SHIVAS

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1 years, 2 days

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These are replies submitted by SHIVAS

@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

@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.

@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

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

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)

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

@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

@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

@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.

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

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

@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

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

@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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