@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 
 

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

@Carl Love for the value of X=0.5 and tau=0.8 and i want to plot eta versus tau for different values of Nr  tau range is 0..2  efficiency_plots.pdf this is the demo figure for the reference here eta is efficiency 

modifed_practice.mw  

waiting for the reply sir 

Thank you

@tomleslie 

I need to plot by using the equations 11,12,13 and 14  from both i need same graphs 

U plotted by using the equations 7,8,9 and 10.

  @tomleslie How to remove error both numeric and Hpm solutions should be nearly same

@tomleslie  sir this is not FDM method 

This is normal dsolve method.

But I m asking about Crank nicolson method 

In that  all terms are mentioned uij uij+1...

With boundary conditions