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How to plot this Equation in Runge-Kutta...

Saha 75 Maple
ode homework dsolve
September 02 2022
1 1
I am trying to solve this equation in RK Method 4th-5th Order but the following error keeps happening.Please help me to solve this equations.
> 

restart
> 

with(PDETools, declare):
> 

with(LinearAlgebra):
> 

with(PolynomialIdeals):
> 

with(plots):
> 

declare(f(x));

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

 (1)
> 

ODE1 := (diff(f(x), x, x, x))/((1-phi)^2.5*(1-phi+phi*rho[s]/rho[f(x)]))+(1/2)(`cosα`)*eta*(diff(f(x), x, x))+(1/2)(`sinα`)*f(x)*(diff(f(x), x, x))-M(diff(f(x), x)-1)/(1-phi+phi*rho[s]/rho[f(x)]) = 0:

(diff(diff(diff(f(x), x), x), x))/((1-phi)^2.5*(1-phi+phi*rho[s]/rho[f(x)]))+(1/2)*cos(alpha)*eta*(diff(diff(f(x), x), x))+(1/2)*sin(alpha)*f(x)*(diff(diff(f(x), x), x))-M(diff(f(x), x)-1)/(1-phi+phi*rho[s]/rho[f(x)]) = 0

 (2)
> 

ODE2 := K[nf]*(diff(theta(x), x, x))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*c[s]/(rho[f]*c[f]))(eta*`cosα`+f*`sinα`)*(diff(theta(x), x)) = 0:

K[nf]*(diff(diff(theta(x), x), x))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*c[s]/(rho[f]*c[f]))*(cos(alpha)*eta+sin(alpha)*f(x))*(diff(theta(x), x)) = 0

 (3)
> 

bcs := f(0) = 0, (D(f))(0) = 0, (D(f))(25) = 1;

f(0) = 0, (D(f))(0) = 0, (D(f))(25) = 1

  

theta(0) = 1

 (4)
> 

a1 := [phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, M = 1, sin(alpha) = 0, cos(alpha) = 1];

[phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, M = 1, sin(alpha) = 0, cos(alpha) = 1]

  

[phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, M = 1, sin(alpha) = 1, cos(alpha) = 0]

  

[phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, M = 1, sin(alpha) = (1/2)*3^(1/2), cos(alpha) = 1/2]

  

[phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, M = 1, sin(alpha) = 0, cos(alpha) = 1]

  

[phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, K[nf] = .6842, Pr = 6.2, M = 1, sin(alpha) = 1, cos(alpha) = 0, K[f] = .613]

  

[phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, K[nf] = .6842, Pr = 6.2, M = 1, sin(alpha) = (1/2)*3^(1/2), cos(alpha) = 1/2, K[f] = .613]

 (5)
> 

b1 := subs(a1, ode1);

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/2)*eta*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

  

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

  

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/4)*eta*(diff(diff(f(x), x), x))+(1/4)*3^(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

  

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/2)*eta*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

  

1.116150082*(diff(diff(theta(x), x), x))+3.049196273*f(x)*(diff(theta(x), x)) = 0

  

1.116150082*(diff(diff(theta(x), x), x))+3.049196273*((1/2)*eta+(1/2)*3^(1/2)*f(x))*(diff(theta(x), x)) = 0

 (6)
> 

c1 := dsolve({b1, bcs}, numeric); c1(0); c2 := dsolve({b2, bcs}, numeric); c2(0); c3 := dsolve({b3, bcs}, numeric); c3(0); c4 := dsolve({b4, bcs}, numeric); c4(0); c5 := dsolve({b5, bcs1}, numeric); c5(0)

Error, (in dsolve/numeric/bvp/convertsys) too few boundary conditions: expected 5, got 3

  

c1(0)

  

Error, (in dsolve/numeric/bvp/convertsys) too few boundary conditions: expected 4, got 3

  

c2(0)

  

Error, (in dsolve/numeric/bvp/convertsys) too few boundary conditions: expected 5, got 3

  

c3(0)

  

Error, (in dsolve/numeric/bvp/convertsys) too few boundary conditions: expected 5, got 3

  

c4(0)

  

Error, (in dsolve/numeric/DAE/make_proc) number of unknown functions and equations must match, got 2 functions {f, theta}, and 1 equations

  

c5(0)

 (7)
> 

NULL

Download HP-Numeric.mw

How to plot this ODE equation in Numeri...

Saha 75 Maple 18
ode syntax homework
June 27 2022
1 3
Could you Please Help me,the Maple code for Plot this equations in any Numerical Method
> 

restart
> 

with(PDETools, declare):
> 

with(LinearAlgebra):
> 

with(PolynomialIdeals):
> 

with(plots):
> 

declare(f(x));

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

 (1)
> 

eq1 := diff(f(x), x, x, x)+(1/2)*(1-phi)^2.5*(1-phi+phi*rho[s]/rho[fl])*(eta*`cosω`+f(x)*`sinω`)*(diff(f(x), x, x))+(1-phi)^2.5*M*sin^2*alpha*(1-(diff(f(x), x)))+(1-phi)^2.5*(1-phi+phi*`ρβ`[s]/`ρβ`[fl])*lambda[T]*theta = 0;

diff(diff(diff(f(x), x), x), x)+(1/2)*(1-phi)^2.5*(1-phi+phi*rho[s]/rho[fl])*(eta*`cosω`+f(x)*`sinω`)*(diff(diff(f(x), x), x))+(1-phi)^2.5*M*sin^2*alpha*(1-(diff(f(x), x)))+(1-phi)^2.5*(1-phi+phi*`ρβ`[s]/`ρβ`[fl])*lambda[T]*theta = 0

 (2)
> 

eq2 := K[nf]*(diff(theta(x), x, x))/K[f]+(1/2)*Pr*(eta*`cosω`+f*`sinω`)*(diff(theta(x), x)) = 0;

K[nf]*(diff(diff(theta(x), x), x))/K[f]+(1/2)*Pr*(eta*`cosω`+f*`sinω`)*(diff(theta(x), x)) = 0

 (3)
> 

bcs := F(0) = 0, F(1) = 0, F(10) = 1, Theta(0) = 1, Theta(10) = 0;

F(0) = 0, F(1) = 0, F(10) = 1, Theta(0) = 1, Theta(10) = 0

 (4)
> 

a1 := [M = 1, alpha = 0, phi = 0.5e-1, `cosω` = 1, `sinω` = 1, sin(alpha) = 0, lambda[T] = 0, Pr = 6.2, rho[s] = 5200, rho[fl] = 997.1, `ρβ`[fl] = 20939.1, K[nf] = .6842, eta = 0];

[M = 1, alpha = 0, phi = 0.5e-1, `cosω` = 1, `sinω` = 1, sin(alpha) = 0, lambda[T] = 0, Pr = 6.2, rho[s] = 5200, rho[fl] = 997.1, `ρβ`[fl] = 20939.1, K[nf] = .6842, eta = 0]

  

[Pr = 6.2, M = 1, phi = 0.5e-1, `cosω` = 1, `sinω` = 1, sin(alpha) = 0, lambda[T] = 0, `ρβ`[fl] = 20939.1, K[nf] = .6842, K[f] = .613, eta = 0]

 (5)
> 

b1 := subs(a1, eq1);

diff(diff(diff(f(x), x), x), x)+.5325197465*f(x)*(diff(diff(f(x), x), x)) = 0

 (6)
> 

c1 := subs(a2, eq2);

1.116150082*(diff(diff(theta(x), x), x))+3.100000000*f*(diff(theta(x), x)) = 0

 (7)
> 

d1 := dsolve({b1, bcs}, {bcs, c1}, numeric); d1(0)

Error, (in dsolve/numeric/process_input) invalid argument: {1.116150082*(diff(diff(theta(x), x), x))+3.100000000*f*(diff(theta(x), x)) = 0, F(0) = 0, F(1) = 0, F(10) = 1, Theta(0) = 1, Theta(10) = 0}

  

d1(0)

 (8)
> 

``

Download  Numerical Method.mw

How to solve this solution in R-K Method...

Saha 75 Maple
ode homework dsolve
June 19 2022
1 4


 

Please Help! To Clear this Error and Plot this series in R-K Method
> 

restart
> 

with(PDETools, declare):
> 

with(LinearAlgebra):
> 

with(PolynomialIdeals):
> 

with(plots):
> 

declare(f(x));

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

 (1)
> 

eq1 := (diff(f(x), `$`(x, 3)))/((1-phi)^2.5*(1-phi+phi*rho[s]/rho[f]))+(1/2)(`cosα`)*eta*(diff(f(x), `$`(x, 2)))+(1/2)(`sinα`)*f(x)*(diff(f(x), `$`(x, 2)))-M(diff(f(x), `$`(x, 1))-1)/(1-phi+phi*rho[s]/rho[f]) = 0;

(diff(diff(diff(f(x), x), x), x))/((1-phi)^2.5*(1-phi+phi*rho[s]/rho[f]))+(1/2)*eta*(diff(diff(f(x), x), x))+(1/2)*f(x)*(diff(diff(f(x), x), x))-M(diff(f(x), x)-1)/(1-phi+phi*rho[s]/rho[f]) = 0

 (2)
> 

eq2 := K[nf]*(diff(Theta(x), `$`(x, 2)))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*c[s]/(rho[f]*c[f]))(eta*`cosα`+f*`sinα`)*(diff(Theta(x), `$`(x, 1))) = 0;

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+(1/2)*Pr*(1-phi(eta*`cosα`+f*`sinα`)+phi(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

 (3)
> 

bcs := eta = 1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, K[s] = 6, K[f] = .613, K[nf] = K[f]*(K[s]+2*K[f]-2*phi*(K[f]-K[s]))/(K[s]+2*K[f]+phi*(K[f]-K[s])):

f(0) = 0, (D(f))(0) = 0, Theta(0) = 1, Theta(10) = 0, (D(f))(10) = 1

 (4)
> 

a1 := [cos(beta) = 1, sin(beta) = 0, M = 1, Pr = 6.2, phi = .1];

[cos(beta) = 1, sin(beta) = 0, M = 1, Pr = 6.2, phi = .1]

  

[cos(beta) = 0, sin(beta) = 1, M = 1, Pr = 6.2, phi = .1]

  

[cos(beta) = 1/2, sin(beta) = (1/2)*3^(1/2), M = 1, Pr = 6.2, phi = .1]

  

[cos(beta) = 1, sin(beta) = 0, M = 1, Pr = 6.2, phi = .1]

  

[cos(beta) = 0, sin(beta) = 1, M = 1, Pr = 6.2, phi = 0.]

  

[cos(beta) = 1/2, sin(beta) = (1/2)*3^(1/2), M = 1, Pr = 6.2, phi = 0.]

 (5)
> 

b1 := subs(a1, eq1);

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+.1*rho[s]/rho[f])+(1/2)*eta*(diff(diff(f(x), x), x))+(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+.1*rho[s]/rho[f]) = 0

  

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+.1*rho[s]/rho[f])+(1/2)*eta*(diff(diff(f(x), x), x))+(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+.1*rho[s]/rho[f]) = 0

  

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+.1*rho[s]/rho[f])+(1/2)*eta*(diff(diff(f(x), x), x))+(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+.1*rho[s]/rho[f]) = 0

  

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+.1*rho[s]/rho[f])+(1/2)*eta*(diff(diff(f(x), x), x))+(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+.1*rho[s]/rho[f]) = 0

  

1.000000000*(diff(diff(diff(f(x), x), x), x))+(1/2)*eta*(diff(diff(f(x), x), x))+(1/2)*f(x)*(diff(diff(f(x), x), x))-1.*(1)(diff(f(x), x)-1) = 0

  

1.000000000*(diff(diff(diff(f(x), x), x), x))+(1/2)*eta*(diff(diff(f(x), x), x))+(1/2)*f(x)*(diff(diff(f(x), x), x))-1.*(1)(diff(f(x), x)-1) = 0

 (6)
> 

c1 := subs(a1, eq2);

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(1-(.1)(eta*`cosα`+f*`sinα`)+(.1)(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

  

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(1-(.1)(eta*`cosα`+f*`sinα`)+(.1)(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

  

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(1-(.1)(eta*`cosα`+f*`sinα`)+(.1)(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

  

eq4

  

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(1-(0.)(eta*`cosα`+f*`sinα`)+(0.)(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

  

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(1-(0.)(eta*`cosα`+f*`sinα`)+(0.)(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

 (7)
> 

d1 := dsolve({b1, bcs}, {bcs, c1}, numeric); d1(0)

Error, (in dsolve/numeric/process_input) invalid argument: {eta = 1, K[f] = .613, K[nf] = K[f]*(K[s]+2*K[f]-2*phi*(K[f]-K[s]))/(K[s]+2*K[f]+phi*(K[f]-K[s])), K[s] = 6, c[f] = 4179, c[s] = 670, rho[f] = 997.1, rho[s] = 5200, K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(.9+.1*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0}

  

d1(0)

 (8)
> 

``

Warning,Solving for expressions other th...

Saha 75 Maple
solve duplicate-question
May 22 2022
2 0


 

Please,Help the Maple code for find the values of F[3],F[4],F[5],........How to rectify this error?
> 

restart
> 

F[0] := 0;

0

  

0

  

A

 (1)
> 

for k from 0 to 10 do F[k+3] := solve(F[k+3]+((1/2)*(sum(cos*beta*delta(k-m-1)*(m+1)*(m+2)*F[m+2], m = 0 .. k))-(1/2)*sin(beta)*(sum(F[k-m]*(m+1)*(m+2)*F[m+2], m = 0 .. k))+M(k+1)*F[k+1])/((k+1)(k+2))(k+3), F[k+3]) end do

Warning, solving for expressions other than names or functions is not recommended.

  

-cos*beta*(table( [( -1 ) = 0, ( 0 ) = 1, ( 1 ) = 0, ( 2 ) = 0, ( 3 ) = 0, ( 4 ) = 0, ( 5 ) = 0, ( 6 ) = 0, ( 7 ) = 0, ( 9 ) = 0, ( 8 ) = 0, ( 10 ) = 0 ] ))('k-1')*A

  

Warning, solving for expressions other than names or functions is not recommended.

  

"F[4]:="

  

Warning, solving for expressions other than names or functions is not recommended.

  

Error, (in thaw) invalid input: thaw uses a 1st argument, e, which is missing

  
> 

``

"How to find the values of this series? ...

Saha 75 Maple
solve recurrence
May 02 2022
3 5

How to find the values of X(1),X(2),..&Y(1),Y(2)...Plese help .
> 

restart;
 

for k from 0 to 5 do
X(k+1):=solve(2*(k+1)*X(k+1)+(k+1)*Y(k+1)-X(k)-Y(k)+(1)/k!,X(k+1));
Y(k+1):=solve((k+1)*X(k+1)+(k+1)*Y(k+1)+2*X(k)+Y(k)+(1)/k!,Y(k+1)); od;
> 


 

2

  

1

  

Warning, solving for expressions other than names or functions is not recommended.

  

Error, (in solve) a constant is invalid as a variable, -(1/2)*Y(1)+1

  
> 

``

Download DE_Using_DTM-Ex-3(1).mw

 

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