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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.

 

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?

 

 

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'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.

 

 

 

Hi,

I have a problem with maple solving diffusion equation. I solved the equation as follows

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;
IBC1 := {u(x, 0) = 0, (D[1](u))(0, t) = 0, (D[1](u))(1, t) = c};
S1 := pdsolve(PDE, IBC1, numeric, time = t);
S1:-plot(t = 0.6e-2);

i am plotting cincentration along the thickness of the material. i want to know why the concentration is negative. It should not be negative because constant flux 'c' is present at x=1. and flux at x=0 is 0, and also the initial concentration is 0.

Thanks

 


Is it possible to somehow extract a derivative from numeric solution of partial differential equation?

I know there is a command that does it for dsolve but i couldn't find the same thing for pdsolve.

The actual problem i have is that i have to take a numeric solution, calculate a derivative from it and later use it somewhere else, but the solution that i have is just a set of numbers an array of some sort and i can't really do that because obviously i will get a zero each time.

Perhaps there is a way to interpolate this numeric solution somehow?

I found that someone asked a similar question earlier but i couldn't find an answer for it.

Hi,

     I'm trying to numerically solve a PDE in Maple for different boundary conditions, however I'm having trouble even getting Maple to numerically solve it for simple boundary conditions.

I have cylindrical coordinates, r, z, theta, and I treat r = r(z, theta) for convenience to plot my solution surface. The initial coundary condition is that at z = epsilon (z = 0 is singular) , r = constant and of course r is periodic in theta. This is just a circle, and the analytical solution is know to be a half-sphere  r = sqrt(R^2 - z^2). I entered my initial boundary conditions into Maple, but it doesn't like the periodic one

IBC := { r(epsilon, theta) = R - epsilon__r,
              r(z, 0) = r(z, 2*Pi) };

pdsolve(
  PDE,
  IBC,
  numeric,
  indepvars = [z, theta],
  time = z,
  range = 0..2*Pi);
Error, (in pdsolve/numeric/par_hyp) Incorrect number of boundary conditions, expected 2, got 1

I'm not sure how to make this work, and then generalize it to more arbitrary intial slices r(epsilon, theta) = f(theta).

Here's the attached worksheet, ForMaplePrimesSUbmission.mw

Any help is appreciated,

Thanks

with(PDEtools);
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);
R := 8.3144621;
T := 298;





PDE := diff(u(x, t), t)-(diff(u(x, t)+o^2*Es*cb*u(x, t)^2/(9*R*T), x, x)) = 0;
IBC := {u(1, t) = 1, u(x, 0) = 0, (D[1](u))(1, t) = l*f/(d*cb)};
S := pdsolve(PDE, IBC, numeric, time = t, range = 0 .. 1, timestep = 0.1e-4, spacestep = 0.1e-6);
p1 := S:-plot(t = .1, numpoints = 100);
Error, (in pdsolve/numeric/plot) unable to compute solution for t>HFloat(0.0):
matrix is singular
p2 := S:-plot(t = .2, numpoints = 50, color = green);
Error, (in pdsolve/numeric/plot) unable to compute solution for t>HFloat(0.0):
matrix is singular
p3 := S:-plot(t = .3, numpoints = 50, color = blue);
Error, (in pdsolve/numeric/plot) unable to compute solution for t>HFloat(0.0):
matrix is singular
plots[display]({p1, p2, p3});
Error, (in plots:-display) expecting plot structures but received: {p1, p2, p3}


Hi, 

     I have a question regarding pdsolve, or Solve from the PDEtools package. I have a set of equations relating partial derivatives, and I'd like to isolate certain terms without explicitly known the functions. I can do this for a single equation, but not multiple ones. I'm curious if Maple can currently handle a system of eqns like these easily, since I will be increasing the number of eqns in the future. Here's the code 

 

 

restart;

PDEtools:-declare(H=H(x,y,t)):

H(x, y, t)*`will now be displayed as`*H

(1)

eq1:= H[tt](x,y,t) = H[xx](x,y,t) + H[yy](x,y,t);

H[tt](x, y, t) = H[xx](x, y, t)+H[yy](x, y, t)

(2)

eq2 := diff(H[tt](x,y,t), t) = diff(H[tx](x,y,t), x) + diff(H[ty](x,y,t), y);

diff(H[tt](x, y, t), t) = diff(H[tx](x, y, t), x)+diff(H[ty](x, y, t), y)

(3)

eq3 := diff(H[tx](x,y,t), t) = diff(H[xx](x,y,t), x) + diff(H[xy](x,y,t), y);

diff(H[tx](x, y, t), t) = diff(H[xx](x, y, t), x)+diff(H[xy](x, y, t), y)

(4)

eq4 :=diff(H[ty](x,y,t), t) = diff(H[xy](x,y,t), x) + diff(H[yy](x,y,t), y);

diff(H[ty](x, y, t), t) = diff(H[xy](x, y, t), x)+diff(H[yy](x, y, t), y)

(5)

PDEtools:-Solve(eq3, H[xy]);

H[xy](x, y, t) = Int(diff(H[tx](x, y, t), t)-(diff(H[xx](x, y, t), x)), y)+_F1(x, t)

(6)

PDEtools:-Solve({eq1, eq2, eq3, eq4}, H[xy]);

Error, (in pdsolve/sys) the input system cannot contain equations in the arbitrary parameters alone; found equation depending only on {H[tt](x,y,t), H[xx](x,y,t), H[yy](x,y,t)}: H[tt](x,y,t)-H[xx](x,y,t)-H[yy](x,y,t)

 

 

 

 

Download PDESolveHelp.mw

 

Hello all,

 

I am trying to solve a simple PDE with one unknow theta and three boundary conditions. Unfortunately, I receive error messages. In this case, the integer s seems to cause troubles. The error message is:

Error, (in pdsolve/numeric/process_PDEs) variable(s) {s} are in the PDE system but are not dependent or independent variables

 

When I assign s a value (such as 1 or 2), no error messages, but still no response. I get:

pds:=module() export plot,plot3d,animate,value,settings; ... end module

Would anyone have an idea how to resolve this issue? Thank you for your suggestions. Below is the small portion of code

restart;
with(PDEtools);

PDE := [(1-y^(s+1))*(diff(theta(y, z), z)) = diff(theta(y, z), `$`(y, 2))+y^(s+1)*(s+1)^(1/s+1)];
IBC := {theta(1, z) = 0, theta(y, 0) = 0, (D[1](theta))(0, z) = 0};

pds := pdsolve(PDE, IBC, numeric);
Error, (in pdsolve/numeric/process_PDEs) variable(s) {s} are in the PDE system but are not dependent or independent variables

Hey people,

 

I am trying to get the following code to run, but it keeps returning an error about too many arguments

 

restart:with(plots):with(PDEtools):
alpha_const := 0.5:gamma_const := 2.5: D1 := 0.05: D2 := 0.002:
A := diff_table(a(x,t)):B := diff_table(b(x,t)):
Selkov[1] := A[t] = 1 - A[]*B[]^(gamma_const) + D1*A[x,x]:
Selkov[2] := B[t] = alpha_const * ( A[]*B[]^(gamma_const) - B[]) + D2 * B[x,x]:

bc[1] := D[1](a)(0,t)=0: bc[2] := D[1](b)(0,t)=0: bc[3] := D[1](a)(4*Pi,t) = 0: bc[4]:=D[1](b)(4*Pi,t)=0:

ic[1] := eval(A[],t=0)=a_0:ic[2] := eval(B[],t=0)=b_0:
case1 := eval(ic,[a_0=1,b_0=1]):
case2 := eval(ic, [a_0=piecewise((x<2*Pi+1) and (x>2*Pi-1), 0.99, 1), b_0=piecewise((x<2*Pi+1) and (x>2*Pi-1), 0.99, 1)]):

Case1Default := pdsolve({Selkov[1],Selkov[2]},{bc[1],bc[2],bc[3],bc[4],case1[1],case1[2]},numerical);


Error, (in pdsolve/sys) too many arguments; some or all of the following are wrong: [{a(x, t), b(x, t)}, {a(x, 0) = 1, b(x, 0) = 1, (D[1](a))(0, t) = 0, (D[1](a))(4*Pi, t) = 0, (D[1](b))(0, t) = 0, (D[1](b))(4*Pi, t) = 0}, numerical]

 

My code worked just earlier today, and now it wont. If i try to run pdsolve({Selkov[1],Selkov[2]}) it says that there is an error with general case of floats. 

 

You help is greatly appreciated!

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