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ΜΕΤΑΠΤΥΧΙΑΚΗ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΓΕΩΡΓΙΑΔΗΣ Α. ΝΙΚΟΛΑΟΣ Επιβλέπουσα: Γουσίδου-Κουτίτα Μαρία Αναπληρώτρια Καθηγήτρια Α.Π.Θ. ΑΡΙΘΜΗΤΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΜΕΘΟΔΩΝ ΕΠΙΛΥΣΗΣ ΔΙΑΦΟΡΙΚΩΝ ΕΞΙΣΩΣΕΩΝ ΜΕ ΕΜΦΑΣΗ ΣΤΙΣ ΜΕΘΟΔΟΥΣ FEHLBERG- ΕΦΑΡΜΟΓΕΣ ΜΕ ΠΡΟΓΡΑΜΜΑΤΑ ΣΕ MATLAB Θεσσαλονίκη, Φεβρουάριος 2011
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ΒΑΣΙΚΟΙ ΟΡΙΣΜΟΙ ΚΑΙ ΘΕΩΡΗΜΑΤΑ
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Η ΔΙΑΦΟΡΙΚΗ ΕΞΙΣΩΣΗ ΚΑΙ ΟΤΡΟΠΟΣ ΕΠΙΛΥΣΗΣ ΤΗΣ Ορισμός: Μια διαφορική εξίσωση είναι μία εξίσωση όπου εμφανίζεται μία άγνωστη συνάρτηση μαζί με τις αντίστοιχες κ-τάξης παραγώγους της. Η λύση αυτής της διαφορικής εξίσωσης είναι μια συνάρτηση που επαληθεύει την εξίσωση μας. Πρόταση: Μια γραμμική διαφορική εξίσωση πρώτης τάξης είναι μια διαφορική εξίσωση της μορφής όπου p και q είναι γνωστές συναρτήσεις της ανεξάρτητης μεταβλητής χ που είναι συνεχείς σε ένα διάστημα της πραγματικής ευθείας. Οι λύσεις της γραμμικής διαφορικής εξίσωσης πρώτης τάξης δίνονται από τον τύπο όπου c σταθερά.
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ΥΠΑΡΞΗ ΚΑΙ ΜΟΝΑΔΙΚΟΤΗΤΑ ΛΥΣΗΣ ΠΡΟΒΛΗΜΑΤΟΣ ΑΡΧΙΚΗΣ ΤΙΜΗΣ ΔΙΑΦΟΡΙΚΩΝ ΕΞΙΣΩΣΕΩΝ Ορισμός:
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Θεώρημα:
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Ορισμός:
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Θεώρημα:
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ΠΕΡΙΓΡΑΦΗ ΤΩΝ ΜΕΘΟΔΩΝ ΠΡΩΤΗΣ ΚΑΙ ΔΥΤΕΡΗΣ ΤΑΞΗΣ
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Η ΜΕΘΟΔΟΣ EULER Παράδειγμα 1: Παράδειγμα 2: Να λυθεί το παραπάνω παράδειγμα και με h=0.1
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format('long') f=inline('2*x-y','x','y'); dsolve('Dy=2*x-y','y(0)=1','x') g=inline('-2+2*x+3*exp(-x)','x') h=0.25; y(1)=1; i=0:0.25:1; nstep=5; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.25:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.25:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp (['e' num2str(j) '=']) disp(e) end format('long') f=inline('2*x-y','x','y'); g=inline('-2+2*x+3*exp(-x)','x') h=0.1; y(1)=1; i=0:0.1:1; nstep=11; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.10:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.1:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( 2*x-y , x , y ); dsolve( Dy=2*x-y , y(0)=1 , x ) g=inline( -2+2*x+3*exp(-x) , x ) h=0.25; y(1)=1; i=0:0.25:1; nstep=5; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( 2*x-y , x , y ); g=inline( -2+2*x+3*exp(-x) , x ) h=0.1; y(1)=1; i=0:0.1:1; nstep=11; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.10:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_10.jpg " format( long ) f=inline( 2*x-y , x , y ); dsolve( Dy=2*x-y , y(0)=1 , x ) g=inline( -2+2*x+3*exp(-x) , x ) h=0.25; y(1)=1; i=0:0.25:1; nstep=5; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( 2*x-y , x , y ); g=inline( -2+2*x+3*exp(-x) , x ) h=0.1; y(1)=1; i=0:0.1:1; nstep=11; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.10:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
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Τα αποτελέσματα των παραπάνω αλγορίθμων ix(i)y(i) 001 10,250,75 20,50,6875 30,750,765625 410,94921875 ix(i)y(i) 001,000000000000000 10,10,900000000000000 20,20,830000000000000 30,30,787000000000000 40,40,768300000000000 50,50,771470000000000 60,60,794323000000000 70,70,834890700000000 80,80,891401630000000 90,90,962261467000000 1011,046035320300000
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ix(i)y(i)g(i)E 001 1,00000000000 00000 10,250,75 0,83640234921 42150,086402349 20,50,6875 0,81959197913 79000,132091979 30,75 0,76562 5 0,91709965822 30440,151474658 41 0,94921 875 1,10363832351 43200,154419574 ix(i)y(i)g(i)E 00 1,000000000 00000010 10,1 0,900000000 0000000,9145122540,014512254 20,2 0,830000000 0000000,8561922590,026192259 30,3 0,787000000 0000000,8224546620,035454662 40,4 0,768300000 0000000,8109601380,042660138 50,5 0,771470000 0000000,8195919790,048121979 60,6 0,794323000 0000000,8464349080,052111908 70,7 0,834890700 0000000,8897559110,054865211 80,8 0,891401630 0000000,9479868920,056585262 90,9 0,962261467 0000001,0197089790,057447512 101 1,046035320 3000001,1036383240,057603003
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Χρησιμοποιώντας και ένα δεύτερο παράδειγμα παρατηρούμε ότι αύξηση του βήματος μας δίνει καλύτερα αποτελέσματα. Στη συνέχεια να λυθεί και με βήμα h=0.1
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format('long') f=inline('x*y+1','x','y'); g=inline('(1/2*pi^(1/2)*2^(1/2)*erf(1/2*2 ^(1/2)*x)+1)*exp(1/2*x^2)','x') h=0.25; y(1)=1; i=0:0.25:1; nstep=5; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.25:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.25:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp (['e' num2str(j) '=']) disp(e) end format('long') f=inline('x*y+1','x','y'); dsolve('Dy=x*y+1','y(0)=1','x') g=inline('(1/2*pi^(1/2)*2^(1/2)*erf(1 /2*2^(1/2)*x)+1)*exp(1/2*x^2)','x') h=0.1; y(1)=1; i=0:0.1:1; nstep=11; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.1:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.1:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( x*y+1 , x , y ); g=inline( (1/2*pi^(1/2)*2^(1/2)*erf(1/2*2 ^(1/2)*x)+1)*exp(1/2*x^2) , x ) h=0.25; y(1)=1; i=0:0.25:1; nstep=5; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( x*y+1 , x , y ); dsolve( Dy=x*y+1 , y(0)=1 , x ) g=inline( (1/2*pi^(1/2)*2^(1/2)*erf(1 /2*2^(1/2)*x)+1)*exp(1/2*x^2) , x ) h=0.1; y(1)=1; i=0:0.1:1; nstep=11; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_14.jpg " format( long ) f=inline( x*y+1 , x , y ); g=inline( (1/2*pi^(1/2)*2^(1/2)*erf(1/2*2 ^(1/2)*x)+1)*exp(1/2*x^2) , x ) h=0.25; y(1)=1; i=0:0.25:1; nstep=5; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( x*y+1 , x , y ); dsolve( Dy=x*y+1 , y(0)=1 , x ) g=inline( (1/2*pi^(1/2)*2^(1/2)*erf(1 /2*2^(1/2)*x)+1)*exp(1/2*x^2) , x ) h=0.1; y(1)=1; i=0:0.1:1; nstep=11; for n=1:nstep y(n+1)=y(n)+h*f(i(n),y(n)); end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
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ix(i)y(i)g(i)E 00110 10,251,251,2870170,03701743 20,51,5781251,6769750,098849973 30,752,0253906252,2325850,207194422 412,6551513673,0594070,404256038 ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25
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ix(i)y(i)g(i)E 001,0000000000000000 0,0000000000000000 10,11,10000000000000001,10534652181284000,0053465218128410 20,21,21100000000000001,22288946247529000,0118894624752930 30,31,33522000000000001,35519196376603000,0199719637660340 40,41,47527660000000001,50531895297066000,0300423529706600 50,51,63428766400000001,67697497251897000,0426873085189770 60,61,81600204720000001,87467899197767000,0586769447776720 70,72,02496217003200002,10398831840077000,0790261483687750 80,82,26670952193424002,37178776967566000,1050782477414200 90,92,54804628368898002,68666585361903000,1386195699300590 1012,87737044922098003,05940740534257000,1820369561215890 ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1
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ΜΕΘΟΔΟΣ RUNGE KUTTA (2,2)
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Παράδειγμα: Να λυθεί με τη μέθοδο Runge-Kutta(2 ης τάξης) η διαφορική εξίσωση από χ=0 μέχρι χ=1 με h=0.25 και με h=0.1.
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format('long') f=inline('2*x-y','x','y'); dsolve('Dy=2*x-y','y(0)=1','x') g=inline('-2+2*x+3*exp(-x)','x') w1=0; w2=1; c1=0; c2=1/2; a21=1/2; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); y(n+1)=y(n)+w1*k1+w2*k2; end for i=0:0.25:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.25:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp (['e' num2str(j) '=']) disp(e) end format('long') f=inline('2*x-y','x','y'); dsolve('Dy=2*x-y','y(0)=1','x') g=inline('-2+2*x+3*exp(-x)','x') w1=0; w2=1; c1=0; c2=1/2; a21=1/2; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); y(n+1)=y(n)+w1*k1+w2*k2; end for i=0:0.1:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.1:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( 2*x-y , x , y ); dsolve( Dy=2*x-y , y(0)=1 , x ) g=inline( -2+2*x+3*exp(-x) , x ) w1=0; w2=1; c1=0; c2=1/2; a21=1/2; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); y(n+1)=y(n)+w1*k1+w2*k2; end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( 2*x-y , x , y ); dsolve( Dy=2*x-y , y(0)=1 , x ) g=inline( -2+2*x+3*exp(-x) , x ) w1=0; w2=1; c1=0; c2=1/2; a21=1/2; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); y(n+1)=y(n)+w1*k1+w2*k2; end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_22.jpg " format( long ) f=inline( 2*x-y , x , y ); dsolve( Dy=2*x-y , y(0)=1 , x ) g=inline( -2+2*x+3*exp(-x) , x ) w1=0; w2=1; c1=0; c2=1/2; a21=1/2; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); y(n+1)=y(n)+w1*k1+w2*k2; end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/4)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( 2*x-y , x , y ); dsolve( Dy=2*x-y , y(0)=1 , x ) g=inline( -2+2*x+3*exp(-x) , x ) w1=0; w2=1; c1=0; c2=1/2; a21=1/2; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); y(n+1)=y(n)+w1*k1+w2*k2; end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/10)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
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ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25 ix(i)y(i)g(i)E 001,00000000000000001,0000000000000000,000000000000000 10,250,84375000000000000,8364023492142150,007347650785785 20,50,83105468750000000,8195919791379000,011462708362100 30,750,93051147460937500,9170996582230440,013411816386331 411,11758708953857001,1036383235143200,013948766024247
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ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 001,0000000000000000 10,10,9150000000000000 20,20,8570750000000000 30,30,8236528750000000 40,40,8124058518750000 50,50,8212272959468750 60,60,8482107028319220 70,70,8916306860628890 80,80,9499257708869150 90,91,0216828226526500 1011,1056229545006500
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g(i)E 1,00000000000000000,0000000000000000 0,91451225410787900,0004880000000000 0,85619225923394500,0008830000000000 0,82245466204515300,0011982129548460 0,81096013810691800,0014457137680820 0,81959197913790000,0016353168089750 0,84643490828207900,0017757945498420 0,88975591137422900,0018747746886610 0,94798689235166500,0019388785352500 1,01970897922179000,0019738434308610 1,10363832351432000,0019846309863290
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ΠΕΡΙΓΡΑΦΗ ΤΩΝ ΜΕΘΟΔΩΝ ΤΡΙΤΗΣ ΚΑΙ ΤΕΤΑΡΤΗΣ ΤΑΞΗΣ
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Η ΜΕΘΟΔΟΣ RUNGE KUTTA
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Παράδειγμα
31
format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=3/8; w2=2/3; w3=1/6; c1=0; c2=1/2; c3=1; a21=1/2; a31=-1; a32=2; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.25:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=3/8; w2=2/3; w3=1/6; c1=0; c2=1/2; c3=1; a21=1/2; a31=-1; a32=2; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.1:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=3/8; w2=2/3; w3=1/6; c1=0; c2=1/2; c3=1; a21=1/2; a31=-1; a32=2; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=3/8; w2=2/3; w3=1/6; c1=0; c2=1/2; c3=1; a21=1/2; a31=-1; a32=2; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_31.jpg " format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=3/8; w2=2/3; w3=1/6; c1=0; c2=1/2; c3=1; a21=1/2; a31=-1; a32=2; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=3/8; w2=2/3; w3=1/6; c1=0; c2=1/2; c3=1; a21=1/2; a31=-1; a32=2; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
32
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25 ix(i)y(i)g(i)E 11,0000000000000000 0,66666666666666700,3333333333333330 21,25000000000000000,97001139322916700,56140350877193000,4086078844572370 31,50000000000000000,87714862191152900,47058823529411800,4065603866174110 41,75000000000000000,75150508262157900,39506172839506200,3564433542265170 52,00000000000000000,62335546844452900,33333333333333300,2900221351111960
33
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 1,0000000000000000 2,00000000000000001,10000000000000000,9950331666666670 3,00000000000000001,20000000000000000,9783455166621170 4,00000000000000001,30000000000000000,9510069226585940 5,00000000000000001,40000000000000000,9147257180144570 6,00000000000000001,50000000000000000,8715939733097660 7,00000000000000001,60000000000000000,8238133186989110 8,00000000000000001,70000000000000000,7734608316822240 9,00000000000000001,80000000000000000,7223294816931410 10,00000000000000001,90000000000000000,6718495190979850 11,00000000000000002,00000000000000000,6230770788952500 g(i)E 0,66666666666666700,3333333333333330 0,62305295950155800,3719802071651090 0,58139534883720900,3969501678249080 0,54200542005420000,4090015026043940 0,50505050505050500,4096752129639520 0,47058823529411800,4010057380156480 0,43859649122807000,3852168274708410 0,40899795501022500,3644628766719990 0,38167938931297700,3406500923801640 0,35650623885918000,3153432802388050 0,33333333333333300,2897437455619170
34
Η ΜΕΘΟΔΟΣ NYSTROM
36
format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/4; w2=3/8; w3=3/8; c1=0; c2=2/3; c3=2/3; a21=2/3; a31=0; a32=2/3; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.25:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/4; w2=3/8; w3=3/8; c1=0; c2=2/3; c3=2/3; a21=2/3; a31=0; a32=2/3; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.1:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=3/8; w3=3/8; c1=0; c2=2/3; c3=2/3; a21=2/3; a31=0; a32=2/3; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=3/8; w3=3/8; c1=0; c2=2/3; c3=2/3; a21=2/3; a31=0; a32=2/3; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_36.jpg " format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=3/8; w3=3/8; c1=0; c2=2/3; c3=2/3; a21=2/3; a31=0; a32=2/3; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=3/8; w3=3/8; c1=0; c2=2/3; c3=2/3; a21=2/3; a31=0; a32=2/3; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
37
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25 ix(i)y(i)g(i)E 11,00000000000000000 0,666666666666667000,33333333333333300 21,250000000000000000,969605999228395000,561403508771930000,40820249045646500 31,500000000000000000,888788475869516000,470588235294118000,41820024057539800 41,750000000000000000,780408616697521000,395061728395062000,38534688830245900 52,000000000000000000,666566020252015000,333333333333333000,33323268691868200
38
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 1,0000000000000000 2,00000000000000001,10000000000000000,9950221728395060 3,00000000000000001,20000000000000000,9803872381293220 4,00000000000000001,30000000000000000,9569314703766760 5,00000000000000001,40000000000000000,9259190333490610 6,00000000000000001,50000000000000000,8888820604986000 7,00000000000000001,60000000000000000,8474511353718690 8,00000000000000001,70000000000000000,8032066201871010 9,00000000000000001,80000000000000000,7575694727326880 10,00000000000000001,90000000000000000,7117370281689460 11,00000000000000002,00000000000000000,6666590928639060 g(i)E 0,66666666666666700,33333333333333300 0,62305295950155800,37196921333794800 0,58139534883720900,39899188929211300 0,54200542005420000,41492605032247600 0,50505050505050500,42086852829855600 0,47058823529411800,41829382520448200 0,43859649122807000,40885464414379900 0,40899795501022500,39420866517687600 0,38167938931297700,37589008341971100 0,35650623885918000,35523078930976600 0,33333333333333300,33332575953057300
39
Η ΜΕΘΟΔΟΣ HEWN
41
format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/4; w2=0; w3=3/4; c1=0; c2=1/3; c3=1/3; a21=1/3; a31=0; a32=2/3; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.25:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/4; w2=0; w3=3/4; c1=0; c2=1/3; c3=1/3; a21=1/3; a31=0; a32=2/3; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.1:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=0; w3=3/4; c1=0; c2=1/3; c3=1/3; a21=1/3; a31=0; a32=2/3; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=0; w3=3/4; c1=0; c2=1/3; c3=1/3; a21=1/3; a31=0; a32=2/3; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_41.jpg " format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=0; w3=3/4; c1=0; c2=1/3; c3=1/3; a21=1/3; a31=0; a32=2/3; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/4; w2=0; w3=3/4; c1=0; c2=1/3; c3=1/3; a21=1/3; a31=0; a32=2/3; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
42
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25 ix(i)y(i)g(i)E 11,00000000000000000 0,666666666666667000,33333333333333300 21,250000000000000000,984806013695988000,561403508771930000,42340250492405800 31,500000000000000000,915233351982195000,470588235294118000,44464511668807700 41,750000000000000000,811902342605009000,395061728395062000,41684061420994700 52,000000000000000000,697873163942227000,333333333333333000,36453983060889400
43
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 1,0000000000000000 2,00000000000000001,10000000000000000,9975110987654320 3,00000000000000001,20000000000000000,9852477541153300 4,00000000000000001,30000000000000000,9639167071764880 5,00000000000000001,40000000000000000,9346855826373710 6,00000000000000001,50000000000000000,8990316832050910 7,00000000000000001,60000000000000000,8585724783185570 8,00000000000000001,70000000000000000,8149100492110770 9,00000000000000001,80000000000000000,7695113167849700 10,00000000000000001,90000000000000000,7236326807502240 11,00000000000000002,00000000000000000,6782866264532730 g(i)E 0,66666666666666700,333333333333333000 0,62305295950155800,374458139263874000 0,58139534883720900,403852405278121000 0,54200542005420000,421911287122288000 0,50505050505050500,429635077586866000 0,47058823529411800,428443447910973000 0,43859649122807000,419975987090487000 0,40899795501022500,405912094200852000 0,38167938931297700,387831927471993000 0,35650623885918000,367126441891044000 0,33333333333333300,344953293119940000
44
ΤΕΤΑΡΤΗ ΤΑΞΗ
45
RUNGE KUTTA (4,4)
49
format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/6; w2=1/3; w3=1/3; w4=1/6; c1=0; c2=1/2; c3=1/2; c4=1; a21=1/2; a31=0; a32=1/2; a41=0; a42=0; a43=1; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a42*k2+ a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4* k4; end for i=1:0.25:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.25:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/6; w2=1/3; w3=1/3; w4=1/6; c1=0; c2=1/2; c3=1/2; c4=1; a21=1/2; a31=0; a32=1/2; a41=0; a42=0; a43=1; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a42*k2+ a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4* k4; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_49.jpg " format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/6; w2=1/3; w3=1/3; w4=1/6; c1=0; c2=1/2; c3=1/2; c4=1; a21=1/2; a31=0; a32=1/2; a41=0; a42=0; a43=1; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a42*k2+ a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4* k4; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
50
format('long') f=inline('-x*y^2','x','y') dsolve('Dy=- x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/6; w2=1/3; w3=1/3; w4=1/6; c1=0; c2=1/2; c3=1/2; c4=1; a21=1/2; a31=0; a32=1/2; a41=0; a42=0; a43=1; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k 1); k3=h*f(i(n)+c3*h,y(n)+a31*k 1+a32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k 1+a42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+ w3*k3+w4*k4; end for i=1:0.1:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.1:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=- x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/6; w2=1/3; w3=1/3; w4=1/6; c1=0; c2=1/2; c3=1/2; c4=1; a21=1/2; a31=0; a32=1/2; a41=0; a42=0; a43=1; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k 1); k3=h*f(i(n)+c3*h,y(n)+a31*k 1+a32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k 1+a42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+ w3*k3+w4*k4; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_50.jpg " format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=- x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/6; w2=1/3; w3=1/3; w4=1/6; c1=0; c2=1/2; c3=1/2; c4=1; a21=1/2; a31=0; a32=1/2; a41=0; a42=0; a43=1; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k 1); k3=h*f(i(n)+c3*h,y(n)+a31*k 1+a32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k 1+a42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+ w3*k3+w4*k4; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
51
ix(i)y(i)g(i)E 11,00000000000000000 0,666666666666667000,33333333333333300 21,250000000000000000,969694281772415000,561403508771930000,40829077300048500 31,500000000000000000,888880444891296000,470588235294118000,41829220959717800 41,750000000000000000,780477845927160000,395061728395062000,38541611753209800 52,000000000000000000,666661297755045000,333333333333333000,33332796442171200 ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25
52
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 1,0000000000000000 2,00000000000000001,10000000000000000,9950248651026070 3,00000000000000001,20000000000000000,9803921161009690 4,00000000000000001,30000000000000000,9569377150604990 5,00000000000000001,40000000000000000,9259257966013530 6,00000000000000001,50000000000000000,8888887236836220 7,00000000000000001,60000000000000000,8474574443936580 8,00000000000000001,70000000000000000,8032126736506880 9,00000000000000001,80000000000000000,7575756064040680 10,00000000000000001,90000000000000000,7117436646314870 11,00000000000000002,00000000000000000,6666666130890390 g(i)E 0,66666666666666700,333333333333333000 0,62305295950155800,371971905601049000 0,58139534883720900,398996767263760000 0,54200542005420000,414932295006299000 0,50505050505050500,420875291550848000 0,47058823529411800,418300488389504000 0,43859649122807000,408860953165588000 0,40899795501022500,394214718640463000 0,38167938931297700,375896217091091000 0,35650623885918000,355237425772307000 0,33333333333333300,333333279755706000
53
KUTTA ΤΥΠΟΣ(4,4)
55
format('long') f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/8; w2=3/8; w3=3/8; w4=1/8; c1=0; c2=1/3; c3=2/3; c4=1; a21=1/3; a31=-1/3; a32=1; a41=1; a42=-1; a43=1; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a 32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a 42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3 *k3+w4*k4; end for i=1:0.1:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.1:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/8; w2=3/8; w3=3/8; w4=1/8; c1=0; c2=1/3; c3=2/3; c4=1; a21=1/3; a31=-1/3; a32=1; a41=1; a42=-1; a43=1; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a 32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a 42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3 *k3+w4*k4; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_55.jpg " format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/8; w2=3/8; w3=3/8; w4=1/8; c1=0; c2=1/3; c3=2/3; c4=1; a21=1/3; a31=-1/3; a32=1; a41=1; a42=-1; a43=1; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a 32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a 42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3 *k3+w4*k4; end for i=1:0.1:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.1:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
56
format('long') f=inline('-x*y^2','x','y') dsolve('Dy=- x*y^2','y(0)=1','x'); g=inline('2/(2+x^2)','x') w1=1/8; w2=3/8; w3=3/8; w4=1/8; c1=0; c2=1/3; c3=2/3; c4=1; a21=1/3; a31=-1/3; a32=1; a41=1; a42=-1; a43=1; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a 32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a 42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3 *k3+w4*k4; end for i=1:0.25:2 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=1:0.25:2 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp (['e' num2str(j) '=']) disp(e) end MATLAB KUTTA (4,4)
![ format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=- x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/8; w2=3/8; w3=3/8; w4=1/8; c1=0; c2=1/3; c3=2/3; c4=1; a21=1/3; a31=-1/3; a32=1; a41=1; a42=-1; a43=1; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a 32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a 42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3 *k3+w4*k4; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end MATLAB KUTTA (4,4)](http://images.slideplayer.gr/15/4851840/slides/slide_56.jpg " format( long ) f=inline( -x*y^2 , x , y ) dsolve( Dy=- x*y^2 , y(0)=1 , x ); g=inline( 2/(2+x^2) , x ) w1=1/8; w2=3/8; w3=3/8; w4=1/8; c1=0; c2=1/3; c3=2/3; c4=1; a21=1/3; a31=-1/3; a32=1; a41=1; a42=-1; a43=1; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a 32*k2); k4=h*f(i(n)+c4*h,y(n)+a41*k1+a 42*k2+a43*k3); y(n+1)=y(n)+w1*k1+w2*k2+w3 *k3+w4*k4; end for i=1:0.25:2 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=1:0.25:2 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g(j)-y(j)); disp ([ e num2str(j) = ]) disp(e) end MATLAB KUTTA (4,4)")
57
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25 ix(i)y(i)g(i)E 11,00000000000000000 0,666666666666667000,33333333333333300 21,250000000000000000,969690082001976000,561403508771930000,40828657323004600 31,500000000000000000,888866928819342000,470588235294118000,41827869352522400 41,750000000000000000,780457844383869000,395061728395062000,38539611598880700 52,000000000000000000,666640176087096000,333333333333333000,33330684275376300
58
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 1,0000000000000000 2,00000000000000001,10000000000000000,9950248477747000 3,00000000000000001,20000000000000000,9803920492233100 4,00000000000000001,30000000000000000,9569375752288080 5,00000000000000001,40000000000000000,9259255732175220 6,00000000000000001,50000000000000000,8888884191546800 7,00000000000000001,60000000000000000,8474570711665200 8,00000000000000001,70000000000000000,8032122497621280 9,00000000000000001,80000000000000000,7575751512448190 10,00000000000000001,90000000000000000,7117431959363590 11,00000000000000002,00000000000000000,6666661453699930 g(i)E 0,66666666666666700,333333333333333000 0,62305295950155800,371971888273142000 0,58139534883720900,398996700386101000 0,54200542005420000,414932155174608000 0,50505050505050500,420875068167017000 0,47058823529411800,418300183860562000 0,43859649122807000,408860579938450000 0,40899795501022500,394214294751903000 0,38167938931297700,375895761931842000 0,35650623885918000,355236957077179000 0,33333333333333300,333332812036660000
59
ΑΝΩΤΕΡΗ ΤΑΞΗ
60
NYSTROM ΤΥΠΟΣ
64
Παράδειγμα:
65
f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('-1+x+2*exp(-x)','x') format('long') c1=0; c2=1/3; c3=2/5; c4=1; c5=2/3; c6=4/5; w1=23/192; w2=0; w3=125/192; w4=-81/192; w5=125/192; a21=1/3; a31=4/25; a32=6/25; a41=1/4; a42=-12/4; a43=15/4; a51=6/81; a52=90/81; a53=-50/81; a54=8/81; a61=6/75; a62=36/75; a63=10/75; a64=8/75; a65=0; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a 43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a 53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a 63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k 4+w5*k5; end for i=0:0.25:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.25:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:6 e=abs(g((j-1)/100)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(-x) , x ) format( long ) c1=0; c2=1/3; c3=2/5; c4=1; c5=2/3; c6=4/5; w1=23/192; w2=0; w3=125/192; w4=-81/192; w5=125/192; a21=1/3; a31=4/25; a32=6/25; a41=1/4; a42=-12/4; a43=15/4; a51=6/81; a52=90/81; a53=-50/81; a54=8/81; a61=6/75; a62=36/75; a63=10/75; a64=8/75; a65=0; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a 43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a 53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a 63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k 4+w5*k5; end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:6 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_65.jpg " f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(-x) , x ) format( long ) c1=0; c2=1/3; c3=2/5; c4=1; c5=2/3; c6=4/5; w1=23/192; w2=0; w3=125/192; w4=-81/192; w5=125/192; a21=1/3; a31=4/25; a32=6/25; a41=1/4; a42=-12/4; a43=15/4; a51=6/81; a52=90/81; a53=-50/81; a54=8/81; a61=6/75; a62=36/75; a63=10/75; a64=8/75; a65=0; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a 43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a 53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a 63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k 4+w5*k5; end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:6 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
66
f=inline('-x*y^2','x','y') dsolve('Dy=- x*y^2','y(0)=1','x'); g=inline('-1+x+2*exp(- x)','x') format('long') c1=0; c2=1/3; c3=2/5; c4=1; c5=2/3; c6=4/5; w1=23/192; w2=0; w3=125/192; w4=-81/192; w5=125/192; a21=1/3; a31=4/25; a32=6/25; a41=1/4; a42=-12/4; a43=15/4; a51=6/81; a52=90/81; a53=-50/81; a54=8/81; a61=6/75; a62=36/75; a63=10/75; a64=8/75; a65=0; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a53*k3+a 54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a63*k3+a 64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k4+w5*k 5; end for i=0:0.1:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.1:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:10 e=abs(g((j-1)/100)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ f=inline( -x*y^2 , x , y ) dsolve( Dy=- x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(- x) , x ) format( long ) c1=0; c2=1/3; c3=2/5; c4=1; c5=2/3; c6=4/5; w1=23/192; w2=0; w3=125/192; w4=-81/192; w5=125/192; a21=1/3; a31=4/25; a32=6/25; a41=1/4; a42=-12/4; a43=15/4; a51=6/81; a52=90/81; a53=-50/81; a54=8/81; a61=6/75; a62=36/75; a63=10/75; a64=8/75; a65=0; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a53*k3+a 54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a63*k3+a 64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k4+w5*k 5; end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:10 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_66.jpg " f=inline( -x*y^2 , x , y ) dsolve( Dy=- x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(- x) , x ) format( long ) c1=0; c2=1/3; c3=2/5; c4=1; c5=2/3; c6=4/5; w1=23/192; w2=0; w3=125/192; w4=-81/192; w5=125/192; a21=1/3; a31=4/25; a32=6/25; a41=1/4; a42=-12/4; a43=15/4; a51=6/81; a52=90/81; a53=-50/81; a54=8/81; a61=6/75; a62=36/75; a63=10/75; a64=8/75; a65=0; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a53*k3+a 54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a63*k3+a 64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k4+w5*k 5; end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:10 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
67
ix(i)y(i)g(i)E 10,000000000000000001,00000000000000000 0,00000000000000000 20,250000000000000000,973921268121264000,807601566142810000,16631970197845400 30,500000000000000000,892778599719703000,713061319425267000,17971728029443600 40,750000000000000000,779891810573625000,694733105482029000,08515870509159600 51,000000000000000000,660448711605436000,735758882342885000,075310170737449100 ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25
68
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 1,0000000000000000 0,00000000000000 00 1,000000000000 0000 2,0000000000000000 0,10000000000000 00 0,995860694497 2650 3,0000000000000000 0,20000000000000 00 0,981909066057 1790 4,0000000000000000 0,30000000000000 00 0,958876358832 1920 5,0000000000000000 0,40000000000000 00 0,927970733149 7740 6,0000000000000000 0,50000000000000 00 0,890722321728 8460 7,0000000000000000 0,60000000000000 00 0,848805325158 5730 8,0000000000000000 0,70000000000000 00 0,803872723532 9100 9,0000000000000000 0,80000000000000 00 0,757428697817 7510 10,000000000000000 0 0,90000000000000 00 0,710749250282 0590 11,000000000000000 0 1,00000000000000 00 0,664848732149 4500 g(i)E 1,00000000000000000,000000000000000000 0,90967483607191900,086185858425346000 0,83746150615596400,144447559901215000 0,78163644136343600,177239917468756000 0,74064009207127900,187330641078495000 0,71306131942526700,177661002303579000 0,69762327218805300,151182052970520000 0,69317060758281900,110702115950091000 0,69865792823444300,058770769583308100 0,71313931948119800,002390069199138980 0,73575888234288500,070910150193435100
69
Η γραφική παράσταση της προσεγγιστικής λύσης δίνεται από το πρώτο γράφημα ενώ η ακριβής λύση δίνεται από το δεύτερο γράφημα.
71
FEHLBERG ΤΥΠΟΣ
76
f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('-1+x+2*exp(-x)','x') format('long') c1=0; c2=1/6; c3=4/15; c4=2/3; c5=4/5; c6=1; w1=31/384; w2=0; w3=1125/2816; w4=9/32; w5=125/768; w6=5/66; a21=1/6; a31=4/75; a32=16/75; a41=5/6; a42=-8/3; a43=5/2; a51=-8/5; a52=144/25; a53=-4; a54=16/25; a61=361/320; a62=-18/5; a63=407/128; a64=-11/80; a65=55/128; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32* k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42* k2+a43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52* k2+a53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62* k2+a63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+ w4*k4+w5*k5+w6*k6; end for i=0:0.25:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.25:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/100)-y(j)); disp (['e' num2str(j) '=']) disp(e) end
![ f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(-x) , x ) format( long ) c1=0; c2=1/6; c3=4/15; c4=2/3; c5=4/5; c6=1; w1=31/384; w2=0; w3=1125/2816; w4=9/32; w5=125/768; w6=5/66; a21=1/6; a31=4/75; a32=16/75; a41=5/6; a42=-8/3; a43=5/2; a51=-8/5; a52=144/25; a53=-4; a54=16/25; a61=361/320; a62=-18/5; a63=407/128; a64=-11/80; a65=55/128; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32* k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42* k2+a43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52* k2+a53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62* k2+a63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+ w4*k4+w5*k5+w6*k6; end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end](http://images.slideplayer.gr/15/4851840/slides/slide_76.jpg " f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(-x) , x ) format( long ) c1=0; c2=1/6; c3=4/15; c4=2/3; c5=4/5; c6=1; w1=31/384; w2=0; w3=1125/2816; w4=9/32; w5=125/768; w6=5/66; a21=1/6; a31=4/75; a32=16/75; a41=5/6; a42=-8/3; a43=5/2; a51=-8/5; a52=144/25; a53=-4; a54=16/25; a61=361/320; a62=-18/5; a63=407/128; a64=-11/80; a65=55/128; h=0.25; i=0:0.25:1; nstep=5; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32* k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42* k2+a43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52* k2+a53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62* k2+a63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+ w4*k4+w5*k5+w6*k6; end for i=0:0.25:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.25:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:5 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end")
77
f=inline('-x*y^2','x','y') dsolve('Dy=-x*y^2','y(0)=1','x'); g=inline('-1+x+2*exp(-x)','x') format('long') c1=0; c2=1/6; c3=4/15; c4=2/3; c5=4/5; c6=1; w1=31/384; w2=0; w3=1125/2816; w4=9/32; w5=125/768; w6=5/66; a21=1/6; a31=4/75; a32=16/75; a41=5/6; a42=-8/3; a43=5/2; a51=-8/5; a52=144/25; a53=-4; a54=16/25; a61=361/320; a62=-18/5; a63=407/128; a64=-11/80; a65=55/128; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a 43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a 53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a 63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k 4+w5*k5+w6*k6; end for i=0:0.1:1 disp (['x' num2str(i) '=']) disp(i) end for n=1:nstep disp (['y' num2str(n) '=']) disp(y(n)) end for i=0:0.1:1 disp (['g' num2str(i) '=' ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/100)-y(j)); disp (['e' num2str(j) '=']) disp(e) end x=0:0.1:1; subplot(1,2,1) plot(y) subplot(1,2,2) plot(g(x))
![ f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(-x) , x ) format( long ) c1=0; c2=1/6; c3=4/15; c4=2/3; c5=4/5; c6=1; w1=31/384; w2=0; w3=1125/2816; w4=9/32; w5=125/768; w6=5/66; a21=1/6; a31=4/75; a32=16/75; a41=5/6; a42=-8/3; a43=5/2; a51=-8/5; a52=144/25; a53=-4; a54=16/25; a61=361/320; a62=-18/5; a63=407/128; a64=-11/80; a65=55/128; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a 43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a 53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a 63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k 4+w5*k5+w6*k6; end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end x=0:0.1:1; subplot(1,2,1) plot(y) subplot(1,2,2) plot(g(x))](http://images.slideplayer.gr/15/4851840/slides/slide_77.jpg " f=inline( -x*y^2 , x , y ) dsolve( Dy=-x*y^2 , y(0)=1 , x ); g=inline( -1+x+2*exp(-x) , x ) format( long ) c1=0; c2=1/6; c3=4/15; c4=2/3; c5=4/5; c6=1; w1=31/384; w2=0; w3=1125/2816; w4=9/32; w5=125/768; w6=5/66; a21=1/6; a31=4/75; a32=16/75; a41=5/6; a42=-8/3; a43=5/2; a51=-8/5; a52=144/25; a53=-4; a54=16/25; a61=361/320; a62=-18/5; a63=407/128; a64=-11/80; a65=55/128; h=0.1; i=0:0.1:1; nstep=11; y(1)=1; for n=1:nstep k1=h*f(i(n)+c1*h,y(n)); k2=h*f(i(n)+c2*h,y(n)+a21*k1); k3=h*f(i(n)+c3*h,y(n)+a31*k1+a32*k2); k4=h*f(i(n)+c5*h,y(n)+a41*k1+a42*k2+a 43*k3); k5=h*f(i(n)+c5*h,y(n)+a51*k1+a52*k2+a 53*k3+a54*k4); k6=h*f(i(n)+c6*h,y(n)+a61*k1+a62*k2+a 63*k3+a64*k4+a65*k5); y(n+1)=y(n)+w1*k1+w2*k2+w3*k3+w4*k 4+w5*k5+w6*k6; end for i=0:0.1:1 disp ([ x num2str(i) = ]) disp(i) end for n=1:nstep disp ([ y num2str(n) = ]) disp(y(n)) end for i=0:0.1:1 disp ([ g num2str(i) = ]) disp(g(i)) end for j=1:1:11 e=abs(g((j-1)/100)-y(j)); disp ([ e num2str(j) = ]) disp(e) end x=0:0.1:1; subplot(1,2,1) plot(y) subplot(1,2,2) plot(g(x))")
78
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.25 ix(i)y(i)g(i)E 10,000000000000000001,00000000000000000 0,00000000000000000 20,250000000000000000,967487258166394000,807601566142810000,15988569202358400 30,500000000000000000,885180544753583000,713061319425267000,17211922532831600 40,750000000000000000,776203891812263000,694733105482029000,08147078633023400 51,000000000000000000,662502743089561000,735758882342885000,07325613925332400
79
ΑΠΟΤΕΛΕΣΜΑΤΑ ΓΙΑ h=0.1 ix(i)y(i) 1,00000000000000000,0000000000000000 1,00000000000000 00 2,00000000000000000,1000000000000000 0,99465345693513 60 3,00000000000000000,2000000000000000 0,97967125903696 70 4,00000000000000000,3000000000000000 0,95590791202513 00 5,00000000000000000,4000000000000000 0,92464069260953 70 6,00000000000000000,5000000000000000 0,88740870082314 10 7,00000000000000000,6000000000000000 0,84584350412954 40 8,00000000000000000,7000000000000000 0,80152154830948 60 9,00000000000000000,8000000000000000 0,75585653628368 40 10,00000000000000000,9000000000000000 0,71003682167456 80 11,00000000000000001,0000000000000000 0,66500286102890 30 g(i)E 1,00000000000000000,000000000000000000 0,90967483607191900,084978620863216900 0,83746150615596400,142209752881003000 0,78163644136343600,174271470661694000 0,74064009207127900,184000600538258000 0,71306131942526700,174347381397874000 0,69762327218805300,148220231941491000 0,69317060758281900,108350940726667000 0,69865792823444300,057198608049241000 0,71313931948119800,003102497806629970 0,73575888234288500,070756021313982000
80
Η γραφική παράσταση της προσεγγιστικής λύσης δίνεται από το πρώτο γράφημα ενώ η ακριβής λύση δίνεται από το δεύτερο γράφημα.
82
Για βήμα h=0.01 στην τιμή χ=1 έχουμε y(1)=0.66650004 0441335 g(1)= 0.735758882342 885 και απόλυτο σφάλμα ίσο με 0.069258841901 550
Παρόμοιες παρουσιάσεις
