@@ -1,403 +1,403 @@ -REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... - - -off echo; - -------------------------------------------------------- -This file is supposed to provide an automatic test of -the program APPLYSYM. On the other hand the application -of APPLYSYM is an interactive process, therefore the -interested user should inspect the example described -in APPLYSYM.TEX which demonstrates the application -of symmetries to integrate a 2nd order ODE. -Here the program QUASILINPDE for integrating first -order quasilinear PDE is demonstrated. - -The following equation comes up in the elimination -of resonant terms in normal forms of singularities -of vector fields (C.Herssens, P.Bonckaert, Limburgs -Universitair Centrum/Belgium, private communication). -------------------------------------------------------- - -The quasilinear PDE: 0 = df(w,x)*x + df(w,y)*y + 2*df(w,z)*z - 2*w - x*y. -The equivalent characteristic system: - -0=2*df(w,z)*z - 2*w - x*y - - -0=2*df(y,z)*z - y - - -0=2*df(x,z)*z - x - -for the functions: y(z) x(z) w(z) . -The general solution of the PDE is given through - - sqrt(z)*y sqrt(z)*x - log(z)*x*y + 2*w -0 = ff(-----------,-----------,---------------------) - z z z - -with arbitrary function ff(..). - -------------------------------------------------------- -Comment: -The result means that w is defined implicitly through - - - log(z)*x*y + 2*w sqrt(z)*x sqrt(z)*y -0 = ff(---------------------,-----------,-----------) - z z z - -with an arbitrary function ff of 3 arguments. As the PDE -was linear, the arguments of ff are such that we can -solve for w: - - sqrt(z)*x sqrt(z)*y -w = log(z)*x*y/2 + z*f(-----------,-----------) - z z - -with an arbitrary function f of 2 arguments. - -------------------------------------------------------- -The following PDEs are taken from E. Kamke, -Loesungsmethoden und Loesungen von Differential- -gleichungen, Partielle Differentialgleichungen -erster Ordnung, B.G. Teubner, Stuttgart (1979). - -------------------- equation 1.4 ---------------------- - -The quasilinear PDE: 0 = df(z,x)*x - y. -The equivalent characteristic system: - -0=df(z,x)*x - y - - -0=df(y,x)*x - -for the functions: y(x) z(x) . -The general solution of the PDE is given through - -0 = ff(y,log(x)*y - z) - -with arbitrary function ff(..). - -------------------- equation 2.5 ---------------------- - - 2 2 -The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y . -The equivalent characteristic system: - - 2 -0=df(z,y)*y - - - 2 2 -0=df(x,y)*y - x - -for the functions: x(y) z(y) . -The general solution of the PDE is given through - - - x + y -0 = ff(----------,z) - x*y - -with arbitrary function ff(..). - -------------------- equation 2.6 ---------------------- - - 2 2 -The quasilinear PDE: 0 = df(z,x)*x - df(z,x)*y + 2*df(z,y)*x*y. -The equivalent characteristic system: - -0=2*df(z,y)*x*y - - - 2 2 -0=2*df(x,y)*x*y - x + y - -for the functions: x(y) z(y) . -The general solution of the PDE is given through - - 2 2 - - x - y -0 = ff(------------,z) - y - -with arbitrary function ff(..). - -------------------- equation 2.7 ---------------------- - -The quasilinear PDE: 0 = df(z,x)*a0*x - df(z,x)*a1 + df(z,y)*a0*y - df(z,y)*a2. -The equivalent characteristic system: - -0=df(z,x)*(a0*x - a1) - - -0=df(y,x)*a0*x - df(y,x)*a1 - a0*y + a2 - -for the functions: y(x) z(x) . -The general solution of the PDE is given through - - a1*y - a2*x -0 = ff(---------------,z) - 2 - a0*a1*x - a1 - -with arbitrary function ff(..). - -------------------- equation 2.14 --------------------- - - 2 2 -The quasilinear PDE: 0 = df(z,x)*a + df(z,y)*b - x + y . -The equivalent characteristic system: - - 2 2 -0=df(z,y)*b - x + y - - -0=df(x,y)*b - a - -for the functions: x(y) z(y) . -The general solution of the PDE is given through - - 2 3 2 3 2 2 2 3 -0 = ff(a*y - b*x,a *y - 3*a*b*x*y - 3*b *z + 3*b *x *y - b *y ) - -with arbitrary function ff(..). - -------------------- equation 2.16 --------------------- - -The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y - a*x. -The equivalent characteristic system: - -0=df(z,y)*y - a*x - - -0=df(x,y)*y - x - -for the functions: x(y) z(y) . -The general solution of the PDE is given through - - x -0 = ff(---,a*x - z) - y - -with arbitrary function ff(..). - -------------------- equation 2.20 --------------------- - -The quasilinear PDE: 0 = df(z,x) + df(z,y) - a*z. -The equivalent characteristic system: - -0=df(z,x) - a*z - - -0=df(y,x) - 1 - -for the functions: y(x) z(x) . -The general solution of the PDE is given through - - z -0 = ff(------,x - y) - a*x - e - -with arbitrary function ff(..). - -------------------- equation 2.21 --------------------- - -The quasilinear PDE: 0 = df(z,x) - df(z,y)*y + z. -The equivalent characteristic system: - -0=df(z,x) + z - - -0=df(y,x) + y - -for the functions: y(x) z(x) . -The general solution of the PDE is given through - - x x -0 = ff(e *z,e *y) - -with arbitrary function ff(..). - -------------------- equation 2.22 --------------------- - -The quasilinear PDE: 0 = 2*df(z,x) - df(z,y)*y + z. -The equivalent characteristic system: - -0=2*df(z,x) + z - - -0=2*df(y,x) + y - -for the functions: y(x) z(x) . -The general solution of the PDE is given through - - x/2 x/2 -0 = ff(e *z,e *y) - -with arbitrary function ff(..). - -------------------- equation 2.23 --------------------- - -The quasilinear PDE: 0 = df(z,x)*a + df(z,y)*y - b*z. -The equivalent characteristic system: - -0=df(z,x)*a - b*z - - -0=df(y,x)*a - y - -for the functions: y(x) z(x) . -The general solution of the PDE is given through - - z y -0 = ff(----------,------) - (b*x)/a x/a - e e - -with arbitrary function ff(..). - -------------------- equation 2.24 --------------------- - -The quasilinear PDE: 0 = df(z,x)*x - df(z,y)*x - df(z,y)*y. -The equivalent characteristic system: - -0=df(z,x)*x - - -0=df(y,x)*x + x + y - -for the functions: y(x) z(x) . -The general solution of the PDE is given through - - 2 -0 = ff(x + 2*x*y,z) - -with arbitrary function ff(..). - -------------------- equation 2.25 --------------------- - -The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y - az. -The equivalent characteristic system: - -0=df(y,z)*az - y - - -0=df(x,z)*az - x - -for the functions: y(z) x(z) . -The general solution of the PDE is given through - - y x -0 = ff(-------,-------) - z/az z/az - e e - -with arbitrary function ff(..). - -------------------- equation 2.26 --------------------- - - 2 2 -The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y + x + y - z - 1. -The equivalent characteristic system: - - 2 2 -0=df(z,y)*y + x + y - z - 1 - - -0=df(x,y)*y - x - -for the functions: x(y) z(y) . -The general solution of the PDE is given through - - 2 2 - x x + y + z + 1 -0 = ff(---,-----------------) - y y - -with arbitrary function ff(..). - -------------------- equation 2.39 --------------------- - - 2 2 2 -The quasilinear PDE: 0 = df(z,x)*a*x + df(z,y)*b*y - c*z . -The equivalent characteristic system: - - 2 2 -0=df(z,y)*b*y - c*z - - - 2 2 -0=df(x,y)*b*y - a*x - -for the functions: x(y) z(y) . -The general solution of the PDE is given through - - b*y - c*z - a*x + b*y -0 = ff(-----------,--------------) - b*y*z b*x*y - -with arbitrary function ff(..). - -------------------- equation 2.40 --------------------- - - 2 3 4 2 -The quasilinear PDE: 0 = df(z,x)*x*y + 2*df(z,y)*y - 2*x + 4*x *y*z - - 2 2 - - 2*y *z . -The equivalent characteristic system: - - 3 4 2 2 2 -0=2*(df(z,y)*y - x + 2*x *y*z - y *z ) - - - 2 -0=y *(2*df(x,y)*y - x) - -for the functions: x(y) z(y) . -The general solution of the PDE is given through - - 4 2 2 - x log(y)*x - log(y)*x *y*z - y *z -0 = ff(---------,----------------------------------) - sqrt(y) 4 2 - x - x *y*z - -with arbitrary function ff(..). - -------------------- equation 3.12 --------------------- - -The quasilinear PDE: 0 = df(w,x)*x + df(w,y)*a*x + df(w,y)*b*y + df(w,z)*c*x - - + df(w,z)*d*y + df(w,z)*f*z. -The equivalent characteristic system: - -0=df(w,x)*x - - -0=df(z,x)*x - c*x - d*y - f*z - - -0=df(y,x)*x - a*x - b*y - -for the functions: z(x) y(x) w(x) . -The general solution of the PDE is given through - - a*x + b*y - y -0 = ff(---------------, - b b - x *b - x - - 2 - - a*d*x + b*c*x + b*f*z - b*z - c*f*x - d*f*y + d*y - f *z + f*z - -------------------------------------------------------------------,w) - f f f 2 f - x *b*f - x *b - x *f + x *f - -with arbitrary function ff(..). - ------------------------- end -------------------------- -(TIME: applysym 7999 8769) +REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... + + +off echo; + +------------------------------------------------------- +This file is supposed to provide an automatic test of +the program APPLYSYM. On the other hand the application +of APPLYSYM is an interactive process, therefore the +interested user should inspect the example described +in APPLYSYM.TEX which demonstrates the application +of symmetries to integrate a 2nd order ODE. +Here the program QUASILINPDE for integrating first +order quasilinear PDE is demonstrated. + +The following equation comes up in the elimination +of resonant terms in normal forms of singularities +of vector fields (C.Herssens, P.Bonckaert, Limburgs +Universitair Centrum/Belgium, private communication). +------------------------------------------------------- + +The quasilinear PDE: 0 = df(w,x)*x + df(w,y)*y + 2*df(w,z)*z - 2*w - x*y. +The equivalent characteristic system: + +0=2*df(w,z)*z - 2*w - x*y + + +0=2*df(y,z)*z - y + + +0=2*df(x,z)*z - x + +for the functions: y(z) x(z) w(z) . +The general solution of the PDE is given through + + sqrt(z)*y sqrt(z)*x - log(z)*x*y + 2*w +0 = ff(-----------,-----------,---------------------) + z z z + +with arbitrary function ff(..). + +------------------------------------------------------- +Comment: +The result means that w is defined implicitly through + + - log(z)*x*y + 2*w sqrt(z)*x sqrt(z)*y +0 = ff(---------------------,-----------,-----------) + z z z + +with an arbitrary function ff of 3 arguments. As the PDE +was linear, the arguments of ff are such that we can +solve for w: + + sqrt(z)*x sqrt(z)*y +w = log(z)*x*y/2 + z*f(-----------,-----------) + z z + +with an arbitrary function f of 2 arguments. + +------------------------------------------------------- +The following PDEs are taken from E. Kamke, +Loesungsmethoden und Loesungen von Differential- +gleichungen, Partielle Differentialgleichungen +erster Ordnung, B.G. Teubner, Stuttgart (1979). + +------------------- equation 1.4 ---------------------- + +The quasilinear PDE: 0 = df(z,x)*x - y. +The equivalent characteristic system: + +0=df(z,x)*x - y + + +0=df(y,x)*x + +for the functions: y(x) z(x) . +The general solution of the PDE is given through + +0 = ff(y,log(x)*y - z) + +with arbitrary function ff(..). + +------------------- equation 2.5 ---------------------- + + 2 2 +The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y . +The equivalent characteristic system: + + 2 +0=df(z,y)*y + + + 2 2 +0=df(x,y)*y - x + +for the functions: x(y) z(y) . +The general solution of the PDE is given through + + - x + y +0 = ff(----------,z) + x*y + +with arbitrary function ff(..). + +------------------- equation 2.6 ---------------------- + + 2 2 +The quasilinear PDE: 0 = df(z,x)*x - df(z,x)*y + 2*df(z,y)*x*y. +The equivalent characteristic system: + +0=2*df(z,y)*x*y + + + 2 2 +0=2*df(x,y)*x*y - x + y + +for the functions: x(y) z(y) . +The general solution of the PDE is given through + + 2 2 + - x - y +0 = ff(------------,z) + y + +with arbitrary function ff(..). + +------------------- equation 2.7 ---------------------- + +The quasilinear PDE: 0 = df(z,x)*a0*x - df(z,x)*a1 + df(z,y)*a0*y - df(z,y)*a2. +The equivalent characteristic system: + +0=df(z,x)*(a0*x - a1) + + +0=df(y,x)*a0*x - df(y,x)*a1 - a0*y + a2 + +for the functions: y(x) z(x) . +The general solution of the PDE is given through + + a1*y - a2*x +0 = ff(---------------,z) + 2 + a0*a1*x - a1 + +with arbitrary function ff(..). + +------------------- equation 2.14 --------------------- + + 2 2 +The quasilinear PDE: 0 = df(z,x)*a + df(z,y)*b - x + y . +The equivalent characteristic system: + + 2 2 +0=df(z,y)*b - x + y + + +0=df(x,y)*b - a + +for the functions: x(y) z(y) . +The general solution of the PDE is given through + + 2 3 2 3 2 2 2 3 +0 = ff(a*y - b*x,a *y - 3*a*b*x*y - 3*b *z + 3*b *x *y - b *y ) + +with arbitrary function ff(..). + +------------------- equation 2.16 --------------------- + +The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y - a*x. +The equivalent characteristic system: + +0=df(z,y)*y - a*x + + +0=df(x,y)*y - x + +for the functions: x(y) z(y) . +The general solution of the PDE is given through + + x +0 = ff(---,a*x - z) + y + +with arbitrary function ff(..). + +------------------- equation 2.20 --------------------- + +The quasilinear PDE: 0 = df(z,x) + df(z,y) - a*z. +The equivalent characteristic system: + +0=df(z,x) - a*z + + +0=df(y,x) - 1 + +for the functions: y(x) z(x) . +The general solution of the PDE is given through + + z +0 = ff(------,x - y) + a*x + e + +with arbitrary function ff(..). + +------------------- equation 2.21 --------------------- + +The quasilinear PDE: 0 = df(z,x) - df(z,y)*y + z. +The equivalent characteristic system: + +0=df(z,x) + z + + +0=df(y,x) + y + +for the functions: y(x) z(x) . +The general solution of the PDE is given through + + x x +0 = ff(e *z,e *y) + +with arbitrary function ff(..). + +------------------- equation 2.22 --------------------- + +The quasilinear PDE: 0 = 2*df(z,x) - df(z,y)*y + z. +The equivalent characteristic system: + +0=2*df(z,x) + z + + +0=2*df(y,x) + y + +for the functions: y(x) z(x) . +The general solution of the PDE is given through + + x/2 x/2 +0 = ff(e *z,e *y) + +with arbitrary function ff(..). + +------------------- equation 2.23 --------------------- + +The quasilinear PDE: 0 = df(z,x)*a + df(z,y)*y - b*z. +The equivalent characteristic system: + +0=df(z,x)*a - b*z + + +0=df(y,x)*a - y + +for the functions: y(x) z(x) . +The general solution of the PDE is given through + + z y +0 = ff(----------,------) + (b*x)/a x/a + e e + +with arbitrary function ff(..). + +------------------- equation 2.24 --------------------- + +The quasilinear PDE: 0 = df(z,x)*x - df(z,y)*x - df(z,y)*y. +The equivalent characteristic system: + +0=df(z,x)*x + + +0=df(y,x)*x + x + y + +for the functions: y(x) z(x) . +The general solution of the PDE is given through + + 2 +0 = ff(x + 2*x*y,z) + +with arbitrary function ff(..). + +------------------- equation 2.25 --------------------- + +The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y - az. +The equivalent characteristic system: + +0=df(y,z)*az - y + + +0=df(x,z)*az - x + +for the functions: y(z) x(z) . +The general solution of the PDE is given through + + y x +0 = ff(-------,-------) + z/az z/az + e e + +with arbitrary function ff(..). + +------------------- equation 2.26 --------------------- + + 2 2 +The quasilinear PDE: 0 = df(z,x)*x + df(z,y)*y + x + y - z - 1. +The equivalent characteristic system: + + 2 2 +0=df(z,y)*y + x + y - z - 1 + + +0=df(x,y)*y - x + +for the functions: x(y) z(y) . +The general solution of the PDE is given through + + 2 2 + x x + y + z + 1 +0 = ff(---,-----------------) + y y + +with arbitrary function ff(..). + +------------------- equation 2.39 --------------------- + + 2 2 2 +The quasilinear PDE: 0 = df(z,x)*a*x + df(z,y)*b*y - c*z . +The equivalent characteristic system: + + 2 2 +0=df(z,y)*b*y - c*z + + + 2 2 +0=df(x,y)*b*y - a*x + +for the functions: x(y) z(y) . +The general solution of the PDE is given through + + b*y - c*z - a*x + b*y +0 = ff(-----------,--------------) + b*y*z b*x*y + +with arbitrary function ff(..). + +------------------- equation 2.40 --------------------- + + 2 3 4 2 +The quasilinear PDE: 0 = df(z,x)*x*y + 2*df(z,y)*y - 2*x + 4*x *y*z + + 2 2 + - 2*y *z . +The equivalent characteristic system: + + 3 4 2 2 2 +0=2*(df(z,y)*y - x + 2*x *y*z - y *z ) + + + 2 +0=y *(2*df(x,y)*y - x) + +for the functions: x(y) z(y) . +The general solution of the PDE is given through + + 4 2 2 + x log(y)*x - log(y)*x *y*z - y *z +0 = ff(---------,----------------------------------) + sqrt(y) 4 2 + x - x *y*z + +with arbitrary function ff(..). + +------------------- equation 3.12 --------------------- + +The quasilinear PDE: 0 = df(w,x)*x + df(w,y)*a*x + df(w,y)*b*y + df(w,z)*c*x + + + df(w,z)*d*y + df(w,z)*f*z. +The equivalent characteristic system: + +0=df(w,x)*x + + +0=df(z,x)*x - c*x - d*y - f*z + + +0=df(y,x)*x - a*x - b*y + +for the functions: z(x) y(x) w(x) . +The general solution of the PDE is given through + + a*x + b*y - y +0 = ff(---------------, + b b + x *b - x + + 2 + - a*d*x + b*c*x + b*f*z - b*z - c*f*x - d*f*y + d*y - f *z + f*z + -------------------------------------------------------------------,w) + f f f 2 f + x *b*f - x *b - x *f + x *f + +with arbitrary function ff(..). + +------------------------ end -------------------------- +(TIME: applysym 7999 8769)