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Question: homotopy pertubation

Question:homotopy pertubation

Gabriel samaila 35 Maple 2016
numeric differential-equations
May 23 2019
0

 Hi guys,
I am trying write a code for homotopy perturbation, i have already generated the polynomial as you can see, i have also  solve for concentration equation since is not couple. But i have a lot of error massages for temperature, velocity and induced magnetic field. can some one please go through the code?
 

NULL

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restart
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PDEtools[declare](f(x),theta(x),u(x),w(x), prime=x):

f(x)*`will now be displayed as`*f

  

theta(x)*`will now be displayed as`*theta

  

u(x)*`will now be displayed as`*u

  

w(x)*`will now be displayed as`*w

  

`derivatives with respect to`*x*`of functions of one variable will now be displayed with '`

 (1)
> 

N := 4:
> 

NULL

NULL

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f(x):=sum((p^(i))*f[i](x),i=0..N);

f[0](x)+p*f[1](x)+p^2*f[2](x)+p^3*f[3](x)+p^4*f[4](x)

 (2)
> 

theta(x) := sum(p^i*theta[i](x), i = 0 .. N);

theta[0](x)+p*theta[1](x)+p^2*theta[2](x)+p^3*theta[3](x)+p^4*theta[4](x)

 (3)
> 

``
> 

u(x) := sum(p^i*u[i](x), i = 0 .. N);

u[0](x)+p*u[1](x)+p^2*u[2](x)+p^3*u[3](x)+p^4*u[4](x)

 (4)
> 

``
> 

w(x) := sum(p^i*w[i](x), i = 0 .. N);

w[0](x)+p*w[1](x)+p^2*w[2](x)+p^3*w[3](x)+p^4*w[4](x)

 (5)
> 

HPMEq := (1-p)*(diff(f(x), `$`(x, 2)))+p*(diff(f(x), `$`(x, 2))-k1*(diff(f(x), x))-k2*f(x));

(1-p)*(diff(diff(f[0](x), x), x)+p*(diff(diff(f[1](x), x), x))+p^2*(diff(diff(f[2](x), x), x))+p^3*(diff(diff(f[3](x), x), x))+p^4*(diff(diff(f[4](x), x), x)))+p*(diff(diff(f[0](x), x), x)+p*(diff(diff(f[1](x), x), x))+p^2*(diff(diff(f[2](x), x), x))+p^3*(diff(diff(f[3](x), x), x))+p^4*(diff(diff(f[4](x), x), x))-k1*(diff(f[0](x), x)+p*(diff(f[1](x), x))+p^2*(diff(f[2](x), x))+p^3*(diff(f[3](x), x))+p^4*(diff(f[4](x), x)))-k2*(f[0](x)+p*f[1](x)+p^2*f[2](x)+p^3*f[3](x)+p^4*f[4](x)))

 (6)
> 

HPMEr := (1-p)*(diff(theta(x), `$`(x, 2)))+p*(diff(theta(x), `$`(x, 2))-k11*(diff(theta(x), x))+k12*(diff(u(x), x))^2+k13*(diff(w(x), x))^2+k14*theta(x));

(1-p)*(diff(diff(theta[0](x), x), x)+p*(diff(diff(theta[1](x), x), x))+p^2*(diff(diff(theta[2](x), x), x))+p^3*(diff(diff(theta[3](x), x), x))+p^4*(diff(diff(theta[4](x), x), x)))+p*(diff(diff(theta[0](x), x), x)+p*(diff(diff(theta[1](x), x), x))+p^2*(diff(diff(theta[2](x), x), x))+p^3*(diff(diff(theta[3](x), x), x))+p^4*(diff(diff(theta[4](x), x), x))-k11*(diff(theta[0](x), x)+p*(diff(theta[1](x), x))+p^2*(diff(theta[2](x), x))+p^3*(diff(theta[3](x), x))+p^4*(diff(theta[4](x), x)))+k12*(diff(u[0](x), x)+p*(diff(u[1](x), x))+p^2*(diff(u[2](x), x))+p^3*(diff(u[3](x), x))+p^4*(diff(u[4](x), x)))^2+k13*(diff(w[0](x), x)+p*(diff(w[1](x), x))+p^2*(diff(w[2](x), x))+p^3*(diff(w[3](x), x))+p^4*(diff(w[4](x), x)))^2+k14*(theta[0](x)+p*theta[1](x)+p^2*theta[2](x)+p^3*theta[3](x)+p^4*theta[4](x)))

 (7)
> 

HPMEs := (1-p)*(diff(u(x), `$`(x, 2)))+p*(diff(u(x), `$`(x, 2))-R*(diff(u(x), x))-A-k8*w(x)-k7*u(x)+k5*theta(x)+k6*f(x));

(1-p)*(diff(diff(u[0](x), x), x)+p*(diff(diff(u[1](x), x), x))+p^2*(diff(diff(u[2](x), x), x))+p^3*(diff(diff(u[3](x), x), x))+p^4*(diff(diff(u[4](x), x), x)))+p*(diff(diff(u[0](x), x), x)+p*(diff(diff(u[1](x), x), x))+p^2*(diff(diff(u[2](x), x), x))+p^3*(diff(diff(u[3](x), x), x))+p^4*(diff(diff(u[4](x), x), x))-R*(diff(u[0](x), x)+p*(diff(u[1](x), x))+p^2*(diff(u[2](x), x))+p^3*(diff(u[3](x), x))+p^4*(diff(u[4](x), x)))-A-k8*(w[0](x)+p*w[1](x)+p^2*w[2](x)+p^3*w[3](x)+p^4*w[4](x))-k7*(u[0](x)+p*u[1](x)+p^2*u[2](x)+p^3*u[3](x)+p^4*u[4](x))+k5*(theta[0](x)+p*theta[1](x)+p^2*theta[2](x)+p^3*theta[3](x)+p^4*theta[4](x))+k6*(f[0](x)+p*f[1](x)+p^2*f[2](x)+p^3*f[3](x)+p^4*f[4](x)))

 (8)
> 

HPMEt := (1-p)*(diff(w(x), `$`(x, 2)))+p*(diff(w(x), `$`(x, 2))-R*(diff(w(x), x))+k9*u(x)-k10*w(x))

(1-p)*(diff(diff(w[0](x), x), x)+p*(diff(diff(w[1](x), x), x))+p^2*(diff(diff(w[2](x), x), x))+p^3*(diff(diff(w[3](x), x), x))+p^4*(diff(diff(w[4](x), x), x)))+p*(diff(diff(w[0](x), x), x)+p*(diff(diff(w[1](x), x), x))+p^2*(diff(diff(w[2](x), x), x))+p^3*(diff(diff(w[3](x), x), x))+p^4*(diff(diff(w[4](x), x), x))-R*(diff(w[0](x), x)+p*(diff(w[1](x), x))+p^2*(diff(w[2](x), x))+p^3*(diff(w[3](x), x))+p^4*(diff(w[4](x), x)))+k9*(u[0](x)+p*u[1](x)+p^2*u[2](x)+p^3*u[3](x)+p^4*u[4](x))-k10*(w[0](x)+p*w[1](x)+p^2*w[2](x)+p^3*w[3](x)+p^4*w[4](x)))

 (9)
> 

for i from 0 to N do equ[1][i] := coeff(HPMEq, p, i) = 0 end do;

diff(diff(f[0](x), x), x) = 0

  

diff(diff(f[1](x), x), x)-k1*(diff(f[0](x), x))-k2*f[0](x) = 0

  

diff(diff(f[2](x), x), x)-k2*f[1](x)-k1*(diff(f[1](x), x)) = 0

  

diff(diff(f[3](x), x), x)-k2*f[2](x)-k1*(diff(f[2](x), x)) = 0

  

diff(diff(f[4](x), x), x)-k1*(diff(f[3](x), x))-k2*f[3](x) = 0

 (10)
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for i from 0 to N do equa[1][i] := coeff(HPMEr, p, i) = 0 end do;

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

  

diff(diff(theta[1](x), x), x)-k11*(diff(theta[0](x), x))+k12*(diff(u[0](x), x))^2+k13*(diff(w[0](x), x))^2+k14*theta[0](x) = 0

  

diff(diff(theta[2](x), x), x)+2*k13*(diff(w[0](x), x))*(diff(w[1](x), x))-k11*(diff(theta[1](x), x))+2*k12*(diff(u[0](x), x))*(diff(u[1](x), x))+k14*theta[1](x) = 0

  

diff(diff(theta[3](x), x), x)+k12*(2*(diff(u[0](x), x))*(diff(u[2](x), x))+(diff(u[1](x), x))^2)+k14*theta[2](x)+k13*(2*(diff(w[0](x), x))*(diff(w[2](x), x))+(diff(w[1](x), x))^2)-k11*(diff(theta[2](x), x)) = 0

  

diff(diff(theta[4](x), x), x)+k12*(2*(diff(u[0](x), x))*(diff(u[3](x), x))+2*(diff(u[1](x), x))*(diff(u[2](x), x)))-k11*(diff(theta[3](x), x))+k14*theta[3](x)+k13*(2*(diff(w[0](x), x))*(diff(w[3](x), x))+2*(diff(w[1](x), x))*(diff(w[2](x), x))) = 0

 (11)
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for i from 0 to N do equat[1][i] := coeff(HPMEs, p, i) = 0 end do;

diff(diff(u[0](x), x), x) = 0

  

diff(diff(u[1](x), x), x)-R*(diff(u[0](x), x))-A-k7*u[0](x)+k5*theta[0](x)+k6*f[0](x)-k8*w[0](x) = 0

  

diff(diff(u[2](x), x), x)-R*(diff(u[1](x), x))-k7*u[1](x)+k6*f[1](x)-k8*w[1](x)+k5*theta[1](x) = 0

  

diff(diff(u[3](x), x), x)-R*(diff(u[2](x), x))+k6*f[2](x)-k7*u[2](x)+k5*theta[2](x)-k8*w[2](x) = 0

  

diff(diff(u[4](x), x), x)-R*(diff(u[3](x), x))+k5*theta[3](x)+k6*f[3](x)-k7*u[3](x)-k8*w[3](x) = 0

 (12)
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``
> 

for i from 0 to N do equati[1][i] := coeff(HPMEt, p, i) = 0 end do;

diff(diff(w[0](x), x), x) = 0

  

diff(diff(w[1](x), x), x)-R*(diff(w[0](x), x))-k10*w[0](x)+k9*u[0](x) = 0

  

diff(diff(w[2](x), x), x)-k10*w[1](x)+k9*u[1](x)-R*(diff(w[1](x), x)) = 0

  

diff(diff(w[3](x), x), x)-k10*w[2](x)+k9*u[2](x)-R*(diff(w[2](x), x)) = 0

  

diff(diff(w[4](x), x), x)+k9*u[3](x)-R*(diff(w[3](x), x))-k10*w[3](x) = 0

 (13)
> 

con[1][0] := f[0](-1) = 1, f[0](1) = 1:

-.5000000000*k2+0.3435019841e-1*k2^4+.5000000000*k2*x^2-.2500000000*k2^2*x^2+.2083333333*k2^2+0.4166666667e-1*k2^2*x^4-0.2083333333e-1*k2^3*x^4+.1041666667*k2^3*x^2+0.4166666667e-1*k1^2*k2+0.5952380952e-3*k2^3*k1*x^7-0.8472222222e-1*k2^3+0.2480158730e-4*k2^4*x^8+0.4166666667e-2*x^6*k1^2*k2^2+0.8333333333e-2*k2*x^5*k1^3-0.9722222222e-2*k1*k2^3*x^5-0.3472222222e-1*k1^2*k2^2*x^4-0.2777777778e-1*k2*x^3*k1^3+0.5046296296e-1*k1*k2^3*x^3+0.6805555556e-1*k2^2*k1^2*x^2-0.4133597884e-1*k1*k2^3*x+0.1944444444e-1*k2*k1^3*x+1.+0.1388888889e-2*k2^3*x^6+0.1666666667e-1*k1*k2^2*x^5+0.4166666667e-1*k2*x^4*k1^2-.1111111111*k2^2*k1*x^3+0.9444444444e-1*k1*k2^2*x-0.8333333333e-1*k2*k1^2*x^2+.1666666667*k2*k1*x^3-0.3750000000e-1*k1^2*k2^2-0.6944444444e-3*k2^4*x^6+0.8680555556e-2*k2^4*x^4-0.4236111111e-1*k2^4*x^2-.1666666667*k1*k2*x

  

1-(1/2)*k2+(277/8064)*k2^4+(1/2)*k2*x^2-(1/4)*k2^2*x^2+(5/24)*k2^2+(1/24)*k2^2*x^4-(1/48)*k2^3*x^4+(5/48)*k2^3*x^2+(1/24)*k1^2*k2+(1/1680)*k2^3*k1*x^7-(61/720)*k2^3+(1/40320)*k2^4*x^8+(1/240)*x^6*k1^2*k2^2+(1/120)*k2*x^5*k1^3-(7/720)*k1*k2^3*x^5-(5/144)*k1^2*k2^2*x^4-(1/36)*k2*x^3*k1^3+(109/2160)*k1*k2^3*x^3+(49/720)*k2^2*k1^2*x^2-(125/3024)*k1*k2^3*x+(7/360)*k2*k1^3*x+(1/720)*k2^3*x^6+(1/60)*k1*k2^2*x^5+(1/24)*k2*x^4*k1^2-(1/9)*k2^2*k1*x^3+(17/180)*k1*k2^2*x-(1/12)*k2*k1^2*x^2+(1/6)*k2*k1*x^3-(3/80)*k1^2*k2^2-(1/1440)*k2^4*x^6+(5/576)*k2^4*x^4-(61/1440)*k2^4*x^2-(1/6)*k1*k2*x

  

2.400000000*k2+0.3589208394e-1*k2^4+1.104000000*k2^2+2.713333334*k1*k2+1.904000000*k1^2*k2+0.115520003e-1*k2^3+.939244445*k1*k2^2+.3973226666*k1^2*k2^2+0.1412642116e-1*k1*k2^3+.9218444444*k1^3*k2

 (14)

NULL

> 

"cond[1][0]:=theta[0](-1)=0.1, theta[0](1)=1,w[0](-1)=0, w[0](1)=0,u[0](-1)=0, u[0](1)=0: for j from 1 to N do: cond[1][j]:=theta[j](-1)=0, theta[j](1)=0,w[j](-1)=0, w[j](1)=0,u[j](-1)=0, u[j](1)=0: end do: for i from 0 to N do: dsolve({equa[1][i],cond[1][i]},theta[i](x)); theta[i](x):=rhs(`%`): end do: theta(x):=evalf(simplify(sum(theta[n](x),n=0..N))); convert(theta(x),'rational'); "

Error, (in dsolve) found the following equations not depending on the unknowns of the input system: {u[0](-1) = 0, u[0](1) = 0, w[0](-1) = 0, w[0](1) = 0}

  

theta[0](x)+theta[1](x)+theta[2](x)+theta[3](x)+theta[4](x)

  

theta[0](x)+theta[1](x)+theta[2](x)+theta[3](x)+theta[4](x)

 (15)
> 

``
> 

"condi[1][0]:=theta[0](-1)=0.1, theta[0](1)=1,w[0](-1)=0, w[0](1)=0,u[0](-1)=0, u[0](1)=0,f[0](-1)=1, f[0](1)=1: for j from 1 to N do: condi[1][j]:=theta[j](-1)=0, theta[j](1)=0,w[j](-1)=0, w[j](1)=0,u[j](-1)=0, u[j](1)=0, f[j](-1)=0, f[j](1)=0: end do: for i from 0 to N do: dsolve({equat[1][i],condi[1][i]},u[i](x)); u[i](x):=rhs(`%`): end do: u(x):=evalf(simplify(sum(u[n](x),n=0..N)))"

Error, (in dsolve) found the following equations not depending on the unknowns of the input system: {f[0](-1) = 1, f[0](1) = 1, w[0](-1) = 0, w[0](1) = 0, theta[0](-1) = 1/10, theta[0](1) = 1}

  

u[0](x)+u[1](x)+u[2](x)+u[3](x)+u[4](x)

 (16)

``

> 

"condit[1][0]:=theta[0](-1)=0.1, theta[0](1)=1,w[0](-1)=0, w[0](1)=0,u[0](-1)=0, u[0](1)=0,f[0](-1)=1, f[0](1)=1: for j from 1 to N do: condit[1][j]:=theta[j](-1)=0, theta[j](1)=0,w[j](-1)=0, w[j](1)=0,u[j](-1)=0, u[j](1)=0, f[j](-1)=0, f[j](1)=0: end do: for i from 0 to N do: dsolve({equati[1][i],condit[1][i]},w[i](x)); w[i](x):=rhs(`%`): end do: w(x):=evalf(simplify(sum(w[n](x),n=0..N)))"

Error, (in dsolve) found the following equations not depending on the unknowns of the input system: {f[0](-1) = 1, f[0](1) = 1, u[0](-1) = 0, u[0](1) = 0, theta[0](-1) = 1/10, theta[0](1) = 1}

  

w[0](x)+w[1](x)+w[2](x)+w[3](x)+w[4](x)

 (17)

NULL

> 

``
> 

``


 

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