@mmcdara

d1 is a centered divided difference approximation of diff(u, x$3) at "point" [i+1/2, j];not an approximation of the laplacian of u at [i, j]

yes you are right, I am using a 2D finite volume code and I need to evaluate on the faces of given cell  grad(laplacian(u)).n*length_of_face where n is the outward normal. On the right face of a cell located at i+1/2,j I need to evaluate diff(u,x,x,x) and diff(u,y,y,x) my comment was about the first part only.

Could you please comment on the syntax you used for the definition of tau ? I do not understand the double right arrow and the *~ operation.

Thank you again

@tomleslie 

thank you very much !

Does anyone have an  idea on how to assign symbolically a series of values, say alpha_n, to be the (infinite number of) roots of a Bessel function ? 

Cheers

I am trying to solve the heat conduction problem in an axisymmetric cylindrical bar with Laplace transforms, for which an analytical solution exists. I end up finding the right solution in Laplace space but I can't compute its inverse as I need to use the inversion theorem and compute residuals at poles. It implies that I have to assign a series of values, say alpha_n to be the roots of the Bessel function (of the first kind or order 0) J_0.  Is there a way to do that ?  Here is my maple page that shows the problem 

Download chaleur_1D_cylindrical_laplace.mw

@dharr 

Yes I also think that this is the right way to go.

Thank you, how would you specify second order derivative with respect to the first variable with this notation ? 

cheers,

Can

@Thomas Richard 

simplified_PDEL1 := eval(PDEL1,ic1);

Thank you, it was indeed a syntax error. Removing the brackets also worked.

I am using Laplace transforms to end up with an ode to solve instead of the original pde.

Cheers,

@Rouben Rostamian  

Applying your suggestion to the above leads to

restart;
with(Physics);
vars := f(x);
sys_ode := diff(xhat(vars), f(x)) = a_1 + a_3*x, diff(yhat(vars), f(x)) = a_2 - a_4*y + f(x), diff(uhat(vars), f(x)) = a_3*u + (2*u)*a_4, diff(vhat(vars), f(x)) = a_4*v + u*diff(f(x), x);
                   
initvars := 0;
ics := xhat(initvars) = x, yhat(initvars) = y, uhat(initvars) = u, vhat(initvars) = v;
                      
dsolve({ics, eval(sys_ode)}, [xhat(f(x)), yhat(f(x)), uhat(f(x)), vhat(f(x))]);

Error, (in dsolve) found the following equations not depending on the unknowns of the input system: 
{uhat(0) = u, vhat(0) = v, xhat(0) = x, yhat(0) = y, 
(D(uhat))(f(x)) = a_3*u+2*a_4*u, 
(D(vhat))(f(x)) = a_4*v+u*(diff(f(x), x)), 
(D(xhat))(f(x)) = a_3*x+a_1, 
(D(yhat))(f(x)) = a_2-a_4*y+f(x)}

If I dont charge the module physics then I cannot go futher than

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

as I get

Error, invalid input: diff received f(x), which is not valid for its 2nd argument

Any idea on how to do ?

Cheers,

Can

@Rouben Rostamian  

Thank you for your help.

Is there a reason for looking at xhat as a function of all those variables?  Why not take xhat to be a function of a_1 only.  The solution will involve the remaining variables as parameters.

Yes, along the example given before, I was also trying to solve the following with maple:

with(Physics);
vars := x, y, u, v, a_1, a_2, a_3, a_4, f(x);
sys_ode := diff(xhat(vars), f(x)) = a_1 + a_3*x, diff(yhat(vars), f(x)) = a_2 - a_4*y + f(x), diff(uhat(vars), f(x)) = a_3*u + (2*u)*a_4, diff(vhat(vars), f(x)) = a_4*v + u*diff(f(x), x);
          vars := x, y, u, v, a_1, a_2, a_3, a_4, f(x)
initvars := x, y, u, v, a_1, a_2, a_3, a_4, 0;
ics := xhat(initvars) = x, yhat(initvars) = y, uhat(initvars) = u, vhat(initvars) = v;
dsolve({ics, eval(sys_ode)}, [xhat, yhat, uhat, vhat]);

it's an ode with respect to f(x) treaten as a derivative variable but f(x) and its diff(f,x) (its  derivative with respect to x) appears also on the RHS.

The derivative with respect to f(x) is writen as D_9 (as it differentiates with respect to the 9th argument of the variable-list) and does not match the argument f(x) that dsolve seems to be looking for. Hence (I guess) the error:

Error, (in dsolve) found the following equations not depending on the unknowns of the input system:
 {uhat(x, y, u, v, a_1, a_2, a_3, a_4, 0) = u, 
vhat(x, y, u, v, a_1, a_2, a_3, a_4, 0) = v, 
xhat(x, y, u, v, a_1, a_2, a_3, a_4, 0) = x, 
yhat(x, y, u, v, a_1, a_2, a_3, a_4, 0) = y, 
(D[9](uhat))(x, y, u, v, a_1, a_2, a_3, a_4, f(x)) = a_3*u+2*a_4*u, 
(D[9](vhat))(x, y, u, v, a_1, a_2, a_3, a_4, f(x)) = a_4*v+(diff(f(x), x))*u, 
(D[9](xhat))(x, y, u, v, a_1, a_2, a_3, a_4, f(x)) = a_3*x+a_1, 
(D[9](yhat))(x, y, u, v, a_1, a_2, a_3, a_4, f(x)) = a_2-a_4*y+f(x)}