How to conver a patial differetial equation to ordinary differential equation with or without dchange?
 

Download pde_to_ode.mw

I have a PDE

eq1 := du/dx+dv/dy = 0; eq2 := du/dt+u*du/dx+v*du/dy-nu*d^2*u/dy^2 = 0

where u(x, y, t), v(x, y, t) and

eta(x, y, t):=y/((nu*t*cos(alpha)+(nu*x)/(U[w])*sin(alpha))^(1/(2)));

psi(x, y, t):=U[w]*(nu*t*cos(alpha)+(nu*x)/(U[w])*sin(alpha))^(1/(2))*f(eta(x,y,t));

u = diff(psi, y); v= -diff(psi, x).

How to substitiute u = diff(psi, y); v= -diff(psi, x) in eq1 and eq2, Also find the value of nu.

How to solve ordinary differemtial equation system with initial conditions and boundary conditions. Here, some initial conditions are unknown variables. So how to find these  values of parameters.

eq1 := diff(f(x), x, x, x)+(1/2)*cos(alpha)*x*(diff(f(x), x, x))+(1/2)*sin(alpha)*f(x)*(diff(f(x), x, x)) = 0;

eq2 := diff(g(x), x, x)+diff(g(x), x)+(diff(g(x), x))*(diff(h(x), x))+cos(alpha)*x*(diff(g(x), x))+sin(alpha)*f(x)*g(x) = 0;

eq3 := diff(g(x), x, x)+diff(h(x), x, x)+1/2*(cos(alpha)*x+sin(alpha)*f(x)) = 0

ics:=f(0)=0, f'(0)=1, f''(0)=a[1], g(0)=1, g'(0)=a[2], h(0)=1, h'(0)=a[3];

bcs:=f(x) , g(x), h(x) tends to 0 ad x tends to infinity

I have some ODE system contains unknown variables with  boundary conditions  (see the attachment). How to solve these system and find the vales of unknown variables.

In this problem, the boundary conditions tends to 0, when x tends to infinity.  BVP_with_parameters.mw
 

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