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KIRAN SAJJAN

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exact solution of ODE...

KIRAN SAJJAN 40 Maple 2018
ode syntax homework
March 04 2024
3 2

exact_solution_error.mw

P1=(D1*ga+C1)/(1+ga) and P2= (D1-C1)/(1+ga), How to substitute in the solution of U_exact and Theta_exact

Please help me to solve

Apply Keller Box method for ODE...

KIRAN SAJJAN 40 Maple 2018
ode homework numeric
February 20 2024
0 2

Dear sir ,

I have implemented Dsolve method the code was executed, but i need to apply Kellor Box method to solve the ODES 

Please can any one help how to implement? 

because there is no post regarding the Kellor box method. 

> 

restart; with(plots)
> 

``
> 

S := 1; Rd := .1; delta := .1; Hs := 1; Sc := .1; Pr := 6.8; n := 1; Rc := .1; E := .1; M := 1
> 

NULL
 
> 

OdeSys := a1*(diff(f(eta), eta, eta, eta, eta))/a2-S*(3*(diff(f(eta), eta, eta))+eta*(diff(f(eta), eta, eta, eta))+(diff(f(eta), eta))*(diff(f(eta), eta, eta))-f(eta)*(diff(f(eta), eta, eta, eta)))-a5*M*(diff(f(eta), eta, eta))/a2-a1*Kp*(diff(f(eta), eta, eta))/a2 = 0, (a4+4*Rd)*(diff(Theta(eta), eta, eta))+12*Rd*delta*((diff(Theta(eta), eta))*(diff(Theta(eta), eta))+Theta(eta)*(diff(Theta(eta), eta, eta)))+Hs*Theta(eta)-a3*Pr*S*(diff(Theta(eta), eta))*(eta-f(eta)) = 0, diff(Phi(eta), eta, eta)-S*Sc*(diff(Phi(eta), eta))*(eta-f(eta))-Sc*Rc*(1+delta*Theta(eta))^n*Phi(eta)*exp(-E/(1+delta*Theta(eta))) = 0; Cond := f(0) = 0, ((D@@2)(f))(0) = 0, (D(Theta))(0) = 0, (D(Phi))(0) = 0, f(1) = 1, (D(f))(1) = 0, Theta(1) = 1, Phi(1) = 1
   
> 

KpVals := [1, 2, 3, 4]
> 

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

 
> 

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

 plotA:= display
  ( [ seq
      ( odeplot
        ( Ans[k],[eta,(f(eta))],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,'f(eta)'],labelfont=[TIMES,BOLD,16]
  );
 
> 

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

plotB:= display( [ seq( odeplot
        ( Ans[k],[eta,Theta(eta)],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,'Phi(eta)'],labelfont=[TIMES,BOLD,16]
  );

 

 
> 

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

plotC:= display( [ seq( odeplot
        ( Ans[k],[eta,Phi(eta)],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,'Phi(eta)'],labelfont=[TIMES,BOLD,16]
  );

 
> 

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

 plotA:= display
  ( [ seq
      ( odeplot
        ( Ans[k],[eta,(diff(f(eta),eta))],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,"f '(eta)"],labelfont=[TIMES,BOLD,16]
  );
> 

 

 
> 

 

Download kellor_box_method.mw

plot error in 3d plots ...

KIRAN SAJJAN 40 Maple 2018
ode syntax homework
December 19 2023
1 3

sachi_stream_error_3d.mw  3d_sachin_p1.mw

Dear sir, there is something missing why it is not able to evaluate?

By reference of some posts I have implemented to my ODE but not getting the graph.

what is the mistake in both files?

error in calculating table values for t...

KIRAN SAJJAN 40 Maple 2018
homework pde numeric
October 27 2023
1 2
> 

restart
> 

with(PDEtools); with(plots)
> 

inf := 6
> 

Gr := 3; beta := 2; sigma1 := .2; alpha0 := .1; Pr := 7.1; E1 := .5; S := .7; Gm := 3; Kp := .2; Lat := .5; Lac := .5; Sc := .5; a1 := .5; Rd := 3
> 

OdeSys := {diff(Phid(eta, t), t)-Lac*(Phi(eta, t)-Phid(eta, t)) = 0, diff(Thetad(eta, t), t)-Lat*(Theta(eta, t)-Thetad(eta, t)) = 0, diff(Phi(eta, t), eta, eta)-Sc*(diff(Phi(eta, t), t))-Sc*a1*R*(Phi(eta, t)-Phid(eta, t)) = 0, diff(Theta(eta, t), eta, eta)-Pr*(diff(Theta(eta, t), t))-Rd*Theta(eta, t)-(2/3)*R*(Theta(eta, t)-Thetad(eta, t)) = 0, (1+1/beta)*(diff(U(eta, t), eta, eta))-(diff(U(eta, t), t))-Kp*U(eta, t)-R*(U(eta, t)-Ud(eta, t))+Gr*Theta(eta, t)+Gm*Phi(eta, t)+E1*exp(-S*eta) = 0, sigma1*(diff(Ud(eta, t), t))-U(eta, t)+Ud(eta, t) = 0}
> 

Cond := {Phi(0, t) = 1, Phi(eta, 0) = 0, Phi(inf, t) = 0, Phid(eta, 0) = 0, Theta(eta, 0) = 0, Theta(inf, t) = 0, Thetad(eta, 0) = 0, U(0, t) = t, U(eta, 0) = 0, U(inf, t) = 0, Ud(eta, 0) = 0, (D[1](Theta))(0, t) = -alpha0*(-1+Theta(0, t))}
> 

OdeSys1 := {diff(Phid1(eta, t), t)-Lac*(Phi1(eta, t)-Phid1(eta, t)) = 0, diff(Thetad1(eta, t), t)-Lat*(Theta1(eta, t)-Thetad1(eta, t)) = 0, diff(Phi1(eta, t), eta, eta)-Sc*(diff(Phi1(eta, t), t))-Sc*a1*R*(Phi1(eta, t)-Phid1(eta, t)) = 0, diff(Theta1(eta, t), eta, eta)-Pr*(diff(Theta1(eta, t), t))-Rd*Theta1(eta, t)-(2/3)*R*(Theta1(eta, t)-Thetad1(eta, t)) = 0, (1+1/beta)*(diff(U1(eta, t), eta, eta))-(diff(U1(eta, t), t))-Kp*U1(eta, t)-R*(U1(eta, t)-Ud1(eta, t))+Gr*Theta1(eta, t)+Gm*Phi1(eta, t)+E1*exp(-S*eta) = 0, sigma1*(diff(Ud1(eta, t), t))-U1(eta, t)+Ud1(eta, t) = 0}
> 

Cond1 := {Phi1(0, t) = 1, Phi1(eta, 0) = 0, Phi1(inf, t) = 0, Phid1(eta, 0) = 0, Theta1(eta, 0) = 0, Theta1(inf, t) = 0, Thetad1(eta, 0) = 0, U1(0, t) = 1, U1(eta, 0) = 0, U1(inf, t) = 0, Ud1(eta, 0) = 0, (D[1](Theta1))(0, t) = -alpha0*(-1+Theta1(0, t))}
> 

``
> 

NULL
> 

NULL
> 

colour := [red, green, blue, black]
> 

RVals := [1, 5, 10, 20]
> 

NULL
> 

for j to numelems(RVals) do Ans := pdsolve((eval([OdeSys, Cond], R = RVals[j]))[], numeric, time = t, spacestep = 0.25e-1, timestep = 1); Ans1 := pdsolve((eval([OdeSys1, Cond1], R = RVals[j]))[], numeric, time = t, spacestep = 0.25e-1, timestep = 1) end do
> 

interface(rtablesize = 100); interface(displayprecision = 5); Matrix([[eta, `Νu1`, `Νud1`, Shr1, Shrd1, `Νu2`, `Νud2`, Shr2, Shrd2, `Νu3`, `Νud3`, Shr3, Shrd3, `Νu4`, `Νud4`, Shr4, Shrd4], seq([seq([(eval(diff(Theta(.5, eta), eta), Ans[k]))(0), (eval(diff(Phi(.5, eta), eta), Ans[k]))(0), (eval(Thetad(.5, eta), Ans[k]))(0), (eval(Phid(.5, eta), Ans[k]))(0)][], k = 1 .. numelems(RVals))])]); interface(rtablesize = 10); interface(displayprecision = -1)

Error, index must evaluate to a name when indexing a module

  
> 

interface(rtablesize = 100); interface(displayprecision = 5); Matrix([[eta, `Νu1`, `Νud1`, Shr1, Shrd1, `Νu2`, `Νud2`, Shr2, Shrd2, `Νu3`, `Νud3`, Shr3, Shrd3, `Νu4`, `Νud4`, Shr4, Shrd4], seq([seq([(eval(diff(Theta1(1.1, eta), eta), Ans1[k]))(0), (eval(diff(Phi1(1.1, eta), eta), Ans1[k]))(0), (eval(Thetad1(1.1, eta), Ans1[k]))(0), (eval(Phid1(1.1, eta), Ans1[k]))(0)][], k = 1 .. numelems(RVals))])]); interface(rtablesize = 10); interface(displayprecision = -1)

Error, index must evaluate to a name when indexing a module

  
> 

NULL

Download error_in_table_value.mw

first and second order derivative plot ...

KIRAN SAJJAN 40 Maple 2018
homework pde numeric
October 17 2023
1 5

How to plot the second order derivative and first oder derivatives plot in time dependent pde and vector plot of  theta(y,t), u(y,t) at y=0..10 and t=0..1

nowhere i found a vector plot of time-dependent pde 

how to plot give me suggestions.

in vector plots, flow patterns should show with arrow marks

> 

  restart;
  inf:=10:
  pdes:= diff(u(y,t),t)-xi*diff(u(y,t),y)=diff(u(y,t),y$2)/(1+lambda__t)+Gr*theta(y,t)+Gc*C(y,t)-M*u(y,t)-K*u(y,t),
         diff(theta(y,t),t)-xi*diff(theta(y,t),y)=1/Pr*diff(theta(y,t),y$2)+phi*theta(y,t),
         diff(C(y,t),t)-xi*diff(C(y,t),y)=1/Sc*diff(C(y,t),y$2)-delta*C(y,t)+nu*theta(y,t):
  conds:= u(y,0)=0, theta(y,0)=0, C(y,0)=0,
          u(0,t)=0, D[1](theta)(0,t)=-1, D[1](C)(0,t)=-1,
          u(inf,t)=0, theta(inf,t)=0, C(inf,t)=0:
  pars:= { Gr=1, Gc=1, M=1, nu=1, lambda__t=0.5,
           Sc=0.78, delta=0.1, phi=0.5, K=0.5, xi=0.5
         }        

{Gc = 1, Gr = 1, K = .5, M = 1, Sc = .78, delta = .1, nu = 1, phi = .5, xi = .5, lambda__t = .5}

 (1)
> 

  PrVals:=[0.71, 1.00, 3.00, 7.00]:
  colors:=[red, green, blue, black]:
  for j from 1 to numelems(PrVals) do
      pars1:=`union`( pars, {Pr=PrVals[j]}):
      pdSol:= pdsolve( eval([pdes], pars1),
                       eval([conds], pars1),
                       numeric
                     );
      plt[j]:=pdSol:-plot( diff(u(y,t),y), y=0, t=0..2, numpoints=200, color=colors[j]);
  od:
  plots:-display( [seq(plt[j], j=1..numelems(PrVals))]);

 
> 

PrVals := [.71, 1.00, 3.00, 7.00]; colors := [red, green, blue, black]; for j to numelems(PrVals) do pars1 := `union`(pars, {Pr = PrVals[j]}); pdSol := pdsolve(eval([pdes], pars1), eval([conds], pars1), numeric); plt[j] := pdSol:-plot(diff(u(y, t), y, y), y = 0, t = 0 .. 2, numpoints = 200, color = colors[j]) end do; plots:-display([seq(plt[j], j = 1 .. numelems(PrVals))])

 
 

 

Download badPDE.mw

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