> restart:with(plots):
Warning, the name changecoords has been redefined
> nexpf:=13:cnames:=<c1|c2|c3|c4|c5|c6|c7|c8|c9|c10|c11|c12|c13>:
c:=Vector[row](nexpf):for n from 1 to nexpf do c[n]:=cnames[n] end
do:
uexp:=<proc(x,y) 1; end proc, # define the Harmonic expansion
functions
proc(x,y) x; end proc,
proc(x,y) y; end proc,
proc(x,y) x^2-y^2; end proc,
proc(x,y) 2*x*y; end proc,
proc(x,y) x^3-3*x*y^2; end proc,
proc(x,y) 3*x^2*y-y^3; end proc,
proc(x,y) x^4-6*x^2*y^2+y^4; end proc,
proc(x,y) 4*x^3*y-4*x*y^3; end proc,
proc(x,y) x^5-10*x^3*y^2+5*x*y^4; end proc,
proc(x,y) 5*x^4*y-10*x^2*y^3+y^5; end proc,
proc(x,y) x^6-15*x^4*y^2+15*x^2*y^4-y^6; end proc,
proc(x,y) 6*x^5*y-20*x^3*y^3+6*x*y^5; end proc>:
bdry:=proc(x,y) x^2+y^2; end proc: # define the boundary function
> u:=proc(c,x,y) sum('c[n]*uexp[n,1](x,y)','n'=1..nexpf); end
proc:u(c,x,y);bdry(x,y);
c1 + c2 x + c3 y + c4 ( x2 − y2 ) + 2 c5 y x + c6 ( x3 − 3 x y2 ) + c7 ( 3 x2 y − y3 )
+ c8 ( x4 − 6 x2 y2 + y4 ) + c9 ( 4 x3 y − 4 x y3 ) + c10 ( x5 − 10 x3 y2 + 5 x y4 )
+ c11 ( 5 x4 y − 10 x2 y3 + y5 ) + c12 ( x6 − 15 x4 y2 + 15 x2 y4 − y6 )
+ c13 ( 6 x5 y − 20 x3 y3 + 6 x y5 )
x2 + y2
> linerr:=<proc (c) int((u(c,0,t)-bdry(0,t))^2,t=0..2); end proc, #
line integrals
proc (c) int((u(c,t,2)-bdry(t,2))^2,t=0..1); end proc,
proc (c) int((u(c,1,t)-bdry(1,t))^2,t=0..2); end proc,
proc (c) int((u(c,t,0)-bdry(t,0))^2,t=0..1); end proc>:
err:=proc(c) sum('linerr[n,1](c)','n'=1..4); end proc: # the
error expression
simplify(err(c));
43 c2 10536 c12 c9 196
360 c12 c4 2148 c72 35048 c7 c12
−24 c1 −
−
+
+ 2650 c12 c13 −
+
+
3
5
5
7
35
105
6699076 c122 112 c3 264 c4
482 c6 328 c7 376 c8 584 c9 4807 c10
+
−
+
− 42 c5 +
+
−
+
−
1001
3
5
5
5
7
3
14
232 c11 2552 c12
72 c1 c8 8 c1 c11 28 c32
2
+
−
− 507 c13 + 6 c1 + 12 c1 c3 +
−
+
7
9
5
3
3
+
472 c3 c8 8 c3 c11 7788 c82 285976 c8 c11 272708 c112
168 c3 c7
+
+
+
+
− 16 c1 c7 −
15
7
35
315
231
5
−
6968 c7 c8 279592 c8 c12 3682376 c11 c12 360 c1 c12 648 c3 c12 17912 c7 c11
−
−
+
+
−
35
165
693
7
7
63
226 c1 c10 8 c22 96 c2 c6 158 c2 c8 368 c2 c10 1623 c2 c12
+ 6 c1 c2 − 23 c1 c6 +
+
−
−
+
+
3
3
5
15
7
14
+
3048 c62 2659 c6 c8 37568 c6 c10 111988 c6 c12 64858 c8 c10 310088 c102
+
−
−
+
+
35
70
63
105
315
231
+
2210543 c10 c12
3041120 c10 c13 4116 c8 c13 40 c1 c4 80 c3 c4
− 642 c10 c11 +
+
−
−
693
693
5
3
3
116 c42 8488 c8 c4 1256 c11 c4 1016 c7 c4
+
−
−
+
+ 108 c1 c13 − 44 c1 c9 + 12 c1 c5
5
105
21
15
−
130 c11 c2 2023 c11 c6 136856 c11 c13 36332 c11 c9 3112 c11 c5 454 c3 c10
+
+
+
−
+
3
6
693
63
21
3
+ 6 c3 c2 − 43 c3 c6 +
1592 c3 c13 1288 c3 c9 56 c3 c5 10907 c4 c10 13 c4 c2
−
+
−
−
7
15
3
42
3
+
926 c4 c6 1295 c4 c13
2063 c7 c10
−
+ 136 c4 c9 − 22 c4 c5 −
− c7 c2 + 54 c7 c6
15
3
6
−
73054 c7 c13 5232 c7 c9 68 c7 c5
10 c8 c9
86528 c10 c9
+
−
+ 425 c12 c5 −
− 28 c8 c5 −
105
35
5
3
63
+
4448 c10 c5 416 c2 c13 512 c2 c9 32 c2 c5 83936 c6 c13 12288 c6 c9
+
−
+
−
+
21
7
15
3
105
35
352 c6 c5 318928 c132 30272 c13 c9 1824 c13 c5 13248 c92 1984 c9 c5
−
+
−
+
+
−
+ 16 c52
5
77
15
7
35
15
> derr:=Vector(nexpf):
# partial derivatives for optimum
for n from 1 to nexpf do derr[n]:=diff(err(c),c[n])=0 end do:derr;
⎡ 13 Element Column Vector ⎤
⎢
⎥
⎢ Data Type: anything
⎥
⎢
⎥
⎢ Storage: rectangular
⎥
⎢
⎥
⎣ Order: Fortran_order
⎦
# use Linear Algebra package to generate
> with(LinearAlgebra):
matrix equations
> eqns:=[seq(derr[n],n=1..nexpf)]:var:=[seq(c[n],n=1..nexpf)]:(A,b):
=GenerateMatrix(eqns,var);
⎡ 13 x 13 Matrix
⎤ ⎡ 13 Element Column Vector ⎤
⎢
⎥ ⎢
⎥
⎢ Data Type: anything ⎥ ⎢ Data Type: anything
⎥
⎥, ⎢
⎥
A, b := ⎢
⎢ Storage: rectangular ⎥ ⎢ Storage: rectangular
⎥
⎢
⎥ ⎢
⎥
⎣ Order: Fortran_order ⎦ ⎣ Order: Fortran_order
⎦
> ans:=LinearSolve(A,b):for k from 1 to nexpf do c[k]:=ans[k] end
do:c;"|Error| ",evalf(sqrt(err(c)));
⎡ 13 Element Row Vector ⎤
⎢
⎥
⎢ Data Type: anything
⎥
⎢
⎥
⎢ Storage: rectangular
⎥
⎢
⎥
⎣ Order: Fortran_order
⎦
"|Error| ", 0.009249985881
> errxy:=proc(x,y) evalf(sqrt((u(c,x,y)-bdry(x,y))^2)); end proc: #
boundary error as a function of (x,y)
plot({u(c,0,y),bdry(0,y),errxy(0,y)},y=0..2,color=[black,blue,red]
,thickness=4,linestyle=[1,2,3]);
# u(x,y) and boundary function
plots
plot({u(c,x,2),bdry(x,2),errxy(x,2)},x=0..1,color=[black,blue,red]
,thickness=4,linestyle=[1,2,3]);
plot({u(c,1,y),bdry(1,y),errxy(1,y)},y=0..2,color=[black,blue,red]
,thickness=4,linestyle=[1,2,3]);
plot({u(c,x,0),bdry(x,0),errxy(x,0)},x=0..1,color=[black,blue,red]
,thickness=4,linestyle=[1,2,3]);
plot3d(u(c,x,y),x=0..1,y=0..2);
contourplot(u(c,x,y),x=0..1,y=0..2,thickness=4);
> evalf(maximize(errxy(0,t),t=0..2));
the boundary
evalf(maximize(errxy(t,2),t=0..1));
evalf(maximize(errxy(1,t),t=0..2));
evalf(maximize(errxy(t,0),t=0..1));
# calculate maximum error on
0.01220150700
0.01220150472
0.01220150700
0.01220150472
> plot({errxy(1,y)},y=0..2,color=black,thickness=4,linestyle=3);
plot({errxy(0,y)},y=0..2,color=black,thickness=4,linestyle=3);
plot({errxy(x,0)},x=0..1,color=black,thickness=4,linestyle=3);
plot({errxy(x,2)},x=0..1,color=black,thickness=4,linestyle=3);
>
>