Items tagged with ode

$$\textbf{x}' = \begin{bmatrix} -4 & -2 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}+\begin{bmatrix} -t \\ -2t-1 \end{bmatrix},\textbf{x}(0)=\begin{bmatrix} 3 \\ -5 \end{bmatrix}$$

As I know firstly, when the matrix is denoted by $A$, we must compute $e^{At}$ by diagonalizing $A$: if $A=PDP^{-1}$ for a diagonal $D$ then $e^{At} = P e^{Dt} P^{-1}$ where $e^{Dt}$ is a diagonal matrix with $(e^{Dt})_{ii} = e^{D_{ii} t}$...
 
How can I write The Maple code? maple.stackexchange)

restart: with(LinearAlgebra):

A := Matrix(2,2,[-4,-2,3,1]);

....

Dear sir in this problem should accept five boundaryconditions but it is not working for five boundary conditions and showing the following error please can you tell why it is like this ??

Error, (in dsolve/numeric/bvp/convertsys) too many boundary conditions: expected 4, got 5
Error, (in plots:-display) expecting plot structures but received: [fplt[1], fplt[2], fplt[3], fplt[4], fplt[5], fplt[6], fplt[7]]
Error, (in plots:-display) expecting plot structures but received: [tplt[1], tplt[2], tplt[3], tplt[4], tplt[5], tplt[6], tplt[7]]
 

and for the progam please check the following link

stretching_cylinder_new1.mw

Dear all

I am trying to solve system of ODEs by TWS command for traveling wave solution, but an error is showing. When I enter sinlge ODE or PDE the command does not show any error. Why it is showing error for system of ODEs ?


 

with(PDEtools, TWSolutions, declare):

with(DEtools):

Sys := {125*xi^3*(diff(f(xi), xi, xi, xi))-90*f(xi)*xi*(diff(h(xi), xi))-180*h(xi)*xi*(diff(f(xi), xi))+750*xi^2*(diff(f(xi), xi, xi))-180*f(xi)*h(xi)+830*xi*(diff(f(xi), xi))+80*f(xi)-108*(diff(h(xi), xi)), 15*f(xi)*xi*(diff(f(xi), xi))+6*f(xi)^2+10*xi*(diff(g(xi), xi))+8*g(xi)+6*(diff(f(xi), xi)) = 0, 5*f(xi)*xi*(diff(g(xi), xi))+10*g(xi)*xi*(diff(f(xi), xi))+8*f(xi)*g(xi)+10*xi*(diff(h(xi), xi))+12*h(xi)+6*(diff(g(xi), xi)) = 0}

TWSolutions(Sys)

Error, (in pdsolve/sys/info) required an indication of the solving variables for the given system

 

``


 

Download ODEs.mw

I try to solve numerically a boundary VP for ODE with different order of discontinuity of right part.

Say, the following BVP is given:

y''(x)+y'(x)+y(x)=F(x)

y(0)=1, y(2)=1

Let's use piecewise right part

F  := piecewise(x<=1, -x, x>1, 2*x+(x-1)^2)

plot(piecewise(x<=1, -x, x>1, 2*x+(x-1)^2), x=0..2,thickness=5)

The function

piecewise(x<=1, 1-x, x>1, (x-1)^2)

plot(piecewise(x<=1, 1-x, x>1, (x-1)^2), x=0..2, color=blue,thickness=5)

as obviuos, satisfies the BVP exclung the point x=1, where its 1st and 2nd derivatives are discontinuos.

Numerical solution

N0:=6:
As:=dsolve([diff(y(x), x$2)+diff(y(x), x)+y(x)=F,  y(0)=1, y(2)=1], y(x), type=numeric, output = Array([seq(2.0*k/N0, k=0..N0)]), 'maxmesh'=500, 'abserr'=1e-3):

provides the solution essentially different to exact one described above:

But if to use the right part

F := piecewise(x<=1, x^2+x+2, x>1, -x^2+x)

plot(piecewise(x<=1, x^2+x+2, x>1, -x^2+x), x=0..2, color=blue,thickness=5)

for which the function

piecewise(x<=1, 1-x+x^2, x>1, -1+3*x-x^2)

plot(piecewise(x<=1, 1-x+x^2, x>1, -1+3*x-x^2), x=0..2, thickness=5)

satisfies the BVP excluding x=1, where this function has discontinuity of 2nd derivative only, the corresponding numerical solution is very similar to this exact solution:

This reason of the difference between these two cases is clear. In the first case both 1st and 2nd derivatives are discontiuos, while in the second one -- 1st derivative is contiuos.

I wonder, if there are numerical methods, implemeted in Maple, for numerical solution of the first type BVP with non-smooth right part?

what does it means and what it will do. Can some one help me for solving this

Shootlib := "C:/Shoot9"; libname := Shootlib, libname; with(Shoot);

while i m receiving the following message:

"Error, invalid input: with expects its 1st argument, pname, to be of type {`module`, package}, but received Shoot "

Full program is :

restart; Shootlib := "C:/Shoot9"; libname := Shootlib, libname; with(Shoot);
with(plots):
N1 := 1.0; N2 := 2.0; N3 := .5; Bt := 6; Re_m := N1*Bt; gamma1 := 1;
FNS := {f(eta), fp(eta), fpp(eta), g(eta), gp(eta), m(eta), mp(eta), n(eta), np(eta), fppp(eta)};
ODE := {diff(f(eta), eta) = fp(eta),
        diff(fp(eta), eta) = fpp(eta),
        diff(fpp(eta), eta) = fppp(eta),
        diff(g(eta), eta) = gp(eta),
        diff(gp(eta), eta) = N1*(2.*g(eta)+(eta-2.*f(eta))*gp(eta)+2.*g(eta)*fp(eta)+2.*N2*N3*(m(eta)*np(eta)-n(eta)*mp(eta))),
        diff(m(eta), eta) = mp(eta),
        diff(mp(eta), eta) = Re_m*(m(eta)+(eta-2.*f(eta))*mp(eta)+2.*m(eta)*fp(eta)),
        diff(n(eta), eta) = np(eta),
        diff(np(eta), eta) = Re_m*(2.*n(eta)+(eta-2.*f(eta))*np(eta)+2.*N2/N3*m(eta)*gp(eta)),
        diff(fppp(eta), eta) = N1*(3.*fpp(eta)+(eta-2.*f(eta))*fppp(eta)-2.*N2*N2*m(eta)*(diff(mp(eta), eta)))
       }:
   
blt := 1.0;
IC := { f(0) = 0,
        fp(0) = 0,
        fpp(0) = alpha1,
        g(0) = 1,
        gp(0) = beta1,
        m(0) = 0,
        mp(0) = beta2,
        n(0) = 0,
        np(0) = beta3,
        fppp(0) = alpha2
      };
BC := { f(blt) = .5,
        fp(blt) = 0,
        g(blt) = 0,
        m(blt) = 1,
        n(blt) = 1};
infolevel[shoot] := 1;
 

Hi, 

I'm being unable to plot the solution to the equation

schro := {-(diff(psi(x), x, x))+(2*a*b*x^4+a^2*x^6+(b^2-a*(2*p+3))*x^2-(2*p+1)*b)*psi(x) = 0};

for the special cases of a=b=1 and p=0 and p=1

I've used dsolve and am getting Heun functions :-(

The claim is that the solutions come out to be exponentials of the form:

psi(x)=(x^p)*exp(-(a*(x^4))/4 - (b*(x^2))/2)

thanks in advance

Am trying to valid a research work done by kuiken(1968)

Kuiken_(1968).pdf

where we have this two eauations:

restart;
Digits := 35;
with(ODETools);
with(student);
with(plots);
inf := 4;
equ1 := diff(f[0](eta), `$`(eta, 3))+theta[0](eta);
equ2 := diff(theta[0](eta), `$`(eta, 2))+3*f[0](eta)*(diff(theta[0](eta), eta));
Bcs1 := f[0](0) = 0, (D(f[0]))(0) = 0, theta[0](0) = 1, theta[0](inf) = 0, (D(D(f[0])))(inf) = 0;
S1 := dsolve({Bcs1, equ1, equ2}, {f[0](eta), theta[0](eta)}, type = numeric, method = bvp[midrich]);
proc(x_bvp)  ...  end;
S1(0);
[                            d                   
[eta = 0., f[0](eta) = 0., ----- f[0](eta) = 0., 
[                           deta                 

    d   /  d            \                                          
  ----- |----- f[0](eta)| = 0.82449782146165697398999365896678734, 
   deta \ deta          /                                          

  theta[0](eta) = 1.0000000000000000000000000000000000, 

    d                                                         ]
  ----- theta[0](eta) = -0.71098574970825563256340736114251047]
   deta                                                       ]
S1(inf);
[                                                            
[eta = 4., f[0](eta) = 1.7815670728545914261072119522795076, 
[                                                            

    d                                                      
  ----- f[0](eta) = 0.51061876174095320088291844433043562, 
   deta                                                    

    d   /  d            \                           
  ----- |----- f[0](eta)| = 0., theta[0](eta) = 0., 
   deta \ deta          /                           

    d                                                             
  ----- theta[0](eta) = -0.000054818176138173095945902421930470836
   deta                                                           

  ]
  ]
  ]
 

 

Pls, I need to find the function of the limit of f[0](eta) at eta tend to infinity. checked equation 45 of the attached document and for the two equation pls checked equation 36 and 37 for the ODE equation solved above.

Kuiken_solution for equation 36 and 37.pdf

staganation_point11.mw
 

``

restart

l := 1:

1

 

1.5

 

.5

 

[blue, green, red, yellow]

(1)

``

for j to nops(A) do R1 := 2*n/(n+1); R2 := 2*p/(n+1); R3 := 2/(n+1); sol1 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R1*(1-(diff(f(eta), eta))^2)-M*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr*f(eta)*(diff(theta(eta), eta))-R2*pr*(diff(f(eta), eta))*theta(eta)+R3*(A[j]*(diff(f(eta), eta))+B*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L+b*((D@@2)(f))(0), (D(f))(20) = 1, theta(0) = 1+s*(D(theta))(0), theta(20) = 0], numeric, method = bvp); plots[odeplot](sol1, [eta, ((D@@2)(f))(eta)], color = red); fplt[j] := plots[odeplot](sol1, [eta, diff(f(eta), eta)], color = K[j], axes = boxed); tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], color = K[j], axes = normal) end do:

 

 

``

l := 1:

1

 

[0, 1, 1.5]

 

.5

 

[blue, green, red, yellow]

(2)

for j to nops(n1) do R4 := 2*n1[j]/(n1[j]+1); R5 := 2*p1/(n1[j]+1); R6 := 2/(n1[j]+1); sol2 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R4*(1-(diff(f(eta), eta))^2)-M1*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr1*f(eta)*(diff(theta(eta), eta))-R5*pr1*(diff(f(eta), eta))*theta(eta)+R6*(A1*(diff(f(eta), eta))+B1*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L1+b1*((D@@2)(f))(0), (D(f))(7) = 1, theta(0) = 1+s1*(D(theta))(0), theta(7) = 0], numeric, method = bvp); plots[odeplot](sol2, [eta, ((D@@2)(f))(eta)], color = red); fplt[j] := plots[odeplot](sol2, sol1, [[eta, diff(f(eta), eta)], [eta, diff(f(eta), eta)]], color = K1[j], axes = boxed); tplt[j] := plots[odeplot](sol2, [[eta, theta(eta)]], color = K1[j], axes = normal) end do; plots:-display([seq(fplt[j], j = 1 .. nops(n1))]); plots:-display([seq(tplt[j], j = 1 .. nops(n1))])

Error, (in plots/odeplot) invalid argument: sol1

 

 

 

``

``

sol1(0);

[eta = 0., f(eta) = 1.00000000000000044, diff(f(eta), eta) = .899599635987511914, diff(diff(f(eta), eta), eta) = -.200800728024976643, theta(eta) = 1.18657688243172332, diff(theta(eta), eta) = .373153764863445037]

(3)

sol1(.1)

[eta = .1, f(eta) = 1.08902313464617162, diff(f(eta), eta) = .881503890693141945, diff(diff(f(eta), eta), eta) = -.162309073227910134, theta(eta) = 1.22040489003745489, diff(theta(eta), eta) = .304232440930656767]

(4)

sol1(.2)

[eta = .2, f(eta) = 1.17641763749368966, diff(f(eta), eta) = .866916683092940898, diff(diff(f(eta), eta), eta) = -.130454301210374102, theta(eta) = 1.24759607023709362, diff(theta(eta), eta) = .240488988787701030]

(5)

sol1(.3)

[eta = .3, f(eta) = 1.19045803452309284, diff(f(eta), eta) = .579367537136023514, diff(diff(f(eta), eta), eta) = -.285675511621370782, theta(eta) = 1.18389221591022696, diff(theta(eta), eta) = .146013974567769960]

(6)

sol1(.4)

[eta = .4, f(eta) = 1.40000000000000034, diff(f(eta), eta) = 1.00000000000000022, diff(diff(f(eta), eta), eta) = -0.243774513041384287e-17, theta(eta) = .625958972186505536, diff(theta(eta), eta) = -.314549395236395634]

(7)

sol1(.5)

[eta = .5, f(eta) = 1.37026161183094430, diff(f(eta), eta) = .874752886901313142, diff(diff(f(eta), eta), eta) = .345911467377074400, theta(eta) = .432494259338694842, diff(theta(eta), eta) = -.382764248064397461]

(8)

sol1(.6)

[eta = .6, f(eta) = 1.36678221814533528, diff(f(eta), eta) = .771028661281065508, diff(diff(f(eta), eta), eta) = .407805382194403932, theta(eta) = .876413930517023876, diff(theta(eta), eta) = -.197648778495384870]

(9)

sol1(2)

[eta = 2., f(eta) = 2.66120522956795602, diff(f(eta), eta) = .991532161353848585, diff(diff(f(eta), eta), eta) = 0.251405465681268682e-1, theta(eta) = .635967939441598018, diff(theta(eta), eta) = -.144641270049362308]

(10)

``

``

``

 

``

``

NULL


 

Download staganation_point11.mw
 

``

restart

l := 1:

1

 

1.5

 

.5

 

[blue, green, red, yellow]

(1)

``

for j to nops(A) do R1 := 2*n/(n+1); R2 := 2*p/(n+1); R3 := 2/(n+1); sol1 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R1*(1-(diff(f(eta), eta))^2)-M*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr*f(eta)*(diff(theta(eta), eta))-R2*pr*(diff(f(eta), eta))*theta(eta)+R3*(A[j]*(diff(f(eta), eta))+B*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L+b*((D@@2)(f))(0), (D(f))(20) = 1, theta(0) = 1+s*(D(theta))(0), theta(20) = 0], numeric, method = bvp); plots[odeplot](sol1, [eta, ((D@@2)(f))(eta)], color = red); fplt[j] := plots[odeplot](sol1, [eta, diff(f(eta), eta)], color = K[j], axes = boxed); tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], color = K[j], axes = normal) end do:

 

 

``

l := 1:

1

 

[0, 1, 1.5]

 

.5

 

[blue, green, red, yellow]

(2)

for j to nops(n1) do R4 := 2*n1[j]/(n1[j]+1); R5 := 2*p1/(n1[j]+1); R6 := 2/(n1[j]+1); sol2 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R4*(1-(diff(f(eta), eta))^2)-M1*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr1*f(eta)*(diff(theta(eta), eta))-R5*pr1*(diff(f(eta), eta))*theta(eta)+R6*(A1*(diff(f(eta), eta))+B1*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L1+b1*((D@@2)(f))(0), (D(f))(7) = 1, theta(0) = 1+s1*(D(theta))(0), theta(7) = 0], numeric, method = bvp); plots[odeplot](sol2, [eta, ((D@@2)(f))(eta)], color = red); fplt[j] := plots[odeplot](sol2, sol1, [[eta, diff(f(eta), eta)], [eta, diff(f(eta), eta)]], color = K1[j], axes = boxed); tplt[j] := plots[odeplot](sol2, [[eta, theta(eta)]], color = K1[j], axes = normal) end do; plots:-display([seq(fplt[j], j = 1 .. nops(n1))]); plots:-display([seq(tplt[j], j = 1 .. nops(n1))])

Error, (in plots/odeplot) invalid argument: sol1

 

 

 

``

``

sol1(0);

[eta = 0., f(eta) = 1.00000000000000044, diff(f(eta), eta) = .899599635987511914, diff(diff(f(eta), eta), eta) = -.200800728024976643, theta(eta) = 1.18657688243172332, diff(theta(eta), eta) = .373153764863445037]

(3)

sol1(.1)

[eta = .1, f(eta) = 1.08902313464617162, diff(f(eta), eta) = .881503890693141945, diff(diff(f(eta), eta), eta) = -.162309073227910134, theta(eta) = 1.22040489003745489, diff(theta(eta), eta) = .304232440930656767]

(4)

sol1(.2)

[eta = .2, f(eta) = 1.17641763749368966, diff(f(eta), eta) = .866916683092940898, diff(diff(f(eta), eta), eta) = -.130454301210374102, theta(eta) = 1.24759607023709362, diff(theta(eta), eta) = .240488988787701030]

(5)

sol1(.3)

[eta = .3, f(eta) = 1.19045803452309284, diff(f(eta), eta) = .579367537136023514, diff(diff(f(eta), eta), eta) = -.285675511621370782, theta(eta) = 1.18389221591022696, diff(theta(eta), eta) = .146013974567769960]

(6)

sol1(.4)

[eta = .4, f(eta) = 1.40000000000000034, diff(f(eta), eta) = 1.00000000000000022, diff(diff(f(eta), eta), eta) = -0.243774513041384287e-17, theta(eta) = .625958972186505536, diff(theta(eta), eta) = -.314549395236395634]

(7)

sol1(.5)

[eta = .5, f(eta) = 1.37026161183094430, diff(f(eta), eta) = .874752886901313142, diff(diff(f(eta), eta), eta) = .345911467377074400, theta(eta) = .432494259338694842, diff(theta(eta), eta) = -.382764248064397461]

(8)

sol1(.6)

[eta = .6, f(eta) = 1.36678221814533528, diff(f(eta), eta) = .771028661281065508, diff(diff(f(eta), eta), eta) = .407805382194403932, theta(eta) = .876413930517023876, diff(theta(eta), eta) = -.197648778495384870]

(9)

sol1(2)

[eta = 2., f(eta) = 2.66120522956795602, diff(f(eta), eta) = .991532161353848585, diff(diff(f(eta), eta), eta) = 0.251405465681268682e-1, theta(eta) = .635967939441598018, diff(theta(eta), eta) = -.144641270049362308]

(10)

``

``

``

 

``

``

NULL


 

Download staganation_point11.mw
 

``

restart

l := 1:

1

 

1.5

 

.5

 

[blue, green, red, yellow]

(1)

``

for j to nops(A) do R1 := 2*n/(n+1); R2 := 2*p/(n+1); R3 := 2/(n+1); sol1 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R1*(1-(diff(f(eta), eta))^2)-M*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr*f(eta)*(diff(theta(eta), eta))-R2*pr*(diff(f(eta), eta))*theta(eta)+R3*(A[j]*(diff(f(eta), eta))+B*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L+b*((D@@2)(f))(0), (D(f))(20) = 1, theta(0) = 1+s*(D(theta))(0), theta(20) = 0], numeric, method = bvp); plots[odeplot](sol1, [eta, ((D@@2)(f))(eta)], color = red); fplt[j] := plots[odeplot](sol1, [eta, diff(f(eta), eta)], color = K[j], axes = boxed); tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], color = K[j], axes = normal) end do:

 

 

``

l := 1:

1

 

[0, 1, 1.5]

 

.5

 

[blue, green, red, yellow]

(2)

for j to nops(n1) do R4 := 2*n1[j]/(n1[j]+1); R5 := 2*p1/(n1[j]+1); R6 := 2/(n1[j]+1); sol2 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R4*(1-(diff(f(eta), eta))^2)-M1*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr1*f(eta)*(diff(theta(eta), eta))-R5*pr1*(diff(f(eta), eta))*theta(eta)+R6*(A1*(diff(f(eta), eta))+B1*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L1+b1*((D@@2)(f))(0), (D(f))(7) = 1, theta(0) = 1+s1*(D(theta))(0), theta(7) = 0], numeric, method = bvp); plots[odeplot](sol2, [eta, ((D@@2)(f))(eta)], color = red); fplt[j] := plots[odeplot](sol2, sol1, [[eta, diff(f(eta), eta)], [eta, diff(f(eta), eta)]], color = K1[j], axes = boxed); tplt[j] := plots[odeplot](sol2, [[eta, theta(eta)]], color = K1[j], axes = normal) end do; plots:-display([seq(fplt[j], j = 1 .. nops(n1))]); plots:-display([seq(tplt[j], j = 1 .. nops(n1))])

Error, (in plots/odeplot) invalid argument: sol1

 

 

 

``

``

sol1(0);

[eta = 0., f(eta) = 1.00000000000000044, diff(f(eta), eta) = .899599635987511914, diff(diff(f(eta), eta), eta) = -.200800728024976643, theta(eta) = 1.18657688243172332, diff(theta(eta), eta) = .373153764863445037]

(3)

sol1(.1)

[eta = .1, f(eta) = 1.08902313464617162, diff(f(eta), eta) = .881503890693141945, diff(diff(f(eta), eta), eta) = -.162309073227910134, theta(eta) = 1.22040489003745489, diff(theta(eta), eta) = .304232440930656767]

(4)

sol1(.2)

[eta = .2, f(eta) = 1.17641763749368966, diff(f(eta), eta) = .866916683092940898, diff(diff(f(eta), eta), eta) = -.130454301210374102, theta(eta) = 1.24759607023709362, diff(theta(eta), eta) = .240488988787701030]

(5)

sol1(.3)

[eta = .3, f(eta) = 1.19045803452309284, diff(f(eta), eta) = .579367537136023514, diff(diff(f(eta), eta), eta) = -.285675511621370782, theta(eta) = 1.18389221591022696, diff(theta(eta), eta) = .146013974567769960]

(6)

sol1(.4)

[eta = .4, f(eta) = 1.40000000000000034, diff(f(eta), eta) = 1.00000000000000022, diff(diff(f(eta), eta), eta) = -0.243774513041384287e-17, theta(eta) = .625958972186505536, diff(theta(eta), eta) = -.314549395236395634]

(7)

sol1(.5)

[eta = .5, f(eta) = 1.37026161183094430, diff(f(eta), eta) = .874752886901313142, diff(diff(f(eta), eta), eta) = .345911467377074400, theta(eta) = .432494259338694842, diff(theta(eta), eta) = -.382764248064397461]

(8)

sol1(.6)

[eta = .6, f(eta) = 1.36678221814533528, diff(f(eta), eta) = .771028661281065508, diff(diff(f(eta), eta), eta) = .407805382194403932, theta(eta) = .876413930517023876, diff(theta(eta), eta) = -.197648778495384870]

(9)

sol1(2)

[eta = 2., f(eta) = 2.66120522956795602, diff(f(eta), eta) = .991532161353848585, diff(diff(f(eta), eta), eta) = 0.251405465681268682e-1, theta(eta) = .635967939441598018, diff(theta(eta), eta) = -.144641270049362308]

(10)

``

``

``

 

``

``

NULL


 

Download staganation_point11.mw

 

in this program im trying to combine the result, but it showing some error can help me please

 

 

 

http://www.sciencedirect.com/science/article/pii/S100757041300508X> restart;
> l := 1; p := 1; A := .5; B := .5; pr := 1; n := [.5, 1, 1.5]; M := 0; b := .5; L := 0; s := .5; K := [blue, green];
                                      1
                                [0.5, 1, 1.5]
                                [blue, green]

> for j to nops(n) do R1 := 2*n[j]/(n[j]+1); R2 := 2*p/(n[j]+1); R3 := 2/(n[j]+1);

sol1 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R1*(1-(diff(f(eta), eta))^2)-M*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr*f(eta)*(diff(theta(eta), eta))-R2*pr*(diff(f(eta), eta))*theta(eta)+R3*(A*(diff(f(eta), eta))+B*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L+b*((D@@2)(f))(0), (D(f))(7) = 1, theta(0) = 1+s*(D(theta))(0), theta(7) = 0], numeric, method = bvp);

plots[odeplot](sol1, [eta, ((D@@2)(f))(eta)], color = red);

fplt[j] := plots[odeplot](sol1, [eta, diff(diff(f(eta), eta), eta)], color = K[j], axes = boxed); fplt[j] := plots[odeplot](sol1, [eta, f(eta)], color = K[j], axes = boxed);

tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], color = K[j], axes = normal) end do;

plots:-display([seq(fplt[j], j = 1 .. nops(n))]);

plots:-display([seq(fplt[j], j = 1 .. nops(n))]);

plots:-display([seq(tplt[j], j = 1 .. nops(n))]);
http://www.sciencedirect.com/science/article/pii/S100757041300508X

Dear sir 

I am trying to plot the following link paper graphs for practice but I getting the plots for only one set of values here in this paper they plotted many so if you dont muned can help in this case. For example in this first graph named as Fig.1. please can you do this favour... and the paper link is  http://www.sciencedirect.com/science/article/pii/S100757041300508X

 

> restart;
> n := [1, 2, 3, 4, 5]; pr := .71; p := 0; q := 0; b := 0; l := 0; s := 0;
> for j to nops(n) do R1 := 2*n[j]/(1+n[j]); R2 := 2*p/(1+n); sys := diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+1-(diff(f(eta), eta))^2 = 0, (diff(diff(theta(eta), eta), eta))/pr+f(eta)*(diff(theta(eta), eta))-R2*(diff(f(eta), eta))*theta(eta) = 0; bcs := f(0) = 0, (D(f))(0) = l+b*((D@@2)(f))(0), (D(f))(-.5) = 1, theta(0) = 1+s*(D(theta))(0), theta(-.5) = 0; proc (f1, th1, { output::name := 'number' }) local res1, fvals, thvals, res2; option remember; res1 := dsolve({sys, f(0) = 0, theta(0) = 1+th1, (D(f))(-2) = f1, (D(theta))(-2) = th1, ((D@@2)(f))(0) = f1-1}, numeric, :-output = listprocedure); fvals := (subs(res1, [seq(diff(f(eta), [`$`(eta, i)]), i = 0 .. 2)]))(0); thvals := (subs(res1, [seq(diff(theta(eta), [`$`(eta, i)]), i = 0 .. 1)]))(0); res2 := dsolve({sys, f(0) = fvals[1], theta(0) = thvals[1], theta(1) = 0, (D(f))(0) = fvals[2], (D(f))(1) = 0}, numeric, :-output = listprocedure); if output = 'number' then [fvals[3]-(subs(res2, diff(f(eta), `$`(eta, 2))))(0), thvals[2]-(subs(res2, diff(theta(eta), eta)))(0)] else res1, res2 end if end proc; p1 := proc (f1, th1) p(args)[1] end proc; p2 := proc (f1, th1) p(args)[2] end proc; p(.3, -.2); par := fsolve([p1, p2], [.3, -.2]); res1, res2 := p(op(par), output = xxx); plots:-display(plots:-odeplot(res1, [[eta, f(eta)], [eta, theta(eta)]]), plots:-odeplot(res2, [[eta, f(eta)], [eta, theta(eta)]])); plots:-display(plots:-odeplot(res1, [[eta, diff(f(eta), eta)], [eta, diff(theta(eta), eta)]]), plots:-odeplot(res2, [[eta, diff(f(eta), eta)], [eta, diff(theta(eta), eta)]])); plots:-display(plots:-odeplot(res1, [[eta, diff(f(eta), eta, eta)]]), plots:-odeplot(res2, [[eta, diff(f(eta), eta, eta)]])); fplt[j] := plots[odeplot](sol1, [eta, diff(diff(f(eta), eta), eta)], color = L[j], axes = boxed); tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], color = L[j], axes = boxed) end do;


Dear Sir

In this above problem it showing that error as  Error, cannot split rhs for multiple assignment please can you tell why it is showing like this  ?? and where i did multiple assignments ??

Dear sir

 

In my ode problem i do not know that how to set range (eta) from -2 to 2 please can  you help me.

> restart;
> with(plots);
> n := [0, .5, 1, 5]; pr := .71; p := 0; l := [1, 2, 3]; b := 0; s := 0; L := [green, blue, black, gold];
                         [green, blue, black, gold]
> R1 := 2*n/(n+1);
                              2 [0, 0.5, 1, 5]
                             ------------------
                             [0, 0.5, 1, 5] + 1
for j from 1 to nops(l) do; for j from 1 to nops(n) do        R1 := 2*n[j]/(1+n[j]);        R2 := 2*p/(1+n[j]); sol1 := dsolve([diff(diff(diff(f(eta),eta),eta),eta)+f(eta)*diff(diff(f(eta),eta),eta)+R1*(1-diff(f(eta),eta)^2) = 0, (1/pr)*diff(diff(theta(eta),eta),eta)+f(eta)*diff(theta(eta),eta)-R2*diff(f(eta),eta)*theta(eta) = 0, f(0) = 0, (D(f))(0) = l+b*((D@@2)(f))(0), (D(f))(-2) =1, theta(0) = 1+s*(D(theta))(0), theta(-2) = 0], numeric, method = bvp); fplt[j]:= plots[odeplot](sol1,[eta,diff(diff(f(eta),eta),eta)],color=["blue","black","orange"]);         tplt[j]:= plots[odeplot](sol1, [eta,theta(eta)],color=L[j]); fplt[j]:= plots[odeplot](sol1,[eta,diff(f(eta),eta)],color=L[j]);      od:od:

 


Error, (in dsolve/numeric/bvp) unable to store 'Limit([eta, 2*eta, 3*eta]+eta^2*[.250000000000000, .500000000000000, .750000000000000]-.250000000000000*eta^2, eta = -2., left)' when datatype=float[8]
> plots:-display([seq(fplt[j], j = 1 .. nops(n))], color = [green, red], [seq(fplt[j], j = 1 .. nops(l))]);

> sol1(0);

Dear sir

In the  above problem i tried to write a nested program but its not executing and showing the error as Error, (in dsolve/numeric/bvp) unable to store 'Limit([eta, 2*eta, 3*eta]+eta^2*, i want the plot range from -2 to 2 but taking only 0 to -2 ,and -2.5 to 3 but taking only 0 to 1

> restart;
> with(plots);
> pr := .72; p := 0; n := [2, 3, 4, 5]; s := 1; a := .2; b := 1;
> R1 := 2*n/(n+1);
                               2 [2, 3, 4, 5]
                              ----------------
                              [2, 3, 4, 5] + 1
> R2 := 2*p/(n+1);
                                      0
>
>
> for j to nops(n) do R1 := 2*n[j]/(1+n[j]); R2 := 2*p/(1+n[j]); sol1 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R1*(1-(diff(f(eta), eta))^2) = 0, diff(diff(theta(eta), eta), eta)+pr*s^f(eta)*(diff(theta(eta), eta))+R2*pr*s*(diff(f(eta), eta))*theta(eta)+2*(a*(diff(f(eta), eta))+b*theta(eta))/(n[j]-1) = 0, f(0) = 0, (D(f))(0) = 1+b*((D@@2)(f))(0), (D(f))(5) = 0, theta(0) = 1+s*(D(theta))(0), theta(5) = 0], numeric, method = bvp); fplt[j] := plots[odeplot](sol1, [eta, diff(diff(f(eta), eta), eta)], axes = boxed); tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], axes = boxed) end do;
>
> plots:-display([seq(fplt[j], j = 1 .. nops(n))]);

 

> plots:-display([seq(tplt[j], j = 1 .. nops(n))]);

 

 

Dear sir

In the above problem graph, i am getting all the lines are in same color then how to identify the lines of different values like n=2,3,4,5,6(or can we set different color for different values of n for each line)

For the ODE system with boundary conditions, I was able to obtain solutions for n=0, but not for n>0. I obtained the error, Initial newton iteration is not converging. Anyone knows the solution for this? I am open to all suggestions and any help would be greatly appreciated:)

 

ODE_solution.mw

I have tried to solve the following ode equation, but I have got error. What is the potencial problem?

http://i65.tinypic.com/xdcl8p.jpg

 

 error

 

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