# Question:how to solve numerical solution of ODE with Bc and including integral terms

## Question:how to solve numerical solution of ODE with Bc and including integral terms

Maple 2018

restart;
kp := .3;

Pr := .3; N := .5; g := .5; A := 1; B := 0; M := .5; lambda := .5; Ec := .5;

rf := 997.1; kf := .613; cpf := 4179; `&sigma;f` := 0.5e-1;
p1 := 0.1e-1; sigma1 := 2380000; rs1 := 4250; ks1 := 8.9538; cps1 := 686.2;
p2 := 0.5e-1; sigma2 := 3500000; rs2 := 10500; ks2 := 429; cps2 := 235;

NULL;
a1 := (1-p1)^2.5*(1-p2)^2.5;
a2 := (1-p2)*(1-p1+p1*rs1/rf)+p2*rs2/rf;
a3 := 1+3*((p1*sigma1+p2*sigma2)/`&sigma;f`-p1-p2)/(2+(p1*sigma1+p2*sigma2)/((p1+p2)*`&sigma;f`)-((p1*sigma1+p2*sigma2)/`&sigma;f`-p1-p2));

a4 := (1-p2)*(1-p1+p1*rs1*cps1/(rf*cpf))+p2*rs2*cps2/(rf*cpf);
a5 := (ks1+2*kf-2*p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2))/((ks1+2*kf+p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))+2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2)));

OdeSys := (diff(U(Y), Y, Y))/(a1*a2)+Theta(Y)+N*(Theta(Y)*Theta(Y))-a3*(M*M)*U(Y)/a2-(kp*kp)*U(Y)/(a1*a2), a5*(diff(Theta(Y), Y, Y))/a4+Pr*Ec*((diff(U(Y), Y))^2+U(Y)^2*(kp*kp))/(a1*a2); Cond := U(0) = lambda*(D(U))(0), Theta(0) = A+g*(D(Theta))(0), U(1) = 0, Theta(1) = B; Ans := dsolve([OdeSys, Cond], numeric, output = listprocedure);
U := proc (Y) options operator, arrow, function_assign; (eval(U(Y), Ans))(0) end proc;
U := Y -> (eval(U(Y), Ans))(0)
Theta := proc (Y) options operator, arrow, function_assign; (eval(Theta(Y), Ans))(0) end proc;
Theta := Y -> (eval(Theta(Y), Ans))(0)
Theta_b := (int(U(Y)*Theta(Y), Y = 0 .. 1))/(int(U(Y), Y = 0 .. 1));
Error, (in Theta) too many levels of recursion
Q := int(U(Y), Y = 0 .. 1, numeric);
Error, (in Theta) too many levels of recursion
NUMERIC := [(eval((diff(U(Y), Y))/a1, Ans))(0), (eval(-(diff(Theta(Y), Y))/(Theta_b*a5), Ans))(0)];
Error, (in Theta) too many levels of recursion

i need the solution  for Y=0 and Y=1

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