P 2 )), ker B (.Q 2 ) and im B (.Q 2 ) = ker B (.(I 2 Q 2 )) are projetive, and thus, free by the Quillen-Suslin theorem. Let us ompute bases of those free B-modules: > X:=SyzygyModule(P2,B):X2:=SyzygyModule(evalm(-P2),B): > X:=stakmatrix(X,X2); > Y:=SyzygyModule(Q2,B): Y2:=SyzygyModule(evalm(-Q2),B): > Y:=stakmatrix(Y,Y2); 2 α u α 3 2 α X := Y := 2 α α u 2 2 α We an easily hek that X P 2 X and Y Q 2 Y are the blok-diagonal matries diag(, ): > VERIF:=simplify(subs(alpha^2=/4/(u^2-^2),Mult(X,P2, > LeftInverse(X,B),B))); > VERIF2:=simplify(subs(alpha^2=/4/(u^2-^2),Mult(Y,Q2, > LeftInverse(Y,B),B))); VERIF := Then, the matrix S is equivalent to the blok-diagonal matrix Y S X defined by: > S_de:=simplify(subs(alpha^2=/4/(u^2-^2),Mult(Y,S,LeftInverse(X,B),B))); S de := This last result an be diretly obtained as follows: d y+2 d x α 2 2 d x α u 2 d y 2 d x α 2 +2 d x α u 2 > simplify(subs(alpha^2=/4/(u^2-^2),heuristideomposition(s,p2,b) > )); If we denote by d y+2 d x α 2 2 d x α u 2 d y 2 d x α 2 +2 d x α u 2 > G:=diag(X,): H:=diag(Y,): Z:=Mult(G,U,B); T:=Mult(H,V,B); 2 α u α 3 2 α 2 α u Z := 2 α α u 2 T := 2 α 2 α u 2 then the matrix R is equivalent to the simple blok-diagonal matrix T R Z defined by: > simplify(subs(alpha^2=/4/(u^2-^2),simplify(mult(t,r,leftinverse(z,b), > B)))); d y+2 d x α 2 2 d x α u 2 d y 2 d x α 2 +2 d x α u 2 d x 4

5 If F denotes an A-module (e.g., F = C (R 2 )), using the relation 2 α ( 2 u 2 ) = /(2 α ), we then obtain that the linear system ker F (R.) is equivalent to the following one (d x 2 α d y ) ζ =, ζ = φ(y + 2 α x), (d x + 2 α d y ) ζ 2 =, ζ 2 = ψ(y 2 α x), d x ζ 3 =, ζ 3 = C, where φ and ψ are two arbitrary funtions of F and C an arbitrary onstant. 5