Hello,

I am trying to discretize high order derivatives (typicaly of order 3-4) and I would like to use maple to avoid errors. Using this thread of discussion I got the following minimum example that illustrates the problem I am facing:

I would like maple to evaluate the first expression for d1 and give the second expression that I wrote manually.

Is there a way to do this ? (using simplify() does't produce anything)

Cheers,

Download high_order_derivatives.mw

Hello,

I am trying to solve the self similarity problem for external natural convection which is the following coupled problem

•     3/4*F(eta)*Theta(eta)'  = Theta(eta)''
•     1/2 (F(eta)')^2 - 3/4 F(eta)*F(eta)'' = Pr*(Theta(eta) - F(eta)''')

where Pr is a constant and  ()' denotes differentiation with respect to eta. The boundary conditions are:

•     at eta = 0, F = 0, F'=0, Theta = 1
•     at eta -> infinity, F' = 0, Theta = 0

I tried the following but I cannot obtain a solution

Download wip.mw

Hello,

I am trying to recover with maple the analytical solution of the problem of conduction within two composite cylinders given in Carslaw and Jeager's book: "Conduction of heat in solids" (link here). Here is my take on the problem

restart;
with(PDEtools);
C := diff_table(c(rho, t));
E := diff_table(e(rho, t));
PDE1 := diff(C[], t) - K_1*(diff(diff(C[], rho), rho) - diff(C[], rho)/rho) = 0;     
PDE2 := diff(E[], t) - K_2*(diff(diff(E[], rho), rho) - diff(E[], rho)/rho) = 0;       

ic1 := eval(C[], t = 0) = V;
ic2 := eval(E[], t = 0) = 0;

bc_1 := eval(C[], rho = a) = eval(E[], rho = a);
bc_2 := K_1*eval(diff(C[], rho), rho = a) = K_2*eval(diff(E[], rho), rho = a);
bc_3 := eval(E[], rho = infinity) = 0;
bc_4 := (diff(C[]/rho, rho), rho = 0) = 0;
pdsolve([PDE1, PDE2],[ic1,ic2, bc_1,bc_2,bc_3,bc_4]); 

Unfortunately I end up with the following errors:

Error, (in pdsolve/sys) too many arguments; some or all of the following are wrong: [{c(rho, t), e(rho, t)}, [c(rho, 0) = V, e(rho, 0) = 0, c(a, t) = e(a, t), K_1*(diff(c(a, t), a)) = K_2*(diff(e(a, t), a)), e(infinity, t) = 0, ((diff(c(rho, t), rho))/rho-c(rho, t)/rho^2, rho = 0) = 0]]

If I call pdsolve without any initial and boundary conditions, I have the general form of the equation which is correct (product of exponential function in time with Bessel functions for space) but I don't know how to determine the coeficients with maple from there.

Note also that the boundary condition "bc_2" does not seem to be correcty evaluated (it is supposed to specify the continuity of the flux ar r=a). Instead it just evaluate the expression with r swapped with a. If one write the same line of code with r=0 instead of r=a then the evaluation seems correct.

Any help would be very appreciated,

Cheers

Hello,

I am trying to solve the following system of odes where the unknowns are xhat, yhat, uhat and vhat. They depend on the variables (x, y, u, v, a_1, a_2, a_3, a_4, f(x)) where f(x) is an arbitrary function and I want to solve it with respect to a_1. The RHS has f(x) and its derivative with respect to x. The dsolve function fails to solve the ode when I add f(x) at the list of variables. 

restart;
vars := x, y, u, v, a_1, a_2, a_3, a_4, f(x);
sys_ode := diff(xhat(vars), a_1) = a_3*x + a_1, diff(yhat(vars), a_1) = a_2 - a_4*y + f(x), diff(uhat(vars), a_1) = a_3*u + 2*a_4*u, diff(vhat(vars), a_1) = a_4*v + diff(f(x), x)*u;

initvars := x, y, u, v, 0, a_2, a_3, a_4, f(x);
ics := xhat(initvars) = x, yhat(initvars) = y, uhat(initvars) = u, vhat(initvars) = v;
solution_a1 = dsolve([sys_ode, ics], [xhat, yhat, uhat, vhat]);

Can anyone help regarding this issue ?