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I have the following PDE:

 

u_xx = u_tt + (2^{1/2}u_x-u)^{1/2}

 

Do you have a proposed algorithm to solve in maple for this PDE? I mean pdsolve won't solve it because it's a nonlinear PDE.

 

I want to solve numerically the PDE:

u_xx + u_yy= = u^{1/2}+(u_x)^2/(u)^{3/2}

 

My assumptions are that  |sqrt(2)u_x/u|<<1 (but I cannot neglect the first term since its in my first order approximation of another PDE.

 

So I tried solving by using pdsolve in maple, but to no cigar.

 

Here's the maple file:

 nonlinear.mw

PDE := diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = u^(1/2)+(diff(u(x, y), x))^2/u^(3/2); IBC := {D[1](u)*(1, t) = 0, D[2](u)*(x, 1) = 0, u(0, t) = 1, u(x, 0) = 1}; pds := pdsolve(PDE, IBC, type = numeric); pds:-plot3d(t = 0 .. 1, x = 0 .. 1, axes = boxed, orientation = [-120, 40], color = [0, 0, u])

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = u^(1/2)+(diff(u(x, y), x))^2/u^(3/2)

 

{D[1](u)*(1, t) = 0, D[2](u)*(x, 1) = 0, u(0, t) = 1, u(x, 0) = 1}

 

Error, (in pdsolve/numeric/process_PDEs) all dependent variables in PDE must have dependencies explicitly declared, got {u}

 

Error, `pds` does not evaluate to a module

 

``

 

Download nonlinear.mw

Hello All,

I looked through the Maple help on PDE systems and pdsolve and the physics problems that appear there. THere are a number of single-PDE cases with initial / boundary conditions; but I couldn't find PDE systems with ics/bcs.

 

Would you have a (simple) example of a PDE system with its initial / boundary conditions? I am attempting to build understanding of the syntax and different options of "pdsolve". Examples seem to be a great way to learn how to solve PDE systems. One can then pdsolve the PDE system without ics; then add them, try different options etc.

 

Thank you!

 

Hello All,

I have the PDE system shown below. It is a simple system for 2 unknown functions f1(x,t) and f2(x,t). Also, say we have x=x(t)=e^t for example. How does one solve such PDE system with Maple? I tried including the condition x=e^t in the PDE system itself, but got "System inconsistent" error message. x=x(t) can be looked at as an additional constraint and I am baffled how do I feed it into the PDE solver. 

 

Perhaps someone has experience with such systems?

 

 

Hi,

I have a small problem. I want to findout area under a curve. I got the plot from solving a partial differential equation. I want to find out area under the curve with out using interpolation. Are there any methods to find this.

 

here i enclose the method i have done.

Es := 0.117108e12:
Ef := 0.78125e11:
l := 0.150e-6:
s := 0.500000e-3:
f := 0.5898334197e-6:
o := 0.9e-5:
d := 0.10e-17:
cb := 0.1e7/(19.9):
c := l*f/(d*cb):

PDE := diff(u(x, t), t)-(diff(u(x, t), x, x)) = 0:
            
with(plots):
with(plottools):
ys := -0.4245333333e-1:
IBC1 := {u(x, 0) = 0, (D[1](u))(0, t) = 0, (D[1](u))(1, t) = c}:
S1 := pdsolve(PDE, IBC1, numeric, time = t, timestep = 0.1e-2);

p2 := S1:-plot(t = .2525);

p3 := getdata(p2);

p3[3]:
co:=CurveFitting[PolynomialInterpolation](p3[3], x):
Area := int(co, x = x[1] .. x[2]):

So this is the procedure i used to find out, but can there be any other procedure to findout area directly from hte solution of PDE.

Thanks.



 

I want to solve numerically the nonlinear pde:

 

u_x+u_t - (u_{xt})^2 = u(x,t)

 

which method do you propose me to use with maple? (I don't mine about which boundary conditions to be used here).

 

Dear all,

I am trying to solve the following partial differential equation (transport or advection equation) with given initial and boundary conditions:

restart: with(PDEtools):
sys := [v*diff(u(x,t), x) + diff(u(x,t), t) = 0, u(x,0) = exp(-x), u(0,t) = sin(t)];
pdsolve(sys);

But it does not work. The solution is (or should be): 

u(x, t) = exp(t*v-x)+Heaviside(t-x/v)*(sin(t-x/v)-exp(t*v-x))

I think the reason is that the interval for t (in [0, inf)) and x (in [0, 1]) is not specified. On the other hand, this works:

restart: with(PDEtools):
sys := [diff(u(x, t), t) = diff(u(x, t), x, x), u(0, t) = 0, u(1, t) = 0, u(x,0) = f(x)];
sol := pdsolve(sys);

How can I solve a PDE like the transport equation with given initial AND boundary conditions?

Thanks a lot

I have the following paper:

http://projecteuclid.org/download/pdf_1/euclid.dmj/1077490637

 

Now I wanted to check Fritz John's claim in the proof of Theorem 1.1, he says that equation (7) can be easily verified for case i=1,k=2.

 

Now at first I tried to calculate by hand, but it's just a lengthy calculation, so now I turned to maple to check its validity, I get that this claim is false, am I wrong in my code? if yes, then how to change it?

 

P.S

I changed between xi and x and eta and y.

In the following is the code:

 

v := (((x_1-y_1)/(x_3-y_3))^2+((x_2-y_2)/(x_3-y_3))^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3)); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1); 1; evalb(diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2) = diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1))

((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))

 

u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(3/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^4)+(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)-u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(-2*x_2+2*y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)+((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(-(D[1, 1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)-(-(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))/(x_3-y_3)-(-(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)+(-(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))/(x_3-y_3))/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(-2*x_2+2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_1-2*y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*(2*x_1-2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+2*((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_1-2*y_1)*(-2*x_2+2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^3

 

u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(3/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^4)+((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), 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(-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*y_3/(x_3-y_3))/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3))*(-2*x_1+2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_2-2*y_2)*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(2*x_2-2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+2*((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_2-2*y_2)*(-2*x_1+2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^3

 

false

(1)

``

``

 

Download ultrhyperbolic_pde.mw

v := (((x_1-y_1)/(x_3-y_3))^2+((x_2-y_2)/(x_3-y_3))^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3)); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1); 1; evalb(diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2) = diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1))

((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))

 

u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(3/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^4)+(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)-u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(-2*x_2+2*y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)+((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(-(D[1, 1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)-(-(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))/(x_3-y_3)-(-(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)+(-(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 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(-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))/(x_3-y_3)-((D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*y_3/(x_3-y_3))/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3))*(-2*x_1+2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_2-2*y_2)*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(2*x_2-2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+2*((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_2-2*y_2)*(-2*x_1+2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^3

 

false

(1)

``

``

 

Download ultrhyperbolic_pde.mw

I'm trying to build a Maple procedure that will generate vector fields on a metric with certain properties. Working with metric g over the coordinates {u,v,w}, call the field X = (a(u,v,w), b(u,v,w), c(u,v,w)). The field should satisfy <X, X> = 0 and have the directional covariant derivative of X in the direction of each coordinate vector field = 0 (with resepct to the Levi-Civita conenction).

Basically, these conditions yield a system of 3 PDEs and an algebraic expressionin terms of a,b,c. I've been trying to solve them using pdsolve, but I'm getting the error message:

>Error, (in pdsolve/sys) the input system cannot contain equations in the arbitrary parameters alone; found equation depending only on _F1(u,v,w): _F1(u,v,w)

I've attached my worksheet. Can anyone help me out?

 

Thanks! ppwaves.mw

I have the following PDE system to solve numerically and I am not sure how to use maple to solve it.

 

v_t = v_{xx} for 0<x<1 , t>0

v(x,0)=1

v_x(1,t)=-hv^4(1,t) (where h is some numerical number);

v_x(0,t)=0

To solve this pde numerically I need to use the following condition on v(1,t):

v(1,t) = 1-h*\int_{0}^t \theta_3(\tau)v(1,t-\tau)^4d\tau

this is the numerical boundary condition, where \theta_3 is Jacobi theta3 function.

 

I don't see how can I use maple for this numerical pde problem.

 

Here's my attempt at solution:

[code]

 PDE := diff(v(x, t), t) = diff(v(x, t), x, x);

JACOBIINTEGRAL := int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t);

IBC := {`&PartialD;`(v(0, t))/`&PartialD;`(x) = 0, `&PartialD;`(v(1, t))/`&PartialD;`(x) = -0.65e-4*v(1, t)^4, v(x, 0) = 1};

pds := pdsolve(PDE, IBC, numeric, time = t, range = 0 .. 1, spacestep = 0.1e-2, timestep = 0.1e-2, numericalbcs = {v(1, t) = 1-0.65e-4*JACOBIINTEGRAL}, method = ForwardTimeCenteredSpace)

[/code]

But I get the next error message:

Error, (in pdsolve/numeric/process_IBCs) improper op or subscript selector

 

How to fix this or suggest me a better way to solve this pde numerically?

 

Hello,

I need to solve the next ode:

diff(u(x, y), x) = -(2/3)*(3*h^3*nu+9*h^2*nu*y-12*nu*y^3+36*x^2*y+56*y^3)/h^3

diff(v(x, y), y) = (2/3)*(36*nu*x^2*y+56*nu*y^3+3*h^3+9*h^2*y-12*y^3)/h^3

diff(u(x, y), y)+diff(v(x, y), x) = -(6*(1+nu))*x*(h^2-4*y^2)/h^3

The inicial conditions are:

u(L, 0) = 0
v(L, 0) = 0
D[1]*u(L, 0) = 0

When I write on Maple this code, he give me a error:


with(PDEtools, casesplit, declare); declare((u, v)(x, y))


sys2 := [diff(u(x, y), x) = -(2/3)*(3*h^3*nu+9*h^2*nu*y-12*nu*y^3+36*x^2*y+56*y^3)/h^3, diff(v(x, y), y) = (2/3)*(36*nu*x^2*y+56*nu*y^3+3*h^3+9*h^2*y-12*y^3)/h^3, diff(u(x, y), y)+diff(v(x, y), x) = -(6*(1+nu))*x*(h^2-4*y^2)/h^3]

sol := pdsolve(sys2)

ics := u(L, 0) = 0, v(L, 0) = 0, D[1]*u(L, 0) = 0

pdsolve([sys2, ics]);

Why Maple can't solve this PDE?

I think that the problem is on sys2. But I don't know how to explain to Maple the function: diff(u(x, y), y)+diff(v(x, y), x) = -(6*(1+nu))*x*(h^2-4*y^2)/h^3; on the system of equations. I think the problem is there.

I'm so sorry by my bad english.

I need to solve this, anyone help me please.

Thanks.

 

 

 

Please I need some assistance on how to solve the below pde:

diff(c(x,t),t)=diff(c(x,t),x,x)-diff(c(x,t),x)-c(x,t)

I have my initial condition and 2 boundary conditions. I just need some clues on how to go about solving the problem. Thank you!

Greetings,

I need some help/advice because I think I am going crazy..!

I am trying to solve numerically a system of two PDE's. My initial condition is a gaussian and I apply cyclic boundaries.

With Maple 18 and pdsolve/numeric I am waiting a considerable amount of time and the result is a bell shaped travelling wave that increases its height over time. The clush between two such waves leaves them almost unaffected (the big pass through the small one).

 

With Mathematica 10 and NDSolve the waiting time is almost zero and the result (after few timesteps) is a shock wave travelling with steady height. Moreover, the clush between two of them is fully plastic (the big "eat" the small one)

 

I double-triple check the equations and the boundary conditions and I assure you that are the same..Any ideas what is hapening? Is there any way to check that the Maple results are correct?

The model of fixed-bed adsorption column

Fluid phase:

PDE:= diff(U(x, tau),tau)+ psi*Theta*diff(U(x, tau),x)-(1/Pe)*psi*Theta*diff(U(x, tau),$(x, 2))=-3*psi*xi*(U(x, tau)-Q/K);

 

IBC:={U(x, 0) = 0,U(0, tau) = 1+(1/Pe)*(D[1](U))(0, tau),(D[1](U))(1, tau)=0};

Particle:

PDE:= diff(Q(r, tau), tau) = diff(Q(r, tau), $(r, 2))+(2/r)*diff(Q(r, tau),r);

IBC:={Q(r, 0) = 0,(D[1](Q))(0, tau) = 0,(1/K)*(D[1](Q))(1, tau)=xi*(U-Q(1, tau)/K)};

Pe:=0.01:

psi:=6780:

Theta:=3.0:

xi:=10000:

I will really appreciate your help. Thanks in anticipation.

now the equation is

d2u/dt2-(2*d2u/x2)+d2u/dxdt=0    

initial condition: u(x,0)=1-(xsign(x)), abslute x<1,0 otherwise. Assume sign(x)=-1 for x<0, 1for x>0 

 Ut(x,0)=cos(pix), bslute x<1, 0 otherwise , he didnt give any B.Cs

so I would like to know the analytical and numerical sols, and plots for the wave at t=2,4

for Numerical:   delta x=0.1, delta t=0.025, range 0..4

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