with(plots); R1 := .1; R0 := .1; m := .1; a := .1; Ha := .1; Nt := .1; Nb := .1; Pr := 6.2; Le := .6; Bi := 1; Ec := .1; k := 1; r := .1; A := 1; fcns := {C(y), T(y), U(y), W(y)}; sys := diff(U(y), `$`(y, 2))+(R1*(diff(U(y), y))-2*R0*W(y))*exp(a*T(y))-a*(diff(U(y), y))*(diff(T(y), y))-Ha*(U(y)+m*W(y))*exp(a*T(y))/(m^2+1)-U(y)/k+A*exp(a*T(y)) = 0, diff(W(y), `$`(y, 2))+(R1*(diff(W(y), y))+2*R0*U(y))*exp(a*T(y))-a*(diff(W(y), y))*(diff(T(y), y))-Ha*(W(y)-m*U(y))*exp(a*T(y))/(m^2+1)-W(y)/k = 0, diff(T(y), `$`(y, 2))+R1*Pr*(diff(T(y), y))+Pr*Ec*exp(-a*T(y))*((diff(U(y), y))*(diff(U(y), y))+(diff(W(y), y))*(diff(W(y), y)))+Nt*(diff(T(y), y))*(diff(T(y), y))+Pr*Ec*(U(y)*U(y)+W(y)*W(y))*exp(-a*T(y))/k = 0, diff(C(y), `$`(y, 2))+Pr*Le*R1*(diff(C(y), y))+Nt*(diff(C(y), `$`(y, 2)))/Nb-r*C(y) = 0; bc := U(0) = 0, W(0) = 0, C(0) = 0, (D(T))(0) = Bi*(T(0)-1), U(1) = 0, W(1) = 0, C(1) = 1, T(1) = 0; L := [.5, 1.0, 1.5, 2.0]; AP := NULL; for k to 4 do R := dsolve(eval({bc, sys}, Ha = L[k]), fcns, type = numeric, method = bvp[midrich], AP); AP := approxsoln = R; p1u[k] := odeplot(R, [y, U(y)], 0 .. 1, numpoints = 100, labels = ["y", "U"], linestyle = dash, color = black) end do; display({p1u[1], p1u[2], p1u[3], p1u[4]})

restart;
N:=4;alpha:=5*3.14/180;r:=10;Ha:=5;H:=1;
dsolve(diff(f(x),x,x,x));
Rf:=diff(f[m-1](x),x,x,x)+2*alpha*r*sum*(f[m-1-n](x)*diff(f[n](x),x),n=0..m-1)
+(4-Ha)*(alpha)^2*diff(f[m-1](x),x);
dsolve(diff(f[m](x),x,x,x)-CHI[m]*(diff(f[m-1](x),x,x,x))=h*H*Rf,f[m](x));
f[0](x):=1-x^2;
for m from 1 by 1 to N do
CHI[m]:='if'(m>1,1,0);
f[m](x):=int(int(int(CHI[m]*(diff(f[m-1](x),x,x,x))+h*H(diff(f[m-1](x),x,x,x))
+2*h*H*alpha*r*(sum(f[m-1-n](x)*(diff(f[n](x),x)),n=0..m-1))+4*h*H*alpha^2*
(diff(f[m-1](x),x))-h*H*alpha^2*(diff(f[m-1](x),x))*Ha,x),x)+_C1*x,x)+_C2*x+_C3;
s1:=evalf(subs(x=0,f[m](x)))=0;
s2:=evalf(subs(x=0,diff(f[m](x),x)))=0;
s1:=evalf(subs(x=1,f[m](x)))=0;
s:={s1,s2,s3}:
f[m](x):=simplify(subs(solve(s,{_C1,_C2,_C3}),f[m](x)));
end do;
f(x):=sum(f[1](x),1=0..N);
hh:=evalf(subs(x=1,diff(f(x),x))):
plot(hh,h=-1.5..-0.2);
A(x):=subs(h=-0.9,f(x));
plot(A(x),x=0..1);

how to change the ode polt color for the below code

with(plots); 

R1 := .1; R0 := .1; m := .1; a := .1; Ha := .1; Nt := .1; Nb := .1; Pr := 6.2; Le := .6; Bi := 1; Ec := .1; k := 1; r := .1; A := 1; 

fcns := {C(y), T(y), U(y), W(y)};

 sys := diff(U(y), `$`(y, 2))+(R1*(diff(U(y), y))-2*R0*W(y))*exp(a*T(y))-a*(diff(U(y), y))*(diff(T(y), y))-Ha = 0, diff(W(y), `$`(y, 2))+(R1*(diff(W(y), y))+2*R0*U(y))*exp(a*T(y))-a*(diff(W(y), y))*(diff(T(y), y))-Ha = 0, diff(T(y), `$`(y, 2))+R1*Pr*(diff(T(y), y))+Pr*Ec*exp(-a*T(y))*((diff(U(y), y))*(diff(U(y), y))+(diff(W(y), y))*(diff(W(y), y)))+Pr*Ha*Ec = 0, diff(C(y), `$`(y, 2))+Pr*Le*R1*(diff(C(y), y))+Nt*(diff(C(y), `$`(y, 2)))/Nb = 0;

bc := U(0) = 0, W(0) = 0, C(0) = 0, (D(T))(0) = Bi*(T(0)-1), U(1) = 0, W(1) = 0, C(1) = 1, T(1) = 0; 

L := [.5, 1.0, 1.5, 2.0]; AP := NULL; 

for k to 4 do R := dsolve(eval({bc, sys}, Ha = L[k]), fcns, type = numeric, AP); AP := approxsoln = R; p1u[k] := odeplot(R, [y, U(y)], 0 .. 1, numpoints = 100, labels = ["y", "U"], style = line, color = ["black", "blue", "red", "pink"]) end do; 

display({p1u[1], p1u[2], p1u[3], p1u[4]})

fcns:=[u(x,y,t),v(x,y,t),h(x,y,t),c(x,y,t)]:

gr:=0.5:pr:=0.71:sc:=0.7:m:=1.0:k:=0.3:
  fcns:=[u(x,y,t),v(x,y,t),h(x,y,t),c(x,y,t)]:
  IC := [u(x,y,0)=0,v(x,y,0)=0,h(x,y,0)=0,c(x,y,0)=0]:
  BC:=[u(0,y,t)=0,h(0,y,t)=0,c(0,y,t)=0,u(x,0,t)=1,v(x,0,t)=0,h(x,0,t)=1,c(x,0,t)=1,u(x,10,t)=0,h(x,10,t)=0,c(x,10,t)=0];
  eq1:={diff(u(x,y,t),t)+u(x,y,t)*diff(u(x,y,t),x)+v(x,y,t)*diff(u(x,y,t),y)=diff(u(x,y,t),y$2)+gr*h(x,y,t)+gr*c(x,y,t)-m*u(x,y,t)
,diff(h(x,y,t),t)+u(x,y,t)*diff(h(x,y,t),x)+v(x,y,t)*diff(h(x,y,t),y)=1/pr*diff(h(x,y,t),y$2),diff(c(x,y,t),t)+u(x,y,t)*diff(c(x,y,t),x)+v(x,y,t)*diff(c(x,y,t),y)=1/sc*diff(h(x,y,t),y$2)-k*c(x,y,t)}:
  pds:= pdsolve(eq1,IC,BC,fcns,numeric):
  pds:= pdsolve(eq1,IC,BC,fcns,numeric,spacestep = 1/100):

for the above problem i made this code.

how to find skin friction value below code

restart

PDEtools[declare]((U, W, T, C)(y), prime = y):

R1 := .1; R0 := .1; m := .1; a := .1; Ha := .1; Nt := .1; Nb := .1; Pr := 6.2; Le := .6; Bi := 1; Ec := .1; k := 1; r := .1; A := 1;

sys := diff(U(y), `$`(y, 2))+(R1*(diff(U(y), y))-2*R0*W(y))*exp(a*T(y))-a*(diff(U(y), y))*(diff(T(y), y))-Ha = 0, diff(W(y), `$`(y, 2))+(R1*(diff(W(y), y))+2*R0*U(y))*exp(a*T(y))-a*(diff(W(y), y)) = 0, diff(T(y), `$`(y, 2))+R1*Pr*(diff(T(y), y))+Pr*Ec*exp(-a*T(y))*((diff(U(y), y))*(diff(U(y), y))+(diff(W(y), y))*(diff(W(y), y)))+Pr*Ha*Ec*((U(y)+m*W(y))*(U(y)+m*W(y))+(W(y)-m*U(y))*(W(y)-m*U(y)))/(m^2+1)^2+Nb*(diff(T(y), y))*(diff(C(y), y))+Nt*(diff(T(y), y))*(diff(T(y), y)) = 0, diff(C(y), `$`(y, 2))+Pr*Le*R1*(diff(C(y), y))+Nt*(diff(C(y), `$`(y, 2)))/Nb = 0:

ba := {sys, C(0) = 0, C(1) = 1, T(1) = 0, U(0) = 0, U(1) = 0, W(0) = 0, W(1) = 0, (D(T))(0) = Bi*(T(0)-1)}:

r1 := dsolve(ba, numeric, output = Array([0., 0.5e-1, .10, .15, .20, .25, .30, .35, .40, .45, .50, .55, .60, .65, .70, .75, .80, .85, .90, .95, 1.00])):

with(plots); 

p1u := odeplot(r1, [y, U(y)], 0 .. 1, numpoints = 100, labels = ["y", "U"], style = line, color = green); 

plots[display]({p1u})