eq1:=( d)/(dt)u+(d^(2))/(dy^(2))u + s*( d)/(dy)u + delta * theta = 0;

eq2:=( d)/(dt)theta + (d^(2))/(dy^(2))theta + s*Pr*( d)/(dy)theta +lambda* exp(theta/(1 +(epsilon*theta))) = 0; 

initial and boundary conditons   

t <=0; u = theta = 0, for 0 <= y  <= 1   

t> 0;  u =0, theta = 0   at  y = 0;  

t> 0;  u =1, theta = 0   at   y = 1  ;

where, s, epsilon, Pr, lambda, delta are arbitrary parameters

Hello 

How to remove this error 

3d_plots.mw

Hello,

Can we impliment Artificial Neural Network for nonlinear coupled ODE equation with boundary conditions.? In maple

I wont seen any post regarding ANN in mapleprime.

Maple code for solving system of ODE using forward-backward sweep method.

When we specify a set (a sequence of objects enclosed in curly braces), Maple removes duplicates, since the elements of the set must be unique, that is, they cannot be repeated. See below for 2 examples. With the first example  {a<=b  and  b>=a}, everything is in order, since they are one and the same. But Maple treats the same equality, written in two ways  {a=b, b=a} , as different objects. It seems to me that this is not very convenient:

restart;
{a<=b, b>=a}; # OK
{a=b, b=a}; # not OK
is((a=b)=(b=a)); # not OK

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

Streamlines, isotherms and microrotations for Re = 1, Pr = 7.2, Gr = 10⁵ and (a) Ha = 0 (b) Ha = 30 (c) Ha = 60 (d) Ha = 100.

for Ra = 10⁵, Ha = 50, Pr = 0.025 and θ = 1 − Y

eqat := {M . (D(theta))(0)+2.*Pr . f(0) = 0, diff(phi(eta), eta, eta)+2.*Sc . f(eta) . (diff(phi(eta), eta))-(1/2)*S . Sc . eta . (diff(phi(eta), eta))+N[t]/N[b] . (diff(theta(eta), eta, eta)) = 0, diff(g(eta), eta, eta)-2.*(diff(f(eta), eta)) . g(eta)+2.*f(eta) . (diff(g(eta), eta))-S . (g(eta)+(1/2)*eta . (diff(g(eta), eta)))-1/(sigma . Re[r]) . ((1+d^%H . exp(-eta))/(1+d . exp(-eta))) . g(eta)-beta^%H . ((1+d^%H . exp(-eta))^2/sqrt(1+d . exp(-eta))) . g(eta) . sqrt((diff(f(eta), eta))^2+g(eta)^2) = 0, diff(theta(eta), eta, eta)+2.*Pr . f(eta) . (diff(theta(eta), eta))-(1/2)*S . Pr . eta . (diff(theta(eta), eta))+N[b] . Pr . ((diff(theta(eta), eta)) . (diff(phi(eta), eta)))+N[t] . Pr . ((diff(theta(eta), eta))^2)+4/3 . N . (diff((C[T]+theta(eta))^3 . (diff(theta(eta), eta)), eta)) = 0, diff(f(eta), eta, eta, eta)-(diff(f(eta), eta))^2+2.*f(eta) . (diff(f(eta), eta))+g(eta)^2-S . (diff(f(eta), eta)+(1/2)*eta . (diff(f(eta), eta, eta)))-1/(sigma . Re[r]) . ((1+d^%H . exp(-eta))/(1+d . exp(-eta))) . (diff(f(eta), eta))-beta^%H . ((1+d^%H . exp(-eta))^2/sqrt(1+d . exp(-eta))) . (diff(f(eta), eta)) . sqrt((diff(f(eta), eta))^2+g(eta)^2) = 0, g(0) = 1, g(6) = 0, phi(0) = 1, phi(6) = 0, theta(0) = 1, theta(6) = 0, (D(f))(0) = 1, (D(f))(6) = 0};
sys1 := eval(eqat, {M = 0, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .2, d^%H = 1.5});
sys2 := eval(eqat, {M = 0, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .4, d^%H = 1.5});
sys3 := eval(eqat, {M = 0, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .6, d^%H = 1.5});
sys4 := eval(eqat, {M = 0, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .8, d^%H = 1.5});
sys5 := eval(eqat, {M = .5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .2, d^%H = 1.5});
sys6 := eval(eqat, {M = .5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .4, d^%H = 1.5});
sys7 := eval(eqat, {M = .5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .6, d^%H = 1.5});
sys8 := eval(eqat, {M = .5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .8, d^%H = 1.5});
sys9 := eval(eqat, {M = 1, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .2, d^%H = 1.5});
sys10 := eval(eqat, {M = 1, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .4, d^%H = 1.5});
sys11 := eval(eqat, {M = 1, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .6, d^%H = 1.5});
sys12 := eval(eqat, {M = 1, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .8, d^%H = 1.5});
sys13 := eval(eqat, {M = 1.5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .2, d^%H = 1.5});
sys14 := eval(eqat, {M = 1.5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .4, d^%H = 1.5});
sys15 := eval(eqat, {M = 1.5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .6, d^%H = 1.5});
sys16 := eval(eqat, {M = 1.5, N = 2, Pr = .8, S = -2.5, Sc = .5, d = 2, sigma = .2, C[T] = .5, N[b] = .4, N[t] = .4, Re[r] = 1.1, beta^%H = .8, d^%H = 1.5});
 

restart;

OdeSys := diff(U(Y), Y, Y)+Theta(Y)+N*(Theta(Y)*Theta(Y))-(M*M)*U(Y) = 0, diff(Theta(Y), Y, Y)+E*(diff(U(Y), Y))^2 = 0;

Cond := U(0) = lambda*(D(U))(0), Theta(0) = A+g*(D(Theta))(0), U(1) = 0, Theta(1) = B; sys := [OdeSys, Cond];
Ans := dsolve(sys);

I find it difficult to use dsolve to solve system of ordinary differential equations with assigned parameters and initial conditions. The error message "Error, (in dsolve/numeric) 'parameters' must be specified as a list of unique unassigned names" kept coming up.

Pls see the uploaded equation for more understanding

Download Covid19_Simulation.mw

odeSys := {diff(Theta(x), x, x)+Pr*(R*(diff(Theta(x), x))*f(x)+Nb*(diff(Theta(x), x))*(diff(Phi(x), x))+Nt*(diff(Theta(x), x))^2), N2*(diff(G(x), x, x))-N1*(2*G(x)+diff(f(x), x, x))-N3*R*((diff(f(x), x))*G(x)-f(x)*(diff(G(x), x))), diff(Phi(x), x, x)+R*Sc*f(x)*(diff(Phi(x), x))+Nt*(diff(Theta(x), x, x))/Nb, (1+N1)*(diff(g(x), x, x))+R*((diff(g(x), x))*f(x)-g(x)*(diff(f(x), x)))-M*g(x)+2*Kr*(diff(f(x), x)), (1+N1)*(diff(f(x), x, x, x, x))-R*((diff(f(x), x))*(diff(f(x), x, x))-f(x)*(diff(f(x), x, x, x)))+N1*(diff(G(x), x, x))-M*(diff(f(x), x, x))-2*Kr*(diff(g(x), x))}; cond := f(0) = 0, (D(f))(0) = 1, g(0) = 0, Theta(0) = 1, G(0) = -n*((D@@2)(f))(0), Phi(0) = 1, f(1) = lambda, (D(f))(1) = 0, g(1) = 0, Theta(1) = 0, G(1) = n*((D@@2)(f))(1), Phi(1) = 0; ans := {};

n := .5; N1 := 0.; N2 := 1.0; N3 := .1; lambda := .1; M := .1; Kr := .1; Sc := 1.0; Nb := .1; Pr := 1.0; Nt := .1; R := .5;

ans := dsolve*{cond, eval*odeSys};

hello these are the pde and Boundary conditions  i want to calculate the value of f''(0) ,Theta(0) and  Phi(0)

what is the proper cammand to get the table values for the given equation
NBVs := [eval(ans(N1*G(x)+(1+N1)*(diff(f(x), x, x))), x = 0), eval(ans(-(diff(Theta(x), x))), x = 0), eval(ans(-(diff(Phi(x), x))), x = 0)];

Hi,

I want to solve system of PDE equations by maple and i dont know how can i write it codes that can solve them for me. Can you create the code for the equation

Thank you

Good day,
 

1. Please I need your greatest help. Can anyone please help me to run the examples on the attached papers on Maple software?

 2. Also help me to plot the graphs along with the exact solution

 3. If possible with tables

 I tried but did not get the results as expected. I shall be very grateful if I can get assistance from you

Thanks
 

Download surface_dinesh_paper.mw  please help me to solve the problem

Dear all,

consider two lists of complex values :

list1 := [l1,l2,l3,l4,l5]

list2 := [s1,s2,s3,s4,s5].

There is a set of second order differential equation

d^2u(k)/dt^2+I*A*du/dt-B*u=0

where A is sum of elements of list1 and list2 and B is multiplication of their element. Therefore,

d^2u[1](k)/dt^2+I*(l1+s1)*du[1]/dt-(l1*s1)*u[1]=0

d^2u[2](k)/dt^2+I*(l2+s2)*du[2]/dt-(l2*s2)*u[2]=0

d^2u[3](k)/dt^2+I*(l3+s3)*du[3]/dt-(l3*s3)*u[3]=0

d^2u[4](k)/dt^2+I*(l4+s4)*du[4]/dt-(l4*s4)*u[4]=0

d^2u[5](k)/dt^2+I*(l5+s5)*du[5]/dt-(l5*s5)*u[5]=0

How can I create a set of differential equations and initial conditions based on nops(list1), then solve this system of differential equations numerically in Maple.

since u[i] are function of k, next step is to transforme them to real space by inverse fourier transform.

finally save the results and plot them.

Note that for simplisity I wrote a linear equation but it is not. so, because of nonlinear terms it is not possible to use superposition of the solution. I have to take them as coupled system of equations.

====

for example

list1 := [ [0., -5.496799068*10^(-15)-0.*I], [.1, 5.201897725*10^(-16)-1.188994754*I], [.2, 6.924043163*10^(-17)-4.747763855*I], [.3, 2.297497722*10^(-17)-10.66272177*I], [.4, 1.159126178*10^(-17)-18.96299588*I] ] 

list2 :=[ [0., -8.634351786*10^(-7)-67.81404036*I], [.1, -0.7387644021e-5-67.76491234*I], [.2, -0.1433025271e-4-67.59922295*I], [.3, -0.2231598645e-4-67.25152449*I], [.4, -0.3280855430e-4-66.56357035*I] ]

where first element is k and the second value is l_i and s_i

the differential equation is

ode_u[i]:= diff(u[i](t),t$2)+I*(list1[i][2]+list2[i][2])*diff(u[i](t),t)-list1[1][2]*list2[2][2]*u[i](t)=0;

eta is in fourier space where k values are in list1[i][1].

We laso know that f(-k)= - f*(k) where f=list[i][2]

and u[i] as function of k, initially has a Gaussian shape at t=0 in fourier space..

Thanks in advance for your help