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Hi!

I am simulate the code for fractional differential equation. But the out put is not wright...
sir_(2).mw

``

S[0] := .8;

.8

(1)

V[0] := .2;

.2

(2)

R[0] := 0;

0

(3)

alpha := 1;

1

 

.4

 

.8

 

gamma = 0.3e-1

(4)

q := .9;

.9

(5)

T := 1;

1

(6)

N := 5;

5

(7)

h := T/N;

1/5

(8)

``

for i from 0 to N do for j from 0 to 0 do a[j, i+1] := i^(alpha+1)-(i-alpha)*(i+1)^alpha; b[j, i+1] := h^alpha*((i+1-j)^alpha-(i-j)^alpha)/alpha end do end do;

for n from 0 to N do Sp[n+1] = S[0]+(sum(b[d, n+1]*(mu*(1-q)-beta*S[d]*V[d]-mu*S[d]), d = 0 .. n))/GAMMA(alpha); Vp[n+1] = V[0]+(sum(b[d, n+1]*(beta*S[d]*V[d]-(mu+gamma)*S[d]), d = 0 .. n))/GAMMA(alpha); Rp[n+1] = R[0]+(sum(b[d, n+1]*(mu*q-mu*R[d]+gamma*V[d]), d = 0 .. n))/GAMMA(alpha); S[n+1] = S[0]+h^alpha*(mu*(1-q)-beta*Sp[n+1]*Vp[n+1]-mu*Sp[n+1])/GAMMA(alpha+2)+h^alpha*(sum(a[e, n+1]*(mu*(1-q)-beta*S[e]*V[e]-mu*S[e]), e = 0 .. n))/GAMMA(alpha+2); V[n+1] = V[0]+h^alpha*(beta*Sp[n+1]*Vp[n+1]-(mu+gamma)*Sp[n+1])/GAMMA(alpha+2)+h^alpha*(sum(a[e, n+1]*(beta*S[e]*V[e]-(mu+gamma)*S[e]), e = 0 .. n))/GAMMA(alpha+2); R[n+1] = R[0]+h^alpha*(mu*q-mu*Rp[n+1]-gamma*Vp[n+1])/GAMMA(alpha+2)+h^alpha*(sum(a[e, n+1]*(mu*q-mu*R[e]-gamma*V[e]), e = 0 .. n))/GAMMA(alpha+2) end do;

Sp[1] = .7184000000

 

Vp[1] = 0.692454936e-1

 

Rp[1] = 0.9508862660e-1

 

S[1] = .7632000000-0.8000000000e-1*Sp[1]*Vp[1]-0.4000000000e-1*Sp[1]

 

V[1] = .1346227468+0.8000000000e-1*Sp[1]*Vp[1]-(1/10)*(.4+gamma)*Sp[1]

 

R[1] = 0.6045568670e-1-(1/10)*gamma*Vp[1]-0.4000000000e-1*Rp[1]

 

Sp[2] = .7264000000-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]

 

Vp[2] = 0.692454936e-1+.1600000000*S[1]*V[1]-.1954431330*S[1]

 

Rp[2] = .1670886266+.1154431330*V[1]-0.8000000000e-1*R[1]

 

S[2] = .7712000000-0.8000000000e-1*Sp[2]*Vp[2]-0.4000000000e-1*Sp[2]-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]

 

V[2] = .1346227468+0.8000000000e-1*Sp[2]*Vp[2]-(1/10)*(.4+gamma)*Sp[2]+.1600000000*S[1]*V[1]-.1954431330*S[1]

 

R[2] = .1324556867-(1/10)*gamma*Vp[2]-0.4000000000e-1*Rp[2]-.1154431330*V[1]-0.8000000000e-1*R[1]

 

Sp[3] = .7344000000-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]

 

Vp[3] = 0.692454936e-1+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]

 

Rp[3] = .2390886266+.1154431330*V[1]-0.8000000000e-1*R[1]+.1154431330*V[2]-0.8000000000e-1*R[2]

 

S[3] = .7792000000-0.8000000000e-1*Sp[3]*Vp[3]-0.4000000000e-1*Sp[3]-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]

 

V[3] = .1346227468+0.8000000000e-1*Sp[3]*Vp[3]-(1/10)*(.4+gamma)*Sp[3]+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]

 

R[3] = .2044556867-(1/10)*gamma*Vp[3]-0.4000000000e-1*Rp[3]-.1154431330*V[1]-0.8000000000e-1*R[1]-.1154431330*V[2]-0.8000000000e-1*R[2]

 

Sp[4] = .7424000000-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]-.1600000000*S[3]*V[3]-0.8000000000e-1*S[3]

 

Vp[4] = 0.692454936e-1+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]+.1600000000*S[3]*V[3]-.1954431330*S[3]

 

Rp[4] = .3110886266+.1154431330*V[1]-0.8000000000e-1*R[1]+.1154431330*V[2]-0.8000000000e-1*R[2]+.1154431330*V[3]-0.8000000000e-1*R[3]

 

S[4] = .7872000000-0.8000000000e-1*Sp[4]*Vp[4]-0.4000000000e-1*Sp[4]-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]-.1600000000*S[3]*V[3]-0.8000000000e-1*S[3]

 

V[4] = .1346227468+0.8000000000e-1*Sp[4]*Vp[4]-(1/10)*(.4+gamma)*Sp[4]+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]+.1600000000*S[3]*V[3]-.1954431330*S[3]

 

R[4] = .2764556867-(1/10)*gamma*Vp[4]-0.4000000000e-1*Rp[4]-.1154431330*V[1]-0.8000000000e-1*R[1]-.1154431330*V[2]-0.8000000000e-1*R[2]-.1154431330*V[3]-0.8000000000e-1*R[3]

 

Sp[5] = .7504000000-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]-.1600000000*S[3]*V[3]-0.8000000000e-1*S[3]-.1600000000*S[4]*V[4]-0.8000000000e-1*S[4]

 

Vp[5] = 0.692454936e-1+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]+.1600000000*S[3]*V[3]-.1954431330*S[3]+.1600000000*S[4]*V[4]-.1954431330*S[4]

 

Rp[5] = .3830886266+.1154431330*V[1]-0.8000000000e-1*R[1]+.1154431330*V[2]-0.8000000000e-1*R[2]+.1154431330*V[3]-0.8000000000e-1*R[3]+.1154431330*V[4]-0.8000000000e-1*R[4]

 

S[5] = .7952000000-0.8000000000e-1*Sp[5]*Vp[5]-0.4000000000e-1*Sp[5]-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]-.1600000000*S[3]*V[3]-0.8000000000e-1*S[3]-.1600000000*S[4]*V[4]-0.8000000000e-1*S[4]

 

V[5] = .1346227468+0.8000000000e-1*Sp[5]*Vp[5]-(1/10)*(.4+gamma)*Sp[5]+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]+.1600000000*S[3]*V[3]-.1954431330*S[3]+.1600000000*S[4]*V[4]-.1954431330*S[4]

 

R[5] = .3484556867-(1/10)*gamma*Vp[5]-0.4000000000e-1*Rp[5]-.1154431330*V[1]-0.8000000000e-1*R[1]-.1154431330*V[2]-0.8000000000e-1*R[2]-.1154431330*V[3]-0.8000000000e-1*R[3]-.1154431330*V[4]-0.8000000000e-1*R[4]

 

Sp[6] = .7584000000-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]-.1600000000*S[3]*V[3]-0.8000000000e-1*S[3]-.1600000000*S[4]*V[4]-0.8000000000e-1*S[4]-.1600000000*S[5]*V[5]-0.8000000000e-1*S[5]

 

Vp[6] = 0.692454936e-1+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]+.1600000000*S[3]*V[3]-.1954431330*S[3]+.1600000000*S[4]*V[4]-.1954431330*S[4]+.1600000000*S[5]*V[5]-.1954431330*S[5]

 

Rp[6] = .4550886266+.1154431330*V[1]-0.8000000000e-1*R[1]+.1154431330*V[2]-0.8000000000e-1*R[2]+.1154431330*V[3]-0.8000000000e-1*R[3]+.1154431330*V[4]-0.8000000000e-1*R[4]+.1154431330*V[5]-0.8000000000e-1*R[5]

 

S[6] = .8032000000-0.8000000000e-1*Sp[6]*Vp[6]-0.4000000000e-1*Sp[6]-.1600000000*S[1]*V[1]-0.8000000000e-1*S[1]-.1600000000*S[2]*V[2]-0.8000000000e-1*S[2]-.1600000000*S[3]*V[3]-0.8000000000e-1*S[3]-.1600000000*S[4]*V[4]-0.8000000000e-1*S[4]-.1600000000*S[5]*V[5]-0.8000000000e-1*S[5]

 

V[6] = .1346227468+0.8000000000e-1*Sp[6]*Vp[6]-(1/10)*(.4+gamma)*Sp[6]+.1600000000*S[1]*V[1]-.1954431330*S[1]+.1600000000*S[2]*V[2]-.1954431330*S[2]+.1600000000*S[3]*V[3]-.1954431330*S[3]+.1600000000*S[4]*V[4]-.1954431330*S[4]+.1600000000*S[5]*V[5]-.1954431330*S[5]

 

R[6] = .4204556867-(1/10)*gamma*Vp[6]-0.4000000000e-1*Rp[6]-.1154431330*V[1]-0.8000000000e-1*R[1]-.1154431330*V[2]-0.8000000000e-1*R[2]-.1154431330*V[3]-0.8000000000e-1*R[3]-.1154431330*V[4]-0.8000000000e-1*R[4]-.1154431330*V[5]-0.8000000000e-1*R[5]

(9)

``

``

 

Download sir_(2).mw

 

Hello guys,

I was just playing around with differential equations, when I noticed that symbolic solution is  different from the numerical.What is the reason for this strange behavior?


ODE := (diff(y(x), x))*(ln(y(x))+x) = 1

sol := dsolve({ODE, y(1) = 1}, y(x))

a := plot(op(2, sol), x = .75 .. 2, color = "Red");
sol2 := dsolve([ODE, y(1) = 1], numeric, range = .75 .. 2);

with(plots);
b := odeplot(sol2, .75 .. 2, thickness = 4);
display({a, b});

 

 

Strange_issue.mw

Mariusz Iwaniuk

Hi everyone. I'm going to solve a problem of an article with hpm. well I wrote some initial codes(I uploaded both codes and article). but now I face with a problem. I cant reach to the correct plot that is in the article. could you please help me???

(dont think I am lazy ;))) I found f and g (by make a system with A1 and B1 and solve it i found f[0] and g[0], with p^3 coefficient in A-->f[1] and then with B2 I foud g[1]) and their plot was correct. but the problem is theta and phi and their plots :(( )

Project.mw

2.pdf   this is article



 

restart;

lambda:=0.5;K[r]:=0.5;Sc:=0.5;Nb:=0.1;Nt:=0.1;Pr:=10;

.5

 

.5

 

.5

 

.1

 

.1

 

10

(1)

EQUATIONS

equ1:=diff(f(eta),eta$4)-R*(diff(f(eta),eta)*diff(f(eta),eta$2)-f(eta)*diff(f(eta),eta$2))-2*K[r]*diff(g(eta),eta)=0;

equ2:=diff(g(eta),eta$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta)=0;

equ3:=diff(theta(eta),eta$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2=0;

equ4:=diff(phi(eta),eta$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta$2)*(Nt/Nb)=0;

diff(diff(diff(diff(f(eta), eta), eta), eta), eta)-R*((diff(f(eta), eta))*(diff(diff(f(eta), eta), eta))-f(eta)*(diff(diff(f(eta), eta), eta)))-1.0*(diff(g(eta), eta)) = 0

 

diff(diff(g(eta), eta), eta)-R*((diff(f(eta), eta))*g(eta)-f(eta)*(diff(g(eta), eta)))+1.0*(diff(f(eta), eta)) = 0

 

diff(diff(theta(eta), eta), eta)+10*R*f(eta)*(diff(theta(eta), eta))+.1*(diff(phi(eta), eta))*(diff(theta(eta), eta))+.1*(diff(theta(eta), eta))^2 = 0

 

diff(diff(phi(eta), eta), eta)+.5*R*f(eta)*(diff(phi(eta), eta))+1.000000000*(diff(diff(theta(eta), eta), eta)) = 0

(2)

BOUNDARY*CONDITIONS

ics:=
f(0)=0,D(f)(0)=1,g(0)=0,theta(0)=1,phi(0)=1;
f(1)=lambda,D(f)(1)=0,g(1)=0,theta(1)=0,phi(1)=0;

f(0) = 0, (D(f))(0) = 1, g(0) = 0, theta(0) = 1, phi(0) = 1

 

f(1) = .5, (D(f))(1) = 0, g(1) = 0, theta(1) = 0, phi(1) = 0

(3)

HPMs

hpm1:=(1-p)*(diff(f(eta),eta$4)-2*K[r]*diff(g(eta),eta))+p*(diff(f(eta),eta$4)-R*(diff(f(eta),eta)*diff(f(eta),eta$2)-f(eta)*diff(f(eta),eta$2))-2*K[r]*diff(g(eta),eta))=0;

hpm2:=(1-p)*(diff(g(eta),eta$2)+2*K[r]*diff(f(eta),eta))+p*(diff(g(eta),eta$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta))=0;

hpm3:=(1-p)*(diff(theta(eta),eta$2))+p*(diff(theta(eta),eta$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2)=0;

hpm4:=(1-p)*(diff(phi(eta),eta$2)+diff(theta(eta),eta$2)*(Nt/Nb))+p*(diff(phi(eta),eta$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta$2)*(Nt/Nb))=0;

(1-p)*(diff(diff(diff(diff(f(eta), eta), eta), eta), eta)-1.0*(diff(g(eta), eta)))+p*(diff(diff(diff(diff(f(eta), eta), eta), eta), eta)-R*((diff(f(eta), eta))*(diff(diff(f(eta), eta), eta))-f(eta)*(diff(diff(f(eta), eta), eta)))-1.0*(diff(g(eta), eta))) = 0

 

(1-p)*(diff(diff(g(eta), eta), eta)+1.0*(diff(f(eta), eta)))+p*(diff(diff(g(eta), eta), eta)-R*((diff(f(eta), eta))*g(eta)-f(eta)*(diff(g(eta), eta)))+1.0*(diff(f(eta), eta))) = 0

 

(1-p)*(diff(diff(theta(eta), eta), eta))+p*(diff(diff(theta(eta), eta), eta)+10*R*f(eta)*(diff(theta(eta), eta))+.1*(diff(phi(eta), eta))*(diff(theta(eta), eta))+.1*(diff(theta(eta), eta))^2) = 0

 

(1-p)*(diff(diff(phi(eta), eta), eta)+1.000000000*(diff(diff(theta(eta), eta), eta)))+p*(diff(diff(phi(eta), eta), eta)+.5*R*f(eta)*(diff(phi(eta), eta))+1.000000000*(diff(diff(theta(eta), eta), eta))) = 0

(4)

f(eta)=sum(f[i](eta)*p^i,i=0..1);

f(eta) = f[0](eta)+f[1](eta)*p

(5)

g(eta)=sum(g[i](eta)*p^i,i=0..1);

g(eta) = g[0](eta)+g[1](eta)*p

(6)

theta(eta)=sum(theta[i](eta)*p^i,i=0..1);

theta(eta) = theta[0](eta)+theta[1](eta)*p

(7)

phi(eta)=sum(phi[i](eta)*p^i,i=0..1);

phi(eta) = phi[0](eta)+phi[1](eta)*p

(8)

FORequ1

A:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm1)),p);

(-1.*R*(diff(f[1](eta), eta))*(diff(diff(f[1](eta), eta), eta))+R*f[1](eta)*(diff(diff(f[1](eta), eta), eta)))*p^3+(-1.*R*(diff(f[0](eta), eta))*(diff(diff(f[1](eta), eta), eta))-1.*R*(diff(f[1](eta), eta))*(diff(diff(f[0](eta), eta), eta))+R*f[0](eta)*(diff(diff(f[1](eta), eta), eta))+R*f[1](eta)*(diff(diff(f[0](eta), eta), eta)))*p^2+(diff(diff(diff(diff(f[1](eta), eta), eta), eta), eta)-1.0*(diff(g[1](eta), eta))-1.*R*(diff(f[0](eta), eta))*(diff(diff(f[0](eta), eta), eta))+R*f[0](eta)*(diff(diff(f[0](eta), eta), eta)))*p+diff(diff(diff(diff(f[0](eta), eta), eta), eta), eta)-1.0*(diff(g[0](eta), eta)) = 0

(9)

A1:=diff(f[0](eta),eta$4)-2*K[r]*(diff(g[0](eta),eta))=0;
A2:=diff(f[1](eta),eta$4)-2*K[r]*(diff(g[1](eta),eta))-R*(diff(f[0](eta),eta))*(diff(f[0](eta),eta$2))+R*f[0](eta)*(diff(f[0](eta),eta$2))=0;

diff(diff(diff(diff(f[0](eta), eta), eta), eta), eta)-1.0*(diff(g[0](eta), eta)) = 0

 

diff(diff(diff(diff(f[1](eta), eta), eta), eta), eta)-1.0*(diff(g[1](eta), eta))-R*(diff(f[0](eta), eta))*(diff(diff(f[0](eta), eta), eta))+R*f[0](eta)*(diff(diff(f[0](eta), eta), eta)) = 0

(10)

icsA1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsA2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;

f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0

 

f[1](0) = 0, (D(f[1]))(0) = 0, g[1](0) = 0, f[1](1) = 0, (D(f[1]))(1) = 0, g[1](1) = 0

(11)

NULLFORequ2

B:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm2)),p);

(-1.*R*(diff(f[1](eta), eta))*g[1](eta)+R*f[1](eta)*(diff(g[1](eta), eta)))*p^3+(-1.*R*(diff(f[0](eta), eta))*g[1](eta)-1.*R*(diff(f[1](eta), eta))*g[0](eta)+R*f[0](eta)*(diff(g[1](eta), eta))+R*f[1](eta)*(diff(g[0](eta), eta)))*p^2+(diff(diff(g[1](eta), eta), eta)+1.0*(diff(f[1](eta), eta))-1.*R*(diff(f[0](eta), eta))*g[0](eta)+R*f[0](eta)*(diff(g[0](eta), eta)))*p+diff(diff(g[0](eta), eta), eta)+1.0*(diff(f[0](eta), eta)) = 0

(12)

B1:=diff(g[0](eta),eta$2)+2*K[r]*(diff(f[0](eta),eta))=0;
B2:=diff(g[1](eta),eta$2)+2*K[r]*(diff(f[1](eta),eta))-R*(diff(f[0](eta),eta))*g[0](eta)+R*f[0](eta)*(diff(g[0](eta),eta))=0;

diff(diff(g[0](eta), eta), eta)+1.0*(diff(f[0](eta), eta)) = 0

 

diff(diff(g[1](eta), eta), eta)+1.0*(diff(f[1](eta), eta))-R*(diff(f[0](eta), eta))*g[0](eta)+R*f[0](eta)*(diff(g[0](eta), eta)) = 0

(13)

icsB1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsB2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;

f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0

 

f[1](0) = 0, (D(f[1]))(0) = 0, g[1](0) = 0, f[1](1) = 0, (D(f[1]))(1) = 0, g[1](1) = 0

(14)

FORequ3

C:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm3)),p);

(10.*R*f[1](eta)*(diff(theta[1](eta), eta))+.1*(diff(phi[1](eta), eta))*(diff(theta[1](eta), eta))+.1*(diff(theta[1](eta), eta))^2)*p^3+(10.*R*f[0](eta)*(diff(theta[1](eta), eta))+10.*R*f[1](eta)*(diff(theta[0](eta), eta))+.1*(diff(phi[0](eta), eta))*(diff(theta[1](eta), eta))+.1*(diff(phi[1](eta), eta))*(diff(theta[0](eta), eta))+.2*(diff(theta[0](eta), eta))*(diff(theta[1](eta), eta)))*p^2+(diff(diff(theta[1](eta), eta), eta)+10.*R*f[0](eta)*(diff(theta[0](eta), eta))+.1*(diff(phi[0](eta), eta))*(diff(theta[0](eta), eta))+.1*(diff(theta[0](eta), eta))^2)*p+diff(diff(theta[0](eta), eta), eta) = 0

(15)

C1:=diff(theta[0](eta),eta$2)=0;
C2:=diff(theta[1](eta), eta, eta)+Pr*R*f[0](eta)*(diff(theta[0](eta), eta))+Nb*(diff(phi[0](eta), eta))*(diff(theta[0](eta), eta))+Nt*(diff(theta[0](eta), eta))^2=0;

diff(diff(theta[0](eta), eta), eta) = 0

 

diff(diff(theta[1](eta), eta), eta)+10*R*f[0](eta)*(diff(theta[0](eta), eta))+.1*(diff(phi[0](eta), eta))*(diff(theta[0](eta), eta))+.1*(diff(theta[0](eta), eta))^2 = 0

(16)

icsC1:=theta[0](0)=1,theta[0](1)=0;
icsC2:=f[0](0)=0,D(f[0])(0)=1,f[1](1)=0,D(f[1])(1)=0,theta[1](0)=0,theta[1](1)=0,phi[0](0)=0,phi[0](1)=0;

theta[0](0) = 1, theta[0](1) = 0

 

f[0](0) = 0, (D(f[0]))(0) = 1, f[1](1) = 0, (D(f[1]))(1) = 0, theta[1](0) = 0, theta[1](1) = 0, phi[0](0) = 0, phi[0](1) = 0

(17)

FORequ4

E:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm4)),p);

.5*R*f[1](eta)*p^3*(diff(phi[1](eta), eta))+(.5*R*f[0](eta)*(diff(phi[1](eta), eta))+.5*R*f[1](eta)*(diff(phi[0](eta), eta)))*p^2+(diff(diff(phi[1](eta), eta), eta)+1.000000000*(diff(diff(theta[1](eta), eta), eta))+.5*R*f[0](eta)*(diff(phi[0](eta), eta)))*p+diff(diff(phi[0](eta), eta), eta)+1.000000000*(diff(diff(theta[0](eta), eta), eta)) = 0

(18)

E1:=diff(phi[0](eta),eta$2)+Nt*(diff(theta[0](eta),eta$2))/Nb=0;
E2:=diff(phi[1](eta),eta$2)+Nt*(diff(theta[1](eta),eta$2))/Nb+R*Sc*f[0](eta)*(diff(phi[0](eta),eta))=0;

diff(diff(phi[0](eta), eta), eta)+1.000000000*(diff(diff(theta[0](eta), eta), eta)) = 0

 

diff(diff(phi[1](eta), eta), eta)+1.000000000*(diff(diff(theta[1](eta), eta), eta))+.5*R*f[0](eta)*(diff(phi[0](eta), eta)) = 0

(19)

icsE1:=phi[0](0)=1,phi[0](1)=0;
icsE2:=f[0](0)=0,D(f[0])(0)=1,f[1](1)=0,D(f[1])(1)=0,theta[1](0)=0,theta[1](1)=0,phi[1](0)=0,phi[1](1)=0;

phi[0](0) = 1, phi[0](1) = 0

 

f[0](0) = 0, (D(f[0]))(0) = 1, f[1](1) = 0, (D(f[1]))(1) = 0, theta[1](0) = 0, theta[1](1) = 0, phi[1](0) = 0, phi[1](1) = 0

(20)

``

NULL



Download Project.mw


Project.mw

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thanks for your favorits

Hi, i am trying to solve my PDEs with HPM method ,but i get strange errors.

first one is :Error, (in trig/reduce/reduce) Maple was unable to allocate enough memory to complete this computation.  Please see ?alloc,

but when i run my last function again,the error chages,let me show you.


restart;
lambda:=0.5;K[r]:=0.5;Sc:=0.5;Nb:=0.1;Nt:=0.1;Pr:=10;
                              0.5
                              0.5
                              0.5
                              0.1
                              0.1
                               10
> EQUATIONS;


equ1:=diff(f(eta),eta$4)-R*(diff(f(eta),eta)*diff(f(eta),eta$2)-f(eta)*diff(f(eta),eta$2))-2*K[r]*diff(g(eta),eta)=0;

equ2:=diff(g(eta),eta$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta)=0;

equ3:=diff(theta(eta),eta$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2=0;

equ4:=diff(phi(eta),eta$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta$2)*(Nt/Nb)=0;
/  d   /  d   /  d   /  d         \\\\     //  d         \ /  d  
|----- |----- |----- |----- f(eta)|||| - R ||----- f(eta)| |-----
\ deta \ deta \ deta \ deta       ////     \\ deta       / \ deta

   /  d         \\          /  d   /  d         \\\
   |----- f(eta)|| - f(eta) |----- |----- f(eta)|||
   \ deta       //          \ deta \ deta       ///

         /  d         \    
   - 1.0 |----- g(eta)| = 0
         \ deta       /    
     /  d   /  d         \\
     |----- |----- g(eta)||
     \ deta \ deta       //

            //  d         \                 /  d         \\
        - R ||----- f(eta)| g(eta) - f(eta) |----- g(eta)||
            \\ deta       /                 \ deta       //

              /  d         \    
        + 1.0 |----- f(eta)| = 0
              \ deta       /    
  /  d   /  d             \\               /  d             \
  |----- |----- theta(eta)|| + 10 R f(eta) |----- theta(eta)|
  \ deta \ deta           //               \ deta           /

           /  d           \ /  d             \
     + 0.1 |----- phi(eta)| |----- theta(eta)|
           \ deta         / \ deta           /

                             2    
           /  d             \     
     + 0.1 |----- theta(eta)|  = 0
           \ deta           /     
    /  d   /  d           \\                /  d           \
    |----- |----- phi(eta)|| + 0.5 R f(eta) |----- phi(eta)|
    \ deta \ deta         //                \ deta         /

                     /  d   /  d             \\    
       + 1.000000000 |----- |----- theta(eta)|| = 0
                     \ deta \ deta           //    
> BOUNDARY*CONDITIONS;


ics:=
f(0)=0,D(f)(0)=1,g(0)=0,theta(0)=1,phi(0)=1;
f(1)=lambda,D(f)(1)=0,g(1)=0,theta(1)=0,phi(1)=0;
   f(0) = 0, D(f)(0) = 1, g(0) = 0, theta(0) = 1, phi(0) = 1
  f(1) = 0.5, D(f)(1) = 0, g(1) = 0, theta(1) = 0, phi(1) = 0
> HPMs;


hpm1:=(1-p)*(diff(f(eta),eta$4)-2*K[r]*diff(g(eta),eta))+p*(diff(f(eta),eta$4)-R*(diff(f(eta),eta)*diff(f(eta),eta$2)-f(eta)*diff(f(eta),eta$2))-2*K[r]*diff(g(eta),eta))=0;

hpm2:=(1-p)*(diff(g(eta),eta$2)+2*K[r]*diff(f(eta),eta))+p*(diff(g(eta),eta$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta))=0;

hpm3:=(1-p)*(diff(theta(eta),eta$2))+p*(diff(theta(eta),eta$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2)=0;

hpm4:=(1-p)*(diff(phi(eta),eta$2)+diff(theta(eta),eta$2)*(Nt/Nb))+p*(diff(phi(eta),eta$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta$2)*(Nt/Nb))=0;

        //  d   /  d   /  d   /  d         \\\\
(1 - p) ||----- |----- |----- |----- f(eta)||||
        \\ deta \ deta \ deta \ deta       ////

         /  d         \\     //  d   /  d   /  d   /  d         \
   - 1.0 |----- g(eta)|| + p ||----- |----- |----- |----- f(eta)|
         \ deta       //     \\ deta \ deta \ deta \ deta       /

  \\\     //  d         \ /  d   /  d         \\
  ||| - R ||----- f(eta)| |----- |----- f(eta)||
  ///     \\ deta       / \ deta \ deta       //

            /  d   /  d         \\\       /  d         \\    
   - f(eta) |----- |----- f(eta)||| - 1.0 |----- g(eta)|| = 0
            \ deta \ deta       ///       \ deta       //    
        //  d   /  d         \\       /  d         \\     //  d  
(1 - p) ||----- |----- g(eta)|| + 1.0 |----- f(eta)|| + p ||-----
        \\ deta \ deta       //       \ deta       //     \\ deta

   /  d         \\
   |----- g(eta)||
   \ deta       //

       //  d         \                 /  d         \\
   - R ||----- f(eta)| g(eta) - f(eta) |----- g(eta)||
       \\ deta       /                 \ deta       //

         /  d         \\    
   + 1.0 |----- f(eta)|| = 0
         \ deta       //    
                                       /                         
        /  d   /  d             \\     |/  d   /  d             \
(1 - p) |----- |----- theta(eta)|| + p ||----- |----- theta(eta)|
        \ deta \ deta           //     \\ deta \ deta           /

  \               /  d             \
  | + 10 R f(eta) |----- theta(eta)|
  /               \ deta           /

         /  d           \ /  d             \
   + 0.1 |----- phi(eta)| |----- theta(eta)|
         \ deta         / \ deta           /

                           2\    
         /  d             \ |    
   + 0.1 |----- theta(eta)| | = 0
         \ deta           / /    
        //  d   /  d           \\
(1 - p) ||----- |----- phi(eta)||
        \\ deta \ deta         //

                 /  d   /  d             \\\     //  d   /  d   
   + 1.000000000 |----- |----- theta(eta)||| + p ||----- |-----
                 \ deta \ deta           ///     \\ deta \ deta

          \\                /  d           \
  phi(eta)|| + 0.5 R f(eta) |----- phi(eta)|
          //                \ deta         /

                 /  d   /  d             \\\    
   + 1.000000000 |----- |----- theta(eta)||| = 0
                 \ deta \ deta           ///    
f(eta)=sum(f[i](eta)*p^i,i=0..1);
                f(eta) = f[0](eta) + f[1](eta) p
g(eta)=sum(g[i](eta)*p^i,i=0..1);
                g(eta) = g[0](eta) + g[1](eta) p
theta(eta)=sum(theta[i](eta)*p^i,i=0..1);
          theta(eta) = theta[0](eta) + theta[1](eta) p
phi(eta)=sum(phi[i](eta)*p^i,i=0..1);
             phi(eta) = phi[0](eta) + phi[1](eta) p
> FORequ1;


A:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm1)),p);
/      /  d            \ /  d   /  d            \\
|-1. R |----- f[1](eta)| |----- |----- f[1](eta)||
\      \ deta          / \ deta \ deta          //

                 /  d   /  d            \\\  3   /
   + R f[1](eta) |----- |----- f[1](eta)||| p  + |
                 \ deta \ deta          ///      \
      /  d            \ /  d   /  d            \\
-1. R |----- f[0](eta)| |----- |----- f[1](eta)||
      \ deta          / \ deta \ deta          //

          /  d            \ /  d   /  d            \\
   - 1. R |----- f[1](eta)| |----- |----- f[0](eta)||
          \ deta          / \ deta \ deta          //

                 /  d   /  d            \\
   + R f[0](eta) |----- |----- f[1](eta)||
                 \ deta \ deta          //

                 /  d   /  d            \\\  2   //  d   /  d   /
   + R f[1](eta) |----- |----- f[0](eta)||| p  + ||----- |----- |
                 \ deta \ deta          ///      \\ deta \ deta \

    d   /  d            \\\\       /  d            \
  ----- |----- f[1](eta)|||| - 1.0 |----- g[1](eta)|
   deta \ deta          ////       \ deta          /

          /  d            \ /  d   /  d            \\
   - 1. R |----- f[0](eta)| |----- |----- f[0](eta)||
          \ deta          / \ deta \ deta          //

                 /  d   /  d            \\\  
   + R f[0](eta) |----- |----- f[0](eta)||| p
                 \ deta \ deta          ///  

     /  d   /  d   /  d   /  d            \\\\
   + |----- |----- |----- |----- f[0](eta)||||
     \ deta \ deta \ deta \ deta          ////

         /  d            \    
   - 1.0 |----- g[0](eta)| = 0
         \ deta          /    
A1:=diff(f[0](eta),eta$4)-2*K[r]*(diff(g[0](eta),eta))=0;
A2:=diff(f[1](eta),eta$4)-2*K[r]*(diff(g[1](eta),eta))-R*(diff(f[0](eta),eta))*(diff(f[0](eta),eta$2))+R*f[0](eta)*(diff(f[0](eta),eta$2))=0;
/  d   /  d   /  d   /  d            \\\\       /  d            \   
|----- |----- |----- |----- f[0](eta)|||| - 1.0 |----- g[0](eta)| =
\ deta \ deta \ deta \ deta          ////       \ deta          /   

  0
/  d   /  d   /  d   /  d            \\\\       /  d            \
|----- |----- |----- |----- f[1](eta)|||| - 1.0 |----- g[1](eta)|
\ deta \ deta \ deta \ deta          ////       \ deta          /

       /  d            \ /  d   /  d            \\
   - R |----- f[0](eta)| |----- |----- f[0](eta)||
       \ deta          / \ deta \ deta          //

                 /  d   /  d            \\    
   + R f[0](eta) |----- |----- f[0](eta)|| = 0
                 \ deta \ deta          //    
icsA1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsA2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
   f[0](0) = 0, D(f[0])(0) = 1, g[0](0) = 0, f[0](1) = 0.5,

     D(f[0])(1) = 0, g[0](1) = 0
    f[1](0) = 0, D(f[1])(0) = 0, g[1](0) = 0, f[1](1) = 0,

      D(f[1])(1) = 0, g[1](1) = 0
>
FORequ2;


B:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm2)),p);
/      /  d            \          
|-1. R |----- f[1](eta)| g[1](eta)
\      \ deta          /          

                 /  d            \\  3   /
   + R f[1](eta) |----- g[1](eta)|| p  + |
                 \ deta          //      \
      /  d            \          
-1. R |----- f[0](eta)| g[1](eta)
      \ deta          /          

          /  d            \          
   - 1. R |----- f[1](eta)| g[0](eta)
          \ deta          /          

                 /  d            \
   + R f[0](eta) |----- g[1](eta)|
                 \ deta          /

                 /  d            \\  2   //  d   /  d            
   + R f[1](eta) |----- g[0](eta)|| p  + ||----- |----- g[1](eta)
                 \ deta          //      \\ deta \ deta          

  \\       /  d            \        /  d            \          
  || + 1.0 |----- f[1](eta)| - 1. R |----- f[0](eta)| g[0](eta)
  //       \ deta          /        \ deta          /          

                 /  d            \\     /  d   /  d            \\
   + R f[0](eta) |----- g[0](eta)|| p + |----- |----- g[0](eta)||
                 \ deta          //     \ deta \ deta          //

         /  d            \    
   + 1.0 |----- f[0](eta)| = 0
         \ deta          /    
B1:=diff(g[0](eta),eta$2)+2*K[r]*(diff(f[0](eta),eta))=0;
B2:=diff(g[1](eta),eta$2)+2*K[r]*(diff(f[1](eta),eta))-R*(diff(f[0](eta),eta))*g[0](eta)+R*f[0](eta)*(diff(g[0](eta),eta))=0;
     /  d   /  d            \\       /  d            \    
     |----- |----- g[0](eta)|| + 1.0 |----- f[0](eta)| = 0
     \ deta \ deta          //       \ deta          /    
       /  d   /  d            \\       /  d            \
       |----- |----- g[1](eta)|| + 1.0 |----- f[1](eta)|
       \ deta \ deta          //       \ deta          /

              /  d            \          
          - R |----- f[0](eta)| g[0](eta)
              \ deta          /          

                        /  d            \    
          + R f[0](eta) |----- g[0](eta)| = 0
                        \ deta          /    
icsB1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsB2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
   f[0](0) = 0, D(f[0])(0) = 1, g[0](0) = 0, f[0](1) = 0.5,

     D(f[0])(1) = 0, g[0](1) = 0
    f[1](0) = 0, D(f[1])(0) = 0, g[1](0) = 0, f[1](1) = 0,

      D(f[1])(1) = 0, g[1](1) = 0
> FORequ3;


C:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm3)),p);
 /                                     
 |                /  d                \
 |10. R f[1](eta) |----- theta[1](eta)|
 \                \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[1](eta)| |----- theta[1](eta)|
          \ deta            / \ deta              /

                               2\                              
          /  d                \ |  3   /                /  d   
    + 0.1 |----- theta[1](eta)| | p  + |10. R f[0](eta) |-----
          \ deta              / /      \                \ deta

                \                   /  d                \
   theta[1](eta)| + 10. R f[1](eta) |----- theta[0](eta)|
                /                   \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[0](eta)| |----- theta[1](eta)|
          \ deta            / \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[1](eta)| |----- theta[0](eta)|
          \ deta            / \ deta              /

                                                            /
          /  d                \ /  d                \\  2   |/
    + 0.2 |----- theta[0](eta)| |----- theta[1](eta)|| p  + ||
          \ deta              / \ deta              //      \\

     d   /  d                \\
   ----- |----- theta[1](eta)||
    deta \ deta              //

                      /  d                \
    + 10. R f[0](eta) |----- theta[0](eta)|
                      \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[0](eta)| |----- theta[0](eta)|
          \ deta            / \ deta              /

                               2\  
          /  d                \ |  
    + 0.1 |----- theta[0](eta)| | p
          \ deta              / /  

      /  d   /  d                \\    
    + |----- |----- theta[0](eta)|| = 0
      \ deta \ deta              //    
C1:=diff(theta[0](eta),eta$2)=0;
C2:=diff(theta[1](eta), eta, eta)+Pr*R*f[0](eta)*(diff(theta[0](eta), eta))+Nb*(diff(phi[0](eta), eta))*(diff(theta[0](eta), eta))+Nt*(diff(theta[0](eta), eta))^2=0;
                  d   /  d                \    
                ----- |----- theta[0](eta)| = 0
                 deta \ deta              /    
       /  d   /  d                \\
       |----- |----- theta[1](eta)||
       \ deta \ deta              //

                           /  d                \
          + 10 R f[0](eta) |----- theta[0](eta)|
                           \ deta              /

                /  d              \ /  d                \
          + 0.1 |----- phi[0](eta)| |----- theta[0](eta)|
                \ deta            / \ deta              /

                                     2    
                /  d                \     
          + 0.1 |----- theta[0](eta)|  = 0
                \ deta              /     
icsC1:=theta[0](0)=1,theta[0](1)=0;
icsC2:=theta[1](0)=0,theta[1](1)=0,phi[0](0)=0,phi[0](1)=0;
                theta[0](0) = 1, theta[0](1) = 0
 theta[1](0) = 0, theta[1](1) = 0, phi[0](0) = 0, phi[0](1) = 0
> FORequ4;


E:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm4)),p);
                 3 /  d              \   /                /  d   
0.5 R f[1](eta) p  |----- phi[1](eta)| + |0.5 R f[0](eta) |-----
                   \ deta            /   \                \ deta

             \                   /  d              \\  2   //
  phi[1](eta)| + 0.5 R f[1](eta) |----- phi[0](eta)|| p  + ||
             /                   \ deta            //      \\

    d   /  d              \\
  ----- |----- phi[1](eta)||
   deta \ deta            //

                 /  d   /  d                \\
   + 1.000000000 |----- |----- theta[1](eta)||
                 \ deta \ deta              //

                     /  d              \\  
   + 0.5 R f[0](eta) |----- phi[0](eta)|| p
                     \ deta            //  

     /  d   /  d              \\
   + |----- |----- phi[0](eta)||
     \ deta \ deta            //

                 /  d   /  d                \\    
   + 1.000000000 |----- |----- theta[0](eta)|| = 0
                 \ deta \ deta              //    
E1:=diff(phi[0](eta),eta$2)+Nt*(diff(theta[0](eta),eta$2))/Nb=0;
E2:=diff(phi[1](eta),eta$2)+Nt*(diff(theta[1](eta),eta$2))/Nb+R*Sc*f[0](eta)*(diff(phi[0](eta),eta))=0;
       /  d   /  d              \\
       |----- |----- phi[0](eta)||
       \ deta \ deta            //

                        /  d   /  d                \\    
          + 1.000000000 |----- |----- theta[0](eta)|| = 0
                        \ deta \ deta              //    
         /  d   /  d              \\
         |----- |----- phi[1](eta)||
         \ deta \ deta            //

                          /  d   /  d                \\
            + 1.000000000 |----- |----- theta[1](eta)||
                          \ deta \ deta              //

                              /  d              \    
            + 0.5 R f[0](eta) |----- phi[0](eta)| = 0
                              \ deta            /    
icsE1:=theta[0](0)=1,theta[0](1)=0,phi[0](0)=1,phi[0](1)=0;
icsE2:=theta[1](0)=0,theta[1](1)=0,phi[1](0)=0,phi[1](1)=0;
 theta[0](0) = 1, theta[0](1) = 0, phi[0](0) = 1, phi[0](1) = 0
 theta[1](0) = 0, theta[1](1) = 0, phi[1](0) = 0, phi[1](1) = 0
       
theta[0](eta) = -(152675527/100000000)*eta+1;
                                152675527        
              theta[0](eta) = - --------- eta + 1
                                100000000        
U:=f[1](eta)=0;
                         f[1](eta) = 0
Dsolve(A1,B1,icsA1,icsB1);
                  Dsolve(A1, B1, icsA1, icsB1)


sys:={ diff(g[0](eta), eta, eta)+1.0*(diff(f[0](eta), eta)) = 0, diff(f[0](eta), eta, eta, eta, eta)-1.0*(diff(g[0](eta), eta)) = 0};
    //  d   /  d   /  d   /  d            \\\\
   { |----- |----- |----- |----- f[0](eta)||||
    \\ deta \ deta \ deta \ deta          ////

            /  d            \      
      - 1.0 |----- g[0](eta)| = 0,
            \ deta          /      

     /  d   /  d            \\       /  d            \    \
     |----- |----- g[0](eta)|| + 1.0 |----- f[0](eta)| = 0 }
     \ deta \ deta          //       \ deta          /    /
IC_1:={ f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0,f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0};
    {f[0](0) = 0, f[0](1) = 0.5, g[0](0) = 0, g[0](1) = 0,

      D(f[0])(0) = 1, D(f[0])(1) = 0}
ans1 := combine(dsolve(sys union IC_1,{f[0](eta),g[0](eta)}),trig);
Error, (in dsolve) expecting an ODE or a set or list of ODEs. Received `union`(IC_1, sys)
>

Hello everybody.

I'm trying to obtain the numerical solution of a differential equation. Unfortunately, this prove to be quite challenging. I was able to obtain a rough solution using mathematica, but nothing more. The function is strictly increasing (for sure).

Any help is really REALLY appreciated, thanks!

 

``

deq1 := 1/(b-f(b)) = (2*(3-(1-f(b)*(diff(f(b), b, b)))/((diff(f(b), b))*(diff(f(b), b)))))/(1-2*(b-(1-f(b))/(diff(f(b), b))))

1/(b-f(b)) = 2*(3-(1-f(b)*(diff(diff(f(b), b), b)))/(diff(f(b), b))^2)/(1-2*b+2*(1-f(b))/(diff(f(b), b)))

(1)

ic1 := eval(f(b), b = 3/8) = 0, eval(f(b), b = 1/2) = 1/2

f(3/8) = 0, f(1/2) = 1/2

(2)

digits := 3

3

(3)

dsol1 := dsolve({deq1, ic1}, method = bvp[middefer], numeric, range = 3/8 .. 1/2)

Error, (in dsolve/numeric/bvp) initial Newton iteration is not converging

 

``

 

Download diffeqn.mw

Hello

I was looking for information on how to solve a differential equation in Maple, but I do not know how to start.

I have yet equations in symbolic form and I want to apply for the calculation of such a procedure ODE4 (Runge Kutta 4) with fixed step integration Tc and time calculations ongoing T.

Where to start?
1. This should be done in another sheet, or I could write command in next line under the symbolic transformations?
2. To enter numeric parameters that certain symbols were they equal?
3. How to perform iterative calculations?

Thank you in advance for your help

I'm working in a tridimensional euclidean space, with vectorial functions of the type:

Fi(t)=<fix(t),fiy(t),fiz(t)>

Fi'(t)=<fix'(t),fiy'(t),fiz'(t)>

The two odes are of the type:

ode1:=K1*F1''(t)=K2*F2(t)&xF3(t)+...

While there are other non-differential vectorial equations like:

eq1:=K4*F4''(t)=(K5*F5(t)&x<0,1,0>)/Norm(F6(t))+..., etc

 

Is there a way i can input this system in dsolve with vectors instead of scalars? And without splitting everything into its 3 vectorial components? I can't make maple realize some of the Fi(t) functions are vectors, it counts them as scalars and says the number of functions and equations are not the same.

 

Thank you!

Hi,

It might be very silly question, but i dont know why it is not working out. So here is the question. In the attached maple shhet when i am trying to substitute eta(t)=epsilon*z(t) then it is not making that susbtitution for differential operator. Apart from that when i m collecting epsilon terms then also it not collecting it.quesiton.mw

 

Regards

Sunit

restart:with(plots):
eq:=(diff(f(eta),eta$2))-a*f(eta)+b*(1+diff(f(eta),eta)^2)^(-1/2)=0;
bc:=f(1)=0,D(f)(0)=0;
ans := dsolve(eq);

i have attcahed my ode with complex bvp

can anyone solved mine

NULL

restart

with(plots):

NULL

Eq1 := (11-10*d)*(diff(h(eta), eta))+2*f(eta) = 0;

(11-10*d)*(diff(h(eta), eta))+2*f(eta) = 0

 

(11-10*d)*(diff(diff(f(eta), eta), eta))-h(eta)*(diff(f(eta), eta))-f(eta)^2+g(eta)^2 = 0

 

diff(diff(g(eta), eta), eta)-h(eta)*(diff(g(eta), eta))-2*f(eta)*g(eta) = 0

 

diff(p(eta), eta)+2*(diff(f(eta), eta))-2*f(eta)*h(eta) = 0

(1)

NULL

NULL

`V&lambda;` := [0.5e-1, 1.5, 1.5]:

etainf := 3:

bcs := h(0) = 0, p(0) = 0, (D(f))(0) = lambda*f(0)^(4/3)/(f(0)^2+(1-g(0))^2)^(1/3), (D(g))(0) = -Typesetting:-delayDotProduct(lambda*f(0)^(1/3)*(1-g(0)), 1/(f(0)^2+(1-g(0))^2)^(1/3)), f(etainf) = 0, g(etainf) = 0;

h(0) = 0, p(0) = 0, (D(f))(0) = lambda*f(0)^(4/3)/(f(0)^2+(1-g(0))^2)^(1/3), (D(g))(0) = -f(0)^(1/3)*(1-g(0))*lambda/(f(0)^2+(1-g(0))^2)^(1/3), f(3) = 0, g(3) = 0

(2)

NULL

dsys := {Eq1, Eq2, Eq3, Eq4, bcs}:

for i to 3 do lambda := `V&lambda;`[i]; dsol[i] := dsolve(dsys, numeric, continuation = d); print(lambda); print(dsol[i](0)) end do

Error, (in dsolve/numeric/bvp) singularity encountered

 

NULL

NULL

NULL

 

Download compre1.mw

and attch back

Been working on a diffy q project, new to maple here. Any help is appreciated. Keep getting a similar error.

 

"Error, (in dsolve/numeric/type_check) insufficient initial/boundary value information for procedure defined problem"

I thought I gave it initial values?

link to screenshot of the error bellow:

http://i.imgur.com/YVE1x7e.jpg

how to convert system of differential equations to differential form for evalDG?

 

[a(t)*(diff(c(t), t))+b(t), a(t)*(diff(b(t), t))+c(t)*(diff(b(t), t)), a(t)*(diff(c(t), t))+a(t)*(diff(b(t), t))+b(t)];

when i try eliminate dt which is the denominator

eliminate([a(t)*dc(t) + b(t)*dt,a(t)*db(t)+dt*c(t)*db(t),a(t)*dc(t)+a(t)*db(t)+b(t)*dt],dt);

[{dt = -a(t)/c(t)}, {a(t)*(c(t)*dc(t)-b(t)), a(t)*(db(t)*c(t)+c(t)*dc(t)-b(t))}]

 

i got two solutions, which one is correct?

a(t)*(c(t)*dc(t)-b(t)), a(t)*(db(t)*c(t)+c(t)*dc(t)-b(t))

does it mean that two have to use together to form a differential form?

 

update1

with(DifferentialGeometry):
DGsetup([a,b,c], M);
X := evalDG({a*(c*D_c-b), a*(D_b*c+c*D_c-b(t))});
Flow(X,t);
Flow(X, t, ode = true);

got error when run with above result

 

I'm trying to plot the varying results to a second degree differential function with different values for one constant in one graph. Here is what I have so far, which is already working.

______________________________________________________________________

> with(plots);
> m := 0.46e-1; d := 0.42e-1; v := 60; alpha0 := convert(12*degrees, radians); g := 9.81; pa := 1.205; cd := .2; n := 4000; omega := 2*Pi*(1/60);
                           
> p := 6*m/(Pi*d^3);
                           
> k1 := (3/4)*cd*pa/(d*p); k2 := (3/8)*omega*n*pa/p;
                  
> gl1 := vx(t) = diff(x(t), t);
                          
> gl2 := vy(t) = diff(y(t), t);
                     
> gl3 := diff(vx(t), t) = -k1*vx(t)*(vx(t)^2+vy(t)^2)^(1/2)-k2*vy(t);
       
> gl4 := diff(vy(t), t) = -g-k1*vy(t)*(vx(t)^2+vy(t)^2)^(1/2)+k2*vx(t);
 
> init1 := x(0) = 0;
> init2 := y(0) = 0;
> init3 := vx(0) = v*cos(alpha0);
> init4 := vy(0) = v*sin(alpha0);
> sol := dsolve({gl1, gl2, gl3, gl4, init1, init2, init3, init4}, {vx(t), vy(t), x(t), y(t)}, type = numeric);

> sol(.5);
> odeplot(sol, [x(t), y(t)], t = 0 .. 6.7);

______________________________________
What I'd like to do now is, for example, plot the solutions in one graph (preferably as a gif) for when n=1500, n=3000, n=4500 etc. Is there a simple way to achieve this? I've tried various methods so far without success.

Hi

Dear friends

I use the command "dsolve(`union`(deq, initial), numeric, method = lsode)" for solving a fourth order ODE.

But for some numerical values of the parameters the bellow error is occurred:

" an excessive amount of work (greater than mxstep) was done ".

I have three questions:

1- how can I increase the mxstep from default amount (i.e. 500) to a greater value?

2- how can I ensure that the absolute error is less than 10E-6?

3- when I use lsode which way of numerical solution is applied (Euler,midpoint, rk3, rk4, rkf, heun, ... )?

 

Thanks a lot for your help

Hello evrey one , I need help for solve these equation with boundary conditions 

 

 Boundary Conditions


My COde + equation 

NULL

restart; with(plots); with(PDEtools)

NULL

NULL

eq := diff(g(Y), `$`(Y, 4))+diff(g(Y), `$`(Y, 2))+g(Y);

diff(diff(diff(diff(g(Y), Y), Y), Y), Y)+diff(diff(g(Y), Y), Y)+g(Y)

(1)

cis := g(1/4) = 0, (D(g))(1/4) = 0, g(0) = 0, (D[2](g))(0) = 0

Error, (in evalapply) too few variables for the derivative with respect to the 2nd variable

 

solut := dsolve([eq, cis], numeric)

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

 

``

NULL

NULL

NULL

``

 

Download mp.mw

 

Thank you 

 

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