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SHIVAS

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3 years, 79 days
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ODE plot animations...

SHIVAS 40
June 03 2024
0 0

@Carl Love 

with the suggestion, I have modified the worksheet but in the plot, i m not obseved any animation only infinity whalve is changing

please check and suggest how to correct it.

``

> 

restart; with(plots)
> 

ga := .2; Gc := .2; n := 2; Sc := .5; Kp := .2; Q := 0.5e-1; Gr := .2
> 

A := .1Pr := 6.2; Nt := .2; alpha := .1; Rd := 1.5; M := .5; E1 := .3; Ec := .3; Thetap := .2; Nb := .2
> 

``
> 

a1 := 1.301348831
``
> 

a2 := 1.298194584
a3 := .9728927630
> 

a4 := .9161173998
> 

a5 := 1.316893419
> 

a6 := 1.333333333``
> 

OdeSys := a1*((diff(f(eta), eta, eta, eta))*(2*eta*ga+1)+2*(diff(f(eta), eta, eta))*ga)/a2+A^2-(diff(f(eta), eta))^2+f(eta)*(diff(f(eta), eta, eta))-a1*Kp*(diff(f(eta), eta))/a2-a6*M*(diff(f(eta), eta))/a2+a4*(Theta(eta)*Gr+Phi(eta)*Gc)/a2, f(eta)*(diff(Theta(eta), eta))+a5*(1+4*Rd*(1/3))*((diff(Theta(eta), eta, eta))*(2*eta*ga+1)+2*(diff(Theta(eta), eta))*ga)/(a3*Pr)+(diff(f(eta), eta, eta))^2*(2*eta*ga+1)*Ec*a1/a3+Theta(eta)*Q/a3, f(eta)*(diff(Phi(eta), eta))+((diff(Phi(eta), eta, eta))*(2*eta*ga+1)+2*(diff(Phi(eta), eta))*ga)/Sc+Nt*((diff(Theta(eta), eta, eta))*(2*eta*ga+1)+2*(diff(Theta(eta), eta))*ga)/(Nb*Sc)-Kr*(1+Theta(eta)*(Thetap-1))^n*exp(-E1/(1+Theta(eta)*(Thetap-1))); Cond := f(0) = 0, (D(f))(0) = 1, a5*(D(Theta))(0) = -alpha*(Theta(0)-1), Phi(0) = 1, (D(f))(10) = A, Theta(10) = 0, Phi(10) = 0
> 

KrVals := [0.1e-1, .1, .2, .3]
> 

for j to numelems(KrVals) do Ans[j] := dsolve(eval([OdeSys, Cond], Kr = KrVals[j]), numeric, output = listprocedure) end do
> 

with(plots):
  cols := [red, blue, green,black]:
> 

F:= [Theta](eta):
Frame:= Inf-> plots:-display(seq
      ( odeplot
        ( Ans[k],[eta,Theta(eta)],
          eta=0..10,
          color=cols[k]
        ),
        k=1..numelems(KrVals)
      )):
    
plots:-animate(Frame, [Inf], Inf= 0.01..10, frames= 50, size= [900,600],'axes'= 'boxed',labels=[eta,'Theta(eta)']);

 

``

Download animation_of_ode.mw

differnet values of Kr in single plot...

SHIVAS 40
June 01 2024
0 0

@Carl Love 

Thank you for Quick reply  sir.

How to impliment for differnet values of Kr in single plot with animation.

i have mentioned the graphs which are not in animation mood i want same graph with animation form.

for different values of parameter i need to check the differences in the plots how it will vary.

substitution of terms in the expression...

SHIVAS 40
May 31 2024
0 0

@Kitonum 

How to substitute the expressions as another variables
 

> 

OdeSys := diff(Phi0(y), y, y)+S*Sc*(diff(Phi0(y), y))-Kr*Sc*Phi0(y) = 0, E1*(diff(Theta0(y), y, y))+Pr*S*C*(diff(Theta0(y), y))-(Q+R)*Theta0(y)+Du*(diff(Phi0(y), y, y))+Pr*C*QL*Phi0(y) = 0
> 

Cond := Phi0(0) = 1, Theta0(0) = 1, Phi0(10) = 0, Theta0(10) = 0
> 

``
> 

``
> 

solutions := dsolve({Cond, OdeSys})
> 

``
> 

Phi0_exact := eval(Phi0(y), solutions)

-exp(-5*S*Sc-5*(Sc*(S^2*Sc+4*Kr))^(1/2))*exp((1/2)*(-S*Sc+(Sc*(S^2*Sc+4*Kr))^(1/2))*y)/(exp(-5*S*Sc+5*(Sc*(S^2*Sc+4*Kr))^(1/2))-exp(-5*S*Sc-5*(Sc*(S^2*Sc+4*Kr))^(1/2)))+exp(-5*S*Sc+5*(Sc*(S^2*Sc+4*Kr))^(1/2))*exp(-(1/2)*(S*Sc+(Sc*(S^2*Sc+4*Kr))^(1/2))*y)/(exp(-5*S*Sc+5*(Sc*(S^2*Sc+4*Kr))^(1/2))-exp(-5*S*Sc-5*(Sc*(S^2*Sc+4*Kr))^(1/2)))

 (1)
> 

subs(m1 = -5*S*Sc-5*sqrt(Sc*(S*Sc*S+4*Kr)), m2 = ((-S*Sc+sqrt(Sc*(S^2*Sc+4*Kr)))*(1/2))*Phi0_exact)
> 

Phi0_exact

-exp(-5*S*Sc-5*(Sc*(S^2*Sc+4*Kr))^(1/2))*exp((1/2)*(-S*Sc+(Sc*(S^2*Sc+4*Kr))^(1/2))*y)/(exp(-5*S*Sc+5*(Sc*(S^2*Sc+4*Kr))^(1/2))-exp(-5*S*Sc-5*(Sc*(S^2*Sc+4*Kr))^(1/2)))+exp(-5*S*Sc+5*(Sc*(S^2*Sc+4*Kr))^(1/2))*exp(-(1/2)*(S*Sc+(Sc*(S^2*Sc+4*Kr))^(1/2))*y)/(exp(-5*S*Sc+5*(Sc*(S^2*Sc+4*Kr))^(1/2))-exp(-5*S*Sc-5*(Sc*(S^2*Sc+4*Kr))^(1/2)))

 (2)

NULL


 

Download substition_values_in_the_expression.mw

Modified equation...

SHIVAS 40
May 29 2024
0 0

Now the exations are executing for exact solution. 

How to simply the equation in some other symbols like m1 m2 m3

To reduce the length of the equation.

> 

restart
> 

with(plots)
> 

OdeSys := diff(Phi0(y), y, y)+S*Sc*(diff(Phi0(y), y))-Kr*Sc*Phi0(y) = 0, E1*(diff(Theta0(y), y, y))+Pr*S*C*(diff(Theta0(y), y))-F*Theta0(y)+Du*(diff(Phi0(y), y, y))+Pr*C*QL*Phi0(y) = 0, D1*(diff(U0(y), y, y))+A*S*(diff(U0(y), y))-(M1+D1/K)*U0(y)+B*G1*Theta0(y) = 0
> 

Cond := Phi0(0) = 1, Theta0(0) = 1, U0(0) = 1, Phi0(10) = 0, Theta0(10) = 0, U0(10) = 0
> 

OdeSys1 := diff(Phi1(y), y, y)+S*Sc*(diff(Phi1(y), y))-(i*omega+Kr)*Sc*Phi1(y) = 0, E1*(diff(Theta1(y), y, y))+Pr*S*C*(diff(Theta1(y), y))-F*Pr*C*i*omega*Theta1(y)+Du*(diff(Phi1(y), y, y))+Pr*C*QL*Phi1(y) = 0, D1*(diff(U1(y), y, y))+A*S*(diff(U1(y), y))-(M1+D1/K+A*i*omega)*U1(y)+B*G1*Theta1(y) = 0; Cond1 := Phi1(0) = 1, Theta1(0) = 1, U1(0) = 0, Phi1(10) = 0, Theta1(10) = 0, U1(10) = 0
> 

solutions := dsolve({Cond, OdeSys})
> 

solutions1 := dsolve({Cond1, OdeSys1})
> 

U0_exact := eval(U0(y), solutions)
> 

Theta0_exact := eval(Theta0(y), solutions)
> 

Phi0_exact := eval(Phi0(y), solutions)
> 

U1_exact := eval(U1(y), solutions1)
> 

Theta1_exact := eval(Theta1(y), solutions1)
> 

Phi1_exact := eval(Phi1(y), solutions1)
 

 

NULL

Download ode_exact.mw

Remaining terms of the equation...

SHIVAS 40
May 28 2024
0 0

If i am calculating seperatly for phi0 and phi1 i am getting the similar result as shown in pdf  and the link Here but if i m coupled with second equation temperature equation Theta0 and Theta1 not exicuting showing null sir. How to get this one

I want all terms exact equation. Phi0(y) and Phi1(y) i am getting similarly i need the remaing Theta0(y) and Theta1(y), U0(y), and U1(y)

In the second equation i wrongly mentioned it should be B*G1*Theta1(y) instard of B*G1*Theta0(y)

details...

SHIVAS 40
May 27 2024
0 0

Here i m attaching the solution mentioned in the given article exact_solution.pdf

bounday can be any value just for reference consider as 10

Try...

SHIVAS 40
March 05 2024
0 0

@KIRAN SAJJAN

Define all the termes before the solution then check once for numerical values it will satisfiy for expression check with it.

@jganding Thank you...

SHIVAS 40
December 15 2023
0 0

@jganding

Thank you

@jganding I need only the solution for ...

SHIVAS 40
December 15 2023
0 0

@jganding

I need only the solution for equation 1 sir

Initial conditions can consider as Q(0)=Q0, E(0)=E0, and S(0)=S0

similar question...

SHIVAS 40
November 09 2023
0 0

@Carl Love 

dear sir i have noticed that same question asked by the person @suryakanth which I heve  previously posted up to now i have not seen any update in my post for that i have replied by the same quesstion.

stream plots...

SHIVAS 40
October 29 2023
0 0

plot of u1(eta,t)

fieldplot((u1(eta, t)), eta = 0 .. 1, t = 0 .. 1.2)

fieldplot(diff((u1(eta, t)),eta), eta = 0 .. 1, t = 0 .. 1.2)

inf := 3;
M := 2.23606797749; Gr := 5; beta := 1; alpha0 := .1; gama := 3; Ra := 2; Pr := 7.1; R := 2; sigma1 := .2;
in that i taken two different cases when in conditions1) u1(0,t)=1,

case-2) u1(0,t)=t

for diffferent cases i need check the flow pattern with the vecotor plot  above mentioned demo graph

vector plot for time dependent pde...

SHIVAS 40
October 29 2023
0 0

some plots i have mentioned but some i m not able to evaluate 

urgent required for me. how to plot this type of graph

How to draw this vector plot suggest any previously done file I will do according to that

vector_field_plot.mw

i have posted a few days back only i wont received any answer for this question

Please give me reply for my post

exact and numerical method answer...

SHIVAS 40
September 12 2023
0 0

@dharr 
 

``

> 

restart
> 

Nc := .2; M := 4; QG := 2.5; Thetaa := .2; n1 := 1; lambda1 := .1; lambda2 := .1; lambda3 := .1
p := 2Nr := 5``
> 

z1 := .1054816; z2 := 1.220152381; z3 := .9949638875; z4 := 0.4675903524e-1; z5 := 3.098688889
> 

OdeSys := z4*(diff(Theta(X, tau), X, X))/z3-(diff(Theta(X, tau), tau))-Nc*(Theta(X, tau)-Thetaa)^2*z5/z1-M^2*(Theta(X, tau)-Thetaa)^(p+1)*(1+n1*z2/z3)/(z3*(1-Thetaa)^p)-Nr*(Theta(X, tau)^4-Thetaa^4)/z3+QG*(1+lambda1*(Theta(X, tau)-Thetaa)+lambda2*(Theta(X, tau)-Thetaa)^2+lambda3*(Theta(X, tau)-Thetaa)^3)/z3; Cond := {Theta(0, tau) = 1, Theta(X, 0) = 0, (D[1](Theta))(1, tau) = 0}
> 

Ans := pdsolve((eval([OdeSys, Cond]))[], numeric, spacestep = 1/200, timestep = 1/100)
> 

ThetaN := eval(Theta(X, tau), Ans:-value(Theta(X, tau), output = listprocedure))
> 

Matrix([["Theta(X,tau)"], seq([ThetaN(1, .8)])])

Matrix(%id = 18446746700596173150)

 (1)
> 

restart
> 

OdeSys := z4*(diff(Theta(X, tau), X, X))/z3-(diff(Theta(X, tau), tau))-Nc*(Theta(X, tau)-Thetaa)^2*z5/z1-M^2*(Theta(X, tau)-Thetaa)^(p+1)*(1+n1*z2/z3)/(z3*(1-Thetaa)^p)-Nr*(Theta(X, tau)^4-Thetaa^4)/z3+QG*(1+lambda1*(Theta(X, tau)-Thetaa)+lambda2*(Theta(X, tau)-Thetaa)^2+lambda3*(Theta(X, tau)-Thetaa)^3)/z3; Cond := {Theta(0, tau) = 1, Theta(X, 0) = 0, (D[1](Theta))(1, tau) = 0}
> 

ans := simplify(eval([OdeSys, Cond], Theta(X, tau) = Theta_exact))

[(-M^2*(Theta_exact-Thetaa)^(p+1)*(n1*z2+z3)*(1-Thetaa)^(-p)*z1-(z5*Nc*(-Theta_exact+Thetaa)^2*z3-z1*((lambda3*Theta_exact^3+(-3*Thetaa*lambda3+lambda2)*Theta_exact^2+(3*Thetaa^2*lambda3-2*Thetaa*lambda2+lambda1)*Theta_exact-Thetaa^3*lambda3+Thetaa^2*lambda2-lambda1*Thetaa+1)*QG+Thetaa^4*Nr-Nr*Theta_exact^4))*z3)/(z1*z3^2), {Theta(0, tau) = 1, Theta(X, 0) = 0, (D[1](Theta))(1, tau) = 0}]

 (2)
> 

``

``


 

Download exact_solution.mw

in this i need exact equation of Theta (X,tau)

after substitution of parameter values i need to check the answers for both method

exact solution...

SHIVAS 40
September 12 2023
0 0

Dear sir can we find exact solution of the Given pde with IC and BC 

restart;
OdeSys := z4*(diff(Theta(X, tau), X, X))/z3-(diff(Theta(X, tau), tau))-Nc*(Theta(X, tau)-Thetaa)^2*z5/z1-M^2*(Theta(X, tau)-Thetaa)^(p+1)*(1+n1*z2/z3)/(z3*(1-Thetaa)^p)-Nr*(Theta(X, tau)^4-Thetaa^4)/z3+QG*(1+lambda1*(Theta(X, tau)-Thetaa)+lambda2*(Theta(X, tau)-Thetaa)^2+lambda3*(Theta(X, tau)-Thetaa)^3)/z3; Cond := {Theta(0, tau) = 1, Theta(X, 0) = 0, (D[1](Theta))(1, tau) = 0};
ans := simplify(eval([OdeSys, Cond], Theta(X, tau) = Theta_exact));
 in the given equation i want the value of Theta(X, tau)

By numerical method i can get the solution by substituting the values of parameters but i need to check with exact solution method also 

is this possible to solve ?

thank you

single plot...

SHIVAS 40
September 07 2023
0 0

@Carl Love according to your suggestions i tried but i m getting only one curve 

for different values of Nr i m not able to calculate the plots please help me to solve. its imp for me and urgent please reply

Thank you

single_plot_practice.mw

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