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Sun Jan 3 23:51:41 MET 1999 REDUCE 3.7, 15-Jan-99 ... 1: 1: 2: 2: 2: 2: 2: 2: 2: 2: 2: 3: 3: off echo, dfprint$ ------------------------------------------------------- The following runs demonstrate the program LIEPDE for the computation of infinitesimal symmetries. Times given below refer to a 8 MB session under LINUX on a 133 MHz Pentium PC with the CRACK version of April 1998. ------------------------------------------------------- The first example is a single ODE with a parametric function f=f(x) for which point symmetries are to be determined. (Time ~ 6 sec.) ------------------------------------------------------- The ODE under investigation is : 2 2 3 df(y,x,2)= - df(f,x)*y - 3*df(y,x)*f - df(y,x)*y - 2*f *y - f*y + y for the function(s) : y(x) The symmetries are: -------- 1. Symmetry: int(f,x) 1 xi_x=e *int(-----------,x) int(f,x) e int(f,x) 1 eta_y= - e *int(-----------,x)*f*y - y int(f,x) e -------- 2. Symmetry: int(f,x) xi_x= - e int(f,x) eta_y=e *f*y -------- ------------------------------------------------------- The following example demonstrates a number of things. The Burgers equation is investigated concerning third order symmetries. The equation is used to substitute df(u,t) and all derivatives of df(u,t). This computation also shows that any equations that remain unsolved are returned, like in this case the heat quation. (Time ~ 15 sec.) ------------------------------------------------------- The PDE under investigation is : 2 df(u,t)=df(u,x,2) + df(u,x) for the function(s) : u(x,t) The symmetries are: -------- 1. Symmetry: xi_t=0 xi_x=0 2 eta_u=df(u,x,2) + df(u,x) -------- 2. Symmetry: xi_t=0 xi_x=0 2 2 2 2 eta_u=4*df(u,x,2)*t + 4*df(u,x) *t + 4*df(u,x)*t*x + 2*t + x -------- 3. Symmetry: xi_t=0 xi_x=0 2 eta_u=4*df(u,x,2)*t + 4*df(u,x) *t + 2*df(u,x)*x - 1 -------- 4. Symmetry: xi_t=0 xi_x=0 3 eta_u=df(u,x,3) + 3*df(u,x,2)*df(u,x) + df(u,x) -------- 5. Symmetry: xi_t=0 xi_x=0 2 2 3 2 eta_u=4*df(u,x,3)*t + 12*df(u,x,2)*df(u,x)*t + 4*df(u,x,2)*t*x + 4*df(u,x) *t 2 2 + 4*df(u,x) *t*x + df(u,x)*x - x -------- 6. Symmetry: xi_t=0 xi_x=0 3 3 2 eta_u=8*df(u,x,3)*t + 24*df(u,x,2)*df(u,x)*t + 12*df(u,x,2)*t *x 3 3 2 2 2 2 + 8*df(u,x) *t + 12*df(u,x) *t *x + 12*df(u,x)*t + 6*df(u,x)*t*x + 6*t*x 3 + x -------- 7. Symmetry: xi_t=0 xi_x=0 eta_u 3 2 =2*df(u,x,3)*t + 6*df(u,x,2)*df(u,x)*t + df(u,x,2)*x + 2*df(u,x) *t + df(u,x) *x -------- 8. Symmetry: xi_t=0 xi_x=0 eta_u=df(u,x) -------- 9. Symmetry: xi_t=0 xi_x=0 eta_u=2*df(u,x)*t + x -------- 10. Symmetry: xi_t=0 xi_x=0 eta_u=1 -------- Further symmetries: xi_t=0 xi_x=0 c_27 + c_32 eta_u=------------- u e with c_27(x,t), c_32(t) which still have to satisfy: 0=2*df( - c_27,t) - 2*df( - c_27,x,2) + df( - 2*c_32,t) ------------------------------------------------------- Now the same equation is investigated, this time only df(u,x,2) and its derivatives are substituted. As a consequence less jet-variables (u-derivatives of lower order) are generated in the process of formulating the symmetry conditions. Less jet-variables in which the conditions have to be fulfilled identically means less overdetermined conditions and more solutions which to compute takes longer than before. (Time ~ 85 sec.) ------------------------------------------------------- The PDE under investigation is : 2 df(u,x,2)=df(u,t) - df(u,x) for the function(s) : u(x,t) The symmetries are: -------- 1. Symmetry: xi_t=0 xi_x=0 eta_u 2 = - 2*df(u,t,x)*df(u,t) - df(u,t,2,x) - df(u,t,2)*df(u,x) - df(u,t) *df(u,x) -------- 2. Symmetry: xi_t=0 xi_x=0 2 2 2 eta_u= - 16*df(u,t,x)*df(u,t)*t - 2*df(u,t,x)*x - 8*df(u,t,2,x)*t 2 2 2 - 8*df(u,t,2)*df(u,x)*t - 8*df(u,t,2)*t*x - 8*df(u,t) *df(u,x)*t 2 2 - 8*df(u,t) *t*x - 2*df(u,t)*df(u,x)*x + 2*df(u,t)*x - df(u,x) -------- 3. Symmetry: xi_t=0 xi_x=0 4 2 2 4 eta_u= - 32*df(u,t,x)*df(u,t)*t - 24*df(u,t,x)*t *x - 16*df(u,t,2,x)*t 4 3 2 4 - 16*df(u,t,2)*df(u,x)*t - 32*df(u,t,2)*t *x - 16*df(u,t) *df(u,x)*t 2 3 2 2 2 - 32*df(u,t) *t *x - 24*df(u,t)*df(u,x)*t *x + 24*df(u,t)*t *x 3 2 2 4 3 - 8*df(u,t)*t*x + 60*df(u,x)*t + 24*df(u,x)*t*x - df(u,x)*x + 36*t*x + 6*x -------- 4. Symmetry: xi_t=0 xi_x=0 5 4 3 2 eta_u= - 64*df(u,t,x)*df(u,t)*t - 160*df(u,t,x)*t - 80*df(u,t,x)*t *x 5 5 4 - 32*df(u,t,2,x)*t - 32*df(u,t,2)*df(u,x)*t - 80*df(u,t,2)*t *x 2 5 2 4 4 - 32*df(u,t) *df(u,x)*t - 80*df(u,t) *t *x - 160*df(u,t)*df(u,x)*t 3 2 3 2 3 - 80*df(u,t)*df(u,x)*t *x - 240*df(u,t)*t *x - 40*df(u,t)*t *x 3 2 2 4 2 3 5 - 120*df(u,x)*t - 120*df(u,x)*t *x - 10*df(u,x)*t*x - 60*t *x - 20*t*x - x -------- 5. Symmetry: xi_t=0 xi_x=0 3 2 3 eta_u= - 32*df(u,t,x)*df(u,t)*t - 12*df(u,t,x)*t*x - 16*df(u,t,2,x)*t 3 2 2 3 - 16*df(u,t,2)*df(u,x)*t - 24*df(u,t,2)*t *x - 16*df(u,t) *df(u,x)*t 2 2 2 3 - 24*df(u,t) *t *x - 12*df(u,t)*df(u,x)*t*x + 12*df(u,t)*t*x - 2*df(u,t)*x 2 - 6*df(u,x)*t + 6*df(u,x)*x - 9*x -------- 6. Symmetry: xi_t=0 xi_x=0 eta_u= - 4*df(u,t,x)*df(u,t)*t - 2*df(u,t,2,x)*t - 2*df(u,t,2)*df(u,x)*t 2 2 - df(u,t,2)*x - 2*df(u,t) *df(u,x)*t - df(u,t) *x -------- 7. Symmetry: xi_t=0 xi_x=0 3 eta_u= - df(u,t,3) - 3*df(u,t,2)*df(u,t) - df(u,t) -------- 8. Symmetry: xi_t=0 xi_x=0 2 eta_u= - 8*df(u,t,x)*df(u,t)*t*x + df(u,t,x)*x - 4*df(u,t,3)*t 2 - 4*df(u,t,2,x)*t*x - 12*df(u,t,2)*df(u,t)*t - 4*df(u,t,2)*df(u,x)*t*x 2 3 2 2 2 2 - df(u,t,2)*x - 4*df(u,t) *t - 4*df(u,t) *df(u,x)*t*x - df(u,t) *x + df(u,t)*df(u,x)*x -------- 9. Symmetry: xi_t=0 xi_x=0 3 2 3 eta_u= - 64*df(u,t,x)*df(u,t)*t *x + 24*df(u,t,x)*t *x - 8*df(u,t,x)*t*x 4 3 4 - 16*df(u,t,3)*t - 32*df(u,t,2,x)*t *x - 48*df(u,t,2)*df(u,t)*t 3 2 2 3 4 - 32*df(u,t,2)*df(u,x)*t *x - 24*df(u,t,2)*t *x - 16*df(u,t) *t 2 3 2 2 2 2 - 32*df(u,t) *df(u,x)*t *x - 24*df(u,t) *t *x + 24*df(u,t)*df(u,x)*t *x 3 2 4 - 8*df(u,t)*df(u,x)*t*x + 24*df(u,t)*t*x - df(u,t)*x - 24*df(u,x)*t*x 3 2 + 6*df(u,x)*x - 30*t - 15*x -------- 10. Symmetry: xi_t=0 xi_x=0 5 4 3 3 eta_u= - 384*df(u,t,x)*df(u,t)*t *x - 960*df(u,t,x)*t *x - 160*df(u,t,x)*t *x 6 5 6 - 64*df(u,t,3)*t - 192*df(u,t,2,x)*t *x - 192*df(u,t,2)*df(u,t)*t 5 5 4 2 - 192*df(u,t,2)*df(u,x)*t *x - 480*df(u,t,2)*t - 240*df(u,t,2)*t *x 3 6 2 5 2 5 - 64*df(u,t) *t - 192*df(u,t) *df(u,x)*t *x - 480*df(u,t) *t 2 4 2 4 3 3 - 240*df(u,t) *t *x - 960*df(u,t)*df(u,x)*t *x - 160*df(u,t)*df(u,x)*t *x 4 3 2 2 4 3 - 720*df(u,t)*t - 720*df(u,t)*t *x - 60*df(u,t)*t *x - 720*df(u,x)*t *x 2 3 5 3 2 2 4 6 - 240*df(u,x)*t *x - 12*df(u,x)*t*x - 120*t - 180*t *x - 30*t*x - x -------- 11. Symmetry: xi_t=0 xi_x=0 4 3 2 3 eta_u= - 160*df(u,t,x)*df(u,t)*t *x + 80*df(u,t,x)*t *x - 40*df(u,t,x)*t *x 5 4 5 - 32*df(u,t,3)*t - 80*df(u,t,2,x)*t *x - 96*df(u,t,2)*df(u,t)*t 4 3 2 3 5 - 80*df(u,t,2)*df(u,x)*t *x - 80*df(u,t,2)*t *x - 32*df(u,t) *t 2 4 2 3 2 3 - 80*df(u,t) *df(u,x)*t *x - 80*df(u,t) *t *x + 80*df(u,t)*df(u,x)*t *x 2 3 3 2 2 - 40*df(u,t)*df(u,x)*t *x + 360*df(u,t)*t + 120*df(u,t)*t *x 4 2 3 5 2 - 10*df(u,t)*t*x + 420*df(u,x)*t *x + 60*df(u,x)*t*x - df(u,x)*x + 120*t 2 4 + 120*t*x + 10*x -------- 12. Symmetry: xi_t=0 xi_x=0 2 3 eta_u= - 48*df(u,t,x)*df(u,t)*t *x + 12*df(u,t,x)*t*x - 2*df(u,t,x)*x 3 2 3 - 16*df(u,t,3)*t - 24*df(u,t,2,x)*t *x - 48*df(u,t,2)*df(u,t)*t 2 2 3 3 - 24*df(u,t,2)*df(u,x)*t *x - 12*df(u,t,2)*t*x - 16*df(u,t) *t 2 2 2 2 - 24*df(u,t) *df(u,x)*t *x - 12*df(u,t) *t*x + 12*df(u,t)*df(u,x)*t*x 3 2 - 2*df(u,t)*df(u,x)*x + 6*df(u,t)*x - 6*df(u,x)*x + 3 -------- 13. Symmetry: xi_t=0 xi_x=0 eta_u= - 2*df(u,t,x)*df(u,t)*x - 2*df(u,t,3)*t - df(u,t,2,x)*x 3 - 6*df(u,t,2)*df(u,t)*t - df(u,t,2)*df(u,x)*x - 2*df(u,t) *t 2 - df(u,t) *df(u,x)*x -------- 14. Symmetry: xi_t=0 xi_x=0 2 eta_u=df(u,t,2) + df(u,t) -------- 15. Symmetry: xi_t=0 xi_x=0 2 2 2 eta_u= - 8*df(u,t,x)*t*x - 8*df(u,t,2)*t - 8*df(u,t) *t 2 - 8*df(u,t)*df(u,x)*t*x - 2*df(u,t)*x + 2*df(u,x)*x - 1 -------- 16. Symmetry: xi_t=0 xi_x=0 3 4 2 4 eta_u= - 32*df(u,t,x)*t *x - 16*df(u,t,2)*t - 16*df(u,t) *t 3 3 2 2 2 - 32*df(u,t)*df(u,x)*t *x - 48*df(u,t)*t - 24*df(u,t)*t *x - 48*df(u,x)*t *x 3 2 2 4 - 8*df(u,x)*t*x - 12*t - 12*t*x - x -------- 17. Symmetry: xi_t=0 xi_x=0 2 3 2 3 eta_u= - 12*df(u,t,x)*t *x - 8*df(u,t,2)*t - 8*df(u,t) *t 2 2 3 - 12*df(u,t)*df(u,x)*t *x - 6*df(u,t)*t*x + 6*df(u,x)*t*x - df(u,x)*x + 6*t 2 + 3*x -------- 18. Symmetry: xi_t=0 xi_x=0 2 eta_u= - df(u,t,x)*x - 2*df(u,t,2)*t - 2*df(u,t) *t - df(u,t)*df(u,x)*x -------- 19. Symmetry: xi_t=0 xi_x=0 eta_u=df(u,t,x) + df(u,t)*df(u,x) -------- 20. Symmetry: xi_t=0 xi_x=0 2 2 eta_u= - 4*df(u,t,x)*t - 4*df(u,t)*df(u,x)*t - 4*df(u,t)*t*x + 2*df(u,x)*t 2 - df(u,x)*x + 2*x -------- 21. Symmetry: xi_t=0 xi_x=0 3 3 2 2 eta_u= - 8*df(u,t,x)*t - 8*df(u,t)*df(u,x)*t - 12*df(u,t)*t *x - 12*df(u,x)*t 2 3 - 6*df(u,x)*t*x - 6*t*x - x -------- 22. Symmetry: xi_t=0 xi_x=0 eta_u= - 4*df(u,t,x)*t - 4*df(u,t)*df(u,x)*t - 2*df(u,t)*x + df(u,x) -------- 23. Symmetry: xi_t=0 xi_x=0 eta_u=df(u,t) -------- 24. Symmetry: xi_t=0 xi_x=0 2 2 eta_u= - 4*df(u,t)*t - 4*df(u,x)*t*x - 2*t - x -------- 25. Symmetry: xi_t=0 xi_x=0 eta_u= - 2*df(u,t)*t - df(u,x)*x + 1 -------- 26. Symmetry: xi_t=0 xi_x=0 eta_u= - 2*df(u,x)*t - x -------- 27. Symmetry: xi_t=0 xi_x=0 eta_u=df(u,x) -------- 28. Symmetry: xi_t=0 xi_x=0 eta_u=1 -------- Further symmetries: xi_t=0 xi_x=0 c_92 eta_u=------ u e with c_92(x,t) which still have to satisfy: 0=df(c_92,t) - df(c_92,x,2) ------------------------------------------------------- The following example includes the Karpman equations for three unknown functions in 4 variables. If point symmetries are to be computed for a single equation or a system of equations of higher than first order then there is the option to formulate at first preliminary conditions for each equation, have CRACK solving these conditions before the full set of conditions is formulated and solved. This strategy is adopted if a lisp flag prelim_ has the value t. The default value is nil. Similarly, if a system of equations is to be investigated and a flag individual_ has the value t then symmetry conditions are formulated and investigated for each individual equation successively. The default value is nil. It is advantageous to split a large set of conditions into smaller sets to be investigated successively if each set is sufficiently overdetermined to be solvable quickly. Then any substitutions are done in the smaller set and the next set of conditions is shorter. For example, for the Karpman equations below the speedup for prelim_:=t; individual_:=t; is a factor of 10. (Time ~ 1 min.) ------------------------------------------------------- Time: 41810 ms plus GC time: 2430 ms The PDE-system under investigation is : 2 2 2 2 2 df(v,x,2)=( - 4*df(f,t)*a2*r - 2*df(f,x) *a2*r *s1 - 2*df(f,y) *a2*r *s1 2 2 2 2 - 2*df(f,z) *a2*r *s2 - 4*df(f,z)*a2*r *w1 - 2*df(r,x) *a2*s1 2 - 2*df(r,y) *a2*s1 - 2*df(r,z,2)*a2*r*s1 + 2*df(r,z,2)*a2*r*s2 2 2 - 2*df(r,z) *a2*s1 + df(v,t,2)*s1 - df(v,y,2)*s1*w2 2 2 2 - df(v,z,2)*s1*w2 - 4*a1*a2*r *v)/(s1*w2 ) 2 2 2 df(r,x,2)=(2*df(f,t)*r + df(f,x) *r*s1 + df(f,y) *r*s1 + df(f,z) *r*s2 + 2*df(f,z)*r*w1 - df(r,y,2)*s1 - df(r,z,2)*s2 + 2*a1*r*v)/s1 df(f,x,2)=( - 2*df(f,x)*df(r,x)*s1 - df(f,y,2)*r*s1 - 2*df(f,y)*df(r,y)*s1 - df(f,z,2)*r*s2 - 2*df(f,z)*df(r,z)*s2 - 2*df(r,t) - 2*df(r,z)*w1)/ (r*s1) for the function(s) : r(t,z,y,x), f(t,z,y,x), v(t,z,y,x) =============== Initializations time for initializations: 340 ms GC time : 40 ms =============== Preconditions for the 1. equation time to formulate conditions: 2180 ms GC time : 90 ms CRACK needed : 5550 ms GC time : 420 ms =============== Preconditions for the 2. equation =============== Preconditions for the 3. equation time to formulate conditions: 800 ms GC time : 50 ms CRACK needed : 1270 ms GC time : 100 ms =============== Full conditions for the 1. equation time to formulate conditions: 510 ms GC time : 50 ms CRACK needed : 11880 ms GC time : 870 ms =============== Full conditions for the 2. equation time to formulate conditions: 170 ms GC time : 0 ms CRACK needed : 520 ms GC time : 50 ms =============== Full conditions for the 3. equation time to formulate conditions: 260 ms GC time : 0 ms CRACK needed : 680 ms GC time : 60 ms The symmetries are: -------- 1. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=0 eta_r=0 - t eta_f=------- s1*s2 1 eta_v=---------- a1*s1*s2 -------- 2. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=0 eta_r=0 2 - t eta_f=------- s1*s2 2*t eta_v=---------- a1*s1*s2 -------- 3. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=0 eta_r=0 1 eta_f=------- s1*s2 eta_v=0 -------- 4. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=1 eta_r=0 eta_f=0 eta_v=0 -------- 5. Symmetry: xi_x=0 xi_y=0 1 xi_z=---- s1 xi_t=0 eta_r=0 - w1 eta_f=------- s1*s2 eta_v=0 -------- 6. Symmetry: xi_x=0 1 xi_y=------- s1*s2 xi_z=0 xi_t=0 eta_r=0 eta_f=0 eta_v=0 -------- 7. Symmetry: y xi_x=------- s1*s2 - x xi_y=------- s1*s2 xi_z=0 xi_t=0 eta_r=0 eta_f=0 eta_v=0 -------- 8. Symmetry: 1 xi_x=------- s1*s2 xi_y=0 xi_z=0 xi_t=0 eta_r=0 eta_f=0 eta_v=0 -------- Time: 23830 ms plus GC time: 1840 ms ------------------------------------------------------- In the following example a system of two equations (by V.Sokolov) is investigated concerning a special ansatz for 4th order symmetries. The ansatz for the symmetries includes two unknown functions f,g. Because x is the second variable in the list of variables {t,x}, the name u!`2 stands for df(u,x). Because higher order symmetries are investigated we have to set prelim_:=nil. The symmetries to be calculated are lengthy and therefore conditions are not very overdetermined. In that case CRACK can take long to solve a single subset of conditions. The complete set of conditions would have been more overdetermined and easier to solve. Therefore the advantage of first formulating all conditions and then solving them together with one CRACK call is that having more equations, the chance of finding short integrable equations among then is higher, i.e. CRACK has more freedom in optimizing the computation. Therefore individual_:=nil is more appropriate in this example. Because 4th order conditions are to be computed the `binding stack size' is increased. (Time ~ 5 min.) ------------------------------------------------------- The PDE-system under investigation is : df(u,t)=df(u,x,2) + df(u,x)*u + df(u,x)*v + df(v,x)*u df(v,t)=df(u,x)*v - df(v,x,2) + df(v,x)*u + df(v,x)*v for the function(s) : u(t,x), v(t,x) The symmetries are: -------- 1. Symmetry: xi_t=0 xi_x=0 eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 10*df(u,x,2)*df(u,x) 2 + 6*df(u,x,2)*df(v,x) + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v 2 2 2 + 3*df(u,x,2)*v + 2*df(u,x,2) + 6*df(u,x) *u + 9*df(u,x) *v + 4*df(u,x)*df(v,x,2) + 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v 3 2 2 + df(u,x)*u + 9*df(u,x)*u *v + 9*df(u,x)*u*v + 2*df(u,x)*u 3 3 + df(u,x)*v + 2*df(u,x)*v + 2*df(v,x,3)*u + 3*df(v,x)*u 2 2 + 9*df(v,x)*u *v + 3*df(v,x)*u*v + 2*df(v,x)*u)/2 eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2) 2 - 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v + 3*df(u,x)*u *v 2 3 + 9*df(u,x)*u*v + 3*df(u,x)*v + 2*df(u,x)*v - 2*df(v,x,4) 2 + 4*df(v,x,3)*u + 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x) - 3*df(v,x,2)*u 2 2 - 9*df(v,x,2)*u*v - 3*df(v,x,2)*v - 2*df(v,x,2) - 9*df(v,x) *u 2 3 2 2 - 6*df(v,x) *v + df(v,x)*u + 9*df(v,x)*u *v + 9*df(v,x)*u*v 3 + 2*df(v,x)*u + df(v,x)*v + 2*df(v,x)*v)/2 -------- 2. Symmetry: xi_t=0 xi_x=0 eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 10*df(u,x,2)*df(u,x) 2 + 6*df(u,x,2)*df(v,x) + 4*df(u,x,2)*t + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v 2 2 2 + 3*df(u,x,2)*v + 6*df(u,x) *u + 9*df(u,x) *v + 4*df(u,x)*df(v,x,2) + 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v + 4*df(u,x)*t*u 3 2 2 + 4*df(u,x)*t*v + df(u,x)*u + 9*df(u,x)*u *v + 9*df(u,x)*u*v 3 + df(u,x)*v + 2*df(u,x)*x + 2*df(v,x,3)*u + 4*df(v,x)*t*u 3 2 2 + 3*df(v,x)*u + 9*df(v,x)*u *v + 3*df(v,x)*u*v + 2*u)/2 eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2) - 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v + 4*df(u,x)*t*v 2 2 3 + 3*df(u,x)*u *v + 9*df(u,x)*u*v + 3*df(u,x)*v - 2*df(v,x,4) + 4*df(v,x,3)*u + 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x) - 4*df(v,x,2)*t 2 2 2 - 3*df(v,x,2)*u - 9*df(v,x,2)*u*v - 3*df(v,x,2)*v - 9*df(v,x) *u 2 3 - 6*df(v,x) *v + 4*df(v,x)*t*u + 4*df(v,x)*t*v + df(v,x)*u 2 2 3 + 9*df(v,x)*u *v + 9*df(v,x)*u*v + df(v,x)*v + 2*df(v,x)*x + 2*v)/2 -------- 3. Symmetry: xi_t=0 xi_x=0 eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 8*df(u,x,3) 2 + 10*df(u,x,2)*df(u,x) + 6*df(u,x,2)*df(v,x) + 3*df(u,x,2)*u 2 + 9*df(u,x,2)*u*v + 12*df(u,x,2)*u + 3*df(u,x,2)*v + 12*df(u,x,2)*v 2 2 2 + 6*df(u,x) *u + 9*df(u,x) *v + 12*df(u,x) + 4*df(u,x)*df(v,x,2) + 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v + 12*df(u,x)*df(v,x) 3 2 2 2 + df(u,x)*u + 9*df(u,x)*u *v + 6*df(u,x)*u + 9*df(u,x)*u*v 3 2 + 24*df(u,x)*u*v + df(u,x)*v + 6*df(u,x)*v + 2*df(v,x,3)*u 3 2 2 2 + 3*df(v,x)*u + 9*df(v,x)*u *v + 12*df(v,x)*u + 3*df(v,x)*u*v + 12*df(v,x)*u*v)/2 eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2) - 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v - 12*df(u,x)*df(v,x) 2 2 3 + 3*df(u,x)*u *v + 9*df(u,x)*u*v + 12*df(u,x)*u*v + 3*df(u,x)*v 2 + 12*df(u,x)*v - 2*df(v,x,4) + 4*df(v,x,3)*u + 4*df(v,x,3)*v 2 + 8*df(v,x,3) + 10*df(v,x,2)*df(v,x) - 3*df(v,x,2)*u - 9*df(v,x,2)*u*v 2 2 - 12*df(v,x,2)*u - 3*df(v,x,2)*v - 12*df(v,x,2)*v - 9*df(v,x) *u 2 2 3 2 - 6*df(v,x) *v - 12*df(v,x) + df(v,x)*u + 9*df(v,x)*u *v 2 2 3 + 6*df(v,x)*u + 9*df(v,x)*u*v + 24*df(v,x)*u*v + df(v,x)*v 2 + 6*df(v,x)*v )/2 -------- 4. Symmetry: xi_t=0 xi_x=0 eta_u=(2*df(u,x,4) + 8*df(u,x,3)*t + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 10*df(u,x,2)*df(u,x) + 6*df(u,x,2)*df(v,x) + 12*df(u,x,2)*t*u 2 2 + 12*df(u,x,2)*t*v + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v + 3*df(u,x,2)*v 2 2 2 + 4*df(u,x,2)*x + 12*df(u,x) *t + 6*df(u,x) *u + 9*df(u,x) *v + 4*df(u,x)*df(v,x,2) + 12*df(u,x)*df(v,x)*t + 9*df(u,x)*df(v,x)*u 2 + 6*df(u,x)*df(v,x)*v + 6*df(u,x)*t*u + 24*df(u,x)*t*u*v 2 3 2 2 + 6*df(u,x)*t*v + df(u,x)*u + 9*df(u,x)*u *v + 9*df(u,x)*u*v 3 + 4*df(u,x)*u*x + df(u,x)*v + 4*df(u,x)*v*x + 16*df(u,x) 2 3 + 2*df(v,x,3)*u + 12*df(v,x)*t*u + 12*df(v,x)*t*u*v + 3*df(v,x)*u 2 2 2 + 9*df(v,x)*u *v + 3*df(v,x)*u*v + 4*df(v,x)*u*x + 2*u + 6*u*v)/2 eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2) - 12*df(u,x)*df(v,x)*t - 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v 2 2 2 + 12*df(u,x)*t*u*v + 12*df(u,x)*t*v + 3*df(u,x)*u *v + 9*df(u,x)*u*v 3 + 3*df(u,x)*v + 4*df(u,x)*v*x - 2*df(v,x,4) + 8*df(v,x,3)*t + 4*df(v,x,3)*u + 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x) 2 - 12*df(v,x,2)*t*u - 12*df(v,x,2)*t*v - 3*df(v,x,2)*u - 9*df(v,x,2)*u*v 2 2 2 - 3*df(v,x,2)*v - 4*df(v,x,2)*x - 12*df(v,x) *t - 9*df(v,x) *u 2 2 2 - 6*df(v,x) *v + 6*df(v,x)*t*u + 24*df(v,x)*t*u*v + 6*df(v,x)*t*v 3 2 2 + df(v,x)*u + 9*df(v,x)*u *v + 9*df(v,x)*u*v + 4*df(v,x)*u*x 3 2 + df(v,x)*v + 4*df(v,x)*v*x + 6*u*v + 2*v )/2 -------- 5. Symmetry: xi_t=0 xi_x=0 eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 10*df(u,x,2)*df(u,x) 2 + 6*df(u,x,2)*df(v,x) + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v 2 2 2 + 3*df(u,x,2)*v + 6*df(u,x) *u + 9*df(u,x) *v + 4*df(u,x)*df(v,x,2) 3 + 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v + df(u,x)*u 2 2 3 + 9*df(u,x)*u *v + 9*df(u,x)*u*v + df(u,x)*v + 2*df(u,x) 3 2 2 + 2*df(v,x,3)*u + 3*df(v,x)*u + 9*df(v,x)*u *v + 3*df(v,x)*u*v )/2 eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2) 2 - 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v + 3*df(u,x)*u *v 2 3 + 9*df(u,x)*u*v + 3*df(u,x)*v - 2*df(v,x,4) + 4*df(v,x,3)*u 2 + 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x) - 3*df(v,x,2)*u 2 2 2 - 9*df(v,x,2)*u*v - 3*df(v,x,2)*v - 9*df(v,x) *u - 6*df(v,x) *v 3 2 2 3 + df(v,x)*u + 9*df(v,x)*u *v + 9*df(v,x)*u*v + df(v,x)*v + 2*df(v,x)) /2 -------- 4: 4: 4: 4: 4: 4: 4: 4: 4: Time for test: 141150 ms, plus GC time: 10660 ms 5: 5: Quitting Sun Jan 3 23:54:19 MET 1999