:ֵԳ΢̵ַĽ(עⱾԱֵԼ΢ַıֵ)

ʽ:
[y,er,n]=ODEBoundary(pf,qf,rf,bc,x,NSplit,Error,LoopN)
[y,er,n]=ODEBoundary(pf,qf,rf,bc,x,NSplit,Error)
[y,er,n]=ODEBoundary(pf,qf,rf,bc,x,NSplit)
[y,er,n]=ODEBoundary(pf,qf,rf,bc,x)

pf:ű洢һ׵ϵ(һxΪԱĺ)
qf:ű洢һϵϵ(һxΪԱĺ)
rf:ű洢ĳϵ(һxΪԱĺ)
bc:ű洢߽,ֻΪy,ʽμԭԼ
x:,ΪҪֵ,עxԪصֵx0Ԫ֮
NSplit:ȷּ,ĬΪ256.ĳʼ=abs(x0(1)-x0(2))/NSplit
Error:Ƶ,Ĭ1E-12
LoopN:ȷ,ĬΪ8,ע,ǲ=abs(x0(1)-x0(2))/(NSplit*4*LoopN)

y:صĶӦxֵ
er:ص
n:صĳȷ

ԭ:
֪ y'' = pf(x)*y' + qf(x)*y + rf(x)µı߽
߽1:y'(a1)=b1*y+c1( y(a1)=b1)
߽2:y'(a2)=b2*y+c2( y(a2)=b2)
//xıֵ߽Ϊa1a2
ע, y''ʾyxĶ׵, y'ʾһ׵
 x = x(1)x(2)x(m)ʱӦ y ֵ

ο:ִӦѧֲ.ֵ[M].廪ѧ,,page:625-928

PS:ֵĵһࡢڶࡢԼǵĻϱ߽.,б.

ע:

1ַ֧顢Գ΢ַ

2ڷʽĳ΢ַ,Ľ

y1''=a11(x)*y1'+a12(x)*y2'+a13(x)*y1+a14(x)*y2+a15(x)
y2''=a21(x)*y1'+a22(x)*y2'+a23(x)*y1+a24(x)*y2+a25(x)
y1'(0)=0.123
y2(0)=0.5
y2(1)=4.56
 x=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9ʱӦy1,y2ֵa11(x)a12(x)a25(x)ʾxĺ

   бʱԽһΡεɢ(м򵥲)y1''y1'y1ʾ,y2''y2'y2ʾڲʱúֹʽɸı߽ȷȻɢݷ˶Ӧķ⼴ɡһַһԽԷ飬ɢСԾάܴ󣨴ϡ󣩣ʱרԽǾĺõѡ

   Ȼ,ʱⲻĶ׵ģһ׵,һ׻ϵ,Ǹ߽׵ġǼףķʽм򵥵ɢ,˷ͿˡΨһҪע,˷ʱ±겻Ūرڲ÷Ǿģʽ洢£ԽǾ󣬴洢ʱķʽбһĴ洢

3ҪԷ̵ĳ΢ַ,Ӿбд.

4رһ£ڷµĳ΢ַ̻ⲻĳ΢ַ,һ,оһķ

:

1
//֪ y'' = -2/x*y' + 2/x^2*y +sin(log(x))/x^2 ,y(1)=1 ,y(2)=2 , x =1.11.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9ֵ

p="-2/x";
q="2/x^2";
r="sin(log(x))/x^2";
c="y(1)=1,y(2)=2";//ע߽д
x=1.1:0.1:1.9;
[y,er]=Odeboundary(p,q,r,c,x)//ִ6,سõ½
y =
[ 1.09262934953363    1.18708493527214    1.28338249883030    1.38144612562790    1.48115962791218    1.58239267620030    1.68501415489777    1.78889867851306    1.89392958528753 ]
er =
[ 4.9094389664E-08    9.1381456623E-08    1.2927403103E-07    1.6711275829E-07    2.0506472853E-07    2.1019460971E-07    1.9245454185E-07    1.4120630864E-07    7.1134900636E-08 ]


2
//֪ y'' = x*y' + cos(x)*y +cos(x)*(2-x^2) - x*sin(x)*(2+cos(x)),y(0)=0,y'(2)=0.0770037537313969, x =1.11.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9ֵ

pf = "x";
qf = "cos(x)";
rf = "cos(x)*(2-x^2)-x*sin(x)*(2+cos(x))";
bc = "y(0)=0,y'(2)=0.0770037537313969";//ע΢ֱֵд
x=1.1:0.1:1.9;
[y,er,n]=ODEBoundary(pf,qf,rf,bc,x)//ִ6,سõy.עⱾȷΪ y(x)=x*sin(x)
y =
[ 0.98032880546962    1.11844770061069    1.25262651811950    1.37963056535091    1.49624347588101    1.59931880039688    1.68583124467335    1.75292682717265    1.79797127438106 ]
er =
[ 6.9293955562E-07    7.8042714068E-07    8.5705729779E-07    9.1280018965E-07    9.5217275042E-07    9.8078258536E-07    1.0072701652E-06    1.0258770597E-06    1.0364979610E-06 ]
n =
[ 8.00000000000000 ]