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Hi, everyone!

I need help.

There are a system of 2 pde's: 

diff(Y(x, t), x$2) = exp(-2*x*b)*(A(x, t)-Y(x, t)), diff(A(x, t), t) = exp(-2*x*b)*(Y(x, t)-A(x, t)) 

and initial and boundary conditions: 

A(x, 0) = 0, Y(0, t) = 0.1, (D[1](Y))(0, t) = 0. 

For each b = 0, 0.05, 0.1. 
1)to plot 3-d  Y(x,t): 0<=x<=20,0<=t<=7. 
2)to plot  Y(x,4). 

Are there any methods with no finite-difference mesh?

I realized the  methods such as  pds1 := pdsolve(sys, ibc, numeric, time = t, range = 0 .. 7)  can't help me:

Error, (in pdsolve/numeric/match_PDEs_BCs) cannot handle systems with multiple PDE describing the time dependence of the same dependent variable, or having no time dependence 

I found something, that can solve my system analytically: 
pds := pdsolve(sys), where sys - my system without initial and boundary conditions. At the end of the output: huge monster, consisted of symbols and numbers :) And I couldn't affiliate init-bound conditions to it.

I use Maple 13. 

Is it possible to solve (numerically or symbolically) the system of PDEs
sys:={diff(Y(x, t), x$2) = exp(-2*x*b)*(A(x, t)-Y(x, t)), diff(A(x, t), t) = exp(-2*x*b)*(Y(x, t)-A(x, t)) }
under the conditions
ibc:={A(x, 0) = 0, Y(0, t) = 0.1, D[1](Y)(0, t) = 0},
 where the parameter b takes the values 0,0.05,0.1, in Maple? The ranges are t=0..7, x=0..20.

i am solving 4 ODE with boundary condition..

> restart;
> with*plots;


then i got this error..

Error, (in dsolve/numeric/bvp/convertsys) unable to convert to an explicit first-order system


i dont know where i need to change.. could you help me..




How can I use separation of variable to solve heat equation ut=uxx on a rod of length pi/4 with u(0,t)=0, u(pi/4)=0 and u(x,0)=x(pi/4-x) 

then solve heat equation by laplace transform

Dear all,

I need your help.

I compute the exact solution u(x,y,t) os PDE.
My code :

I would like to plot the exact solution of my PDE.

My method work well but  when I put a:=1; b:=1; In the next lines there is no change, always I have a and b in my equation.

If i fix the time, "t=1 or 2 " for example; In the last lines i plot( u(x,y,2)); but doesn't work also.

Then animation in "t" how....

Thanks for your help.




Does anyone has any maple worksheet that generate surface using the PDE method described in this article?  I am trying to learn this method but I am not familiar with the mathematics to do it although the paper gives some description of it.  I hope someone can demonstrate the procedure in Maple.  Thanks

Dear All, I need your help to plot the numerical solution. many thanks.

The variable t in [0,T], x in [0,1], b in [0,2].

Difference finie for waves equation is :

pde:=diff(u(x, y,t), t$2) = c^2*(diff(u(x, y,t),x$2)+diff(u(x,y,t),y$2));

i: according to x, j according to y, and k according to t.

u[i,j,k+1]=2*u[i,j,k]-u[i,j,k-1]+(c*dt/dx)^2*(u[i-1,j,k]-2*u[i,j,k]+u[i+1,j,k])+ (c*dt/dy)^2*(u[i,j-1,k]-2*u[i,j,k]+u[i,j+1,k])


Boundary condition: u(t=0)=1, diff(u(x,y,t),t=0)=0, and the normal derivative on the boundary of Omega =0.

How can solve this problem and plot the numerical solution.




Pls, I tried solving the system of PDE numerically.... When I did for just 1 plot, the graph was plotted but When I varied some parameters its not coming out.... Also, its not bringing any error so I can't trace out my PLS HELP ME OUT with the multiple plots... Attached are my source codes links....


dchange pde maple...

March 13 2014 sarra 100

Dear all,

I need your help

I have a problem with dchange ....  use the dchange command to transform pde to the
                                 x[1], x[2].


> restart;
> with(PDEtools); with(LinearAlgebra);
> pde := diff(u(t), t, t)+2*GAMMA*(diff(u(t), t))+omega^2*u(t) = 0;
           / d  / d      \\           / d      \        2         
           |--- |--- u(t)|| + 2 GAMMA |--- u(t)| + omega  u(t) = 0
           \ dt \ dt     //           \ dt     /                  
> deq1 := diff(u(t), t) = v(t);
                               --- u(t) = v(t)
> deq2 := subs(deq1, pde);
                 / d      \                       2         
                 |--- v(t)| + 2 GAMMA v(t) + omega  u(t) = 0
                 \ dt     /                                 
> dsolve({deq1, deq2}, {u(t), v(t)});
> eqns := [rhs(deq1) = lhs(deq1), rhs(deq2) = lhs(deq2)];
       [        d            / d      \                       2     ]
       [v(t) = --- u(t), 0 = |--- v(t)| + 2 GAMMA v(t) + omega  u(t)]
       [        dt           \ dt     /                             ]
> y := [u, v]; b := diff(y(t), t);
                                   [u, v]
                            [ d         d      ]
                            [--- u(t), --- v(t)]
                            [ dt        dt     ]
> A, b := GenerateMatrix(eqns, y(t));
          Matrix(%id = 122038892), Vector[column](%id = 135944696)
 # Return a vector of eigenvalue of A and matrix  whose columns are eigenvectors of A
> gnat := Eigenvectors(A);
> lambda := gnat[1]; Lambda := gnat[2];
                       Vector[column](%id = 135975976)
                           Matrix(%id = 136787860)
> Y := Vector([y]);
                       Vector[column](%id = 123771808)
> tr := solve(GenerateEquations(Lambda, [x[1], x[2]], Y), {u, v});
     /           /                                    (1/2)             
     |      1    |                   /     2        2\                  
    < u = ------ \-x[1] GAMMA - x[1] \GAMMA  - omega /      - x[2] GAMMA
     |         2                                                        
     \    omega                                                         

                               (1/2)\                 \
              /     2        2\     |                 |
       + x[2] \GAMMA  - omega /     /, v = x[1] + x[2] >
> dchange(tr, pde, [x[1], x[2]]);

hi,i want to take differential with respect to another differential using physics package,but using D instead of diff,could anyone help me do that ? for example :

restart; with(Physics):
A1 := -(1/24)*1*rho*((diff(phi[1](x, t), t))^2)*(h^3)-(1/2)*1*rho*((diff(u[ref](x, t), t))^2)*h-(1/2)*rho*((diff(w(x, t), t))^2)*h+(1/24)*1*1*((diff(phi[1](x, t), x))^2)*(h^3)+(1/2)*1*(1*((diff(u[ref](x, t), x)+(1/2)*(diff(w(x, t), x))^2)^2)+K*1*((diff(w(x, t), x)+phi[1](x, t))^2))*h-1*q*w(x, t):


here i want to compute A2 using D command,not diff and i do not want use convert command ! i just need to calculate A2 directly using D command. tnx for your help.


i want to solve this pde,but i do know how to write my boundary or initial conditons,actually my dependent variables were x and y, i changed x to t,since i faced an error that pde solver needs time=varname , but i could not solve my problem,do anyone know how to insert my boundary conditions ?! please help,thnx

I am trying to get a solution to the heat equation with multiple boundary conditions.

Most of them work but I am having trouble with two things: a Robin boundary condition and initial conditions.

First, here are my equations that work:

returns a solution (actually two including u(x,y,z,t)=0).


However, when I try to add:



I no longer get a solution.


Any guidance would be appreciated.




I have uploaded a worksheet with the equations...


Please, I solved a pde system of equation problem numerically, using maple 17.

But I dont know how to plot multiple solutions on one graph.

I want to vary one of the parameters....

e.g Pr=0.71, Pr=7, Pr=10 where other parameters are kept constant


My working is


M := 1:

pde1 := diff(u(y, t), t)+Typesetting:-delayDotProduct(S, diff(u(y, t), y))-2*k^2*u(y, t) = diff(u(y, t), y, y)+theta(y, t)+Typesetting:-delayDotProduct(N, C(y, t))+Typesetting:-delayDotProduct(M, u(y, t))+u(y, t)/K:

                pde2 := theta(y, t)+t*(diff(theta(y, t), t))+S*(diff(theta(y, t), y)) = (diff(theta(y, t), y, y))/Pr-Typesetting:-delayDotProduct(alpha, theta(y, t)):

pde3 := C(y, t)+t*(diff(C(y, t), t))+S*(diff(C(y, t), y)) = (diff(C(y, t), y, y))/Sh-Typesetting:-delayDotProduct(R, C(y, t)):

PDE := {pde1, pde2, pde3}:

IBC := {C(0, t) = 1, C(1, t) = 0, C(y, 0) = 0, u(0, t) = 0, u(1, t) = 0, u(y, 0) = 0, theta(0, t) = 1, theta(1, t) = 0, theta(y, 0) = 0}:

pds := pdsolve(PDE, IBC, numeric)

module () local INFO; export plot, plot3d, animate, value, settings; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; end module


pds:-plot[display](u(y, t), t = .5, linestyle = "solid", colour = "blue", legend = "Pr=0.71", title = "Velocity Profile", labels = ["y", "theta"])





Please, Any help will be gracefully appreciated


Hi everyone,

I have been  trying to solve a coupled system of 2 differencial equations, 1 PDEs with 1 ODE.

The code is below.

> restart;

> with(linalg):with(plots):
> PDE:=[diff(x(z,t),t)=(a/Pe)*diff(x(z,t),z$2)-a*diff(x(z,t),z)-(1-epsilon)/epsilon*3*Bi*(x(z,t)-1)/(1-Bi*(1-1/ksi(z,t)))]:
> a=2:Pe=3:Bi=5:epsilon=0.85:

> ODE := [diff(ksi(z,t),t) = (b*Bi*x(z,t)-1)/ksi(z,t)^2/(1-Bi*(1-1/ksi(z,t)))]:
> IC1:=c(z,0)=0:
> IC2:=ksi(z,0)=1:
> bc2:=diff(x(z,t),z):
> bc1:=x(z,t)-1/pe*diff(x(z,t),z):
> N:=10:
> L:=1:
> dyduf:=1/2*(-x[2](t)-3*x[0](t)+4*x[1](t))/h:
> dydub:=1/2*(-x[N-1](t)+3*x[N+1](t)+4*x[N](t))/h:
> dydu:=1/2/h*(x[m+1](t)-x[m-1](t)):
> d2ydu2:=1/h^2*(x[m-1](t)-2*x[m](t)+x[m+1](t)):
> bc1:=subs(diff(x(z,t),z)=dyduf,x(z,t)=x[0](t),z=1,bc1):
> bc2:=subs(x(z,t)-1/pe*diff(x(z,t),z)=dydub,x(z,t)=x[N+1](t),t=0,bc2):
> eq[0]:=bc1:
> eq[N+1]:=bc2:
> eq[m]:=subs(diff(x(z,t),z$2)=d2ydu2,diff(x(z,t),z)=dydu,diff(x(z,t),t)=dydu,x(z,t)=x[m](t),z=m*h,PDE):
> for i from 1 to N do eq[i]:=subs (m=i,eq[m]);od:
> x[0](t):=(solve(eq[0],x[0](t)));

> x[N+1](t):=solve(eq[N+1],x[N+1](t));

> for i from 1 to N do eq[i]:=eval(eq[i]);od:
> eqs:=[seq((eq[i]),i=1..N)]:
> Y:=[seq(x[i](t),i=1..N)];

> A:=genmatrix(eqs,Y,'B1'):
Error, (in linalg:-genmatrix) equations are not linear

> evalm(B1);

[B1[1], B1[2], B1[3], B1[4], B1[5], B1[6], B1[7], B1[8], B1[9],


> B:=matrix(N,1):for i to N do B[i,1]:=B1[i]:od:evalm(B);

> h:=eval(L/(N+1));


> A:=map(eval,A);


> if N > 10 then A:=map(evalf,A);end:
> evalm(A);


> mat:=exponential(A,t);
Error, (in linalg:-matfunc) input must be a matrix

> mat:=map(evalf,mat):
> Y0:=matrix(N,1):for i from 1 to N do
> Y0[i,1]:=evalf(subs(x=i*h,rhs(IC)));od:evalm(Y0);
Error, invalid input: rhs received IC, which is not valid for its 1st argument, expr


> s1:=evalm(Y0+inverse(A)&*b):
Error, (in linalg:-inverse) expecting a matrix

> Y:=evalm(mat&*s1-inverse(A)&*b):
Error, (in linalg:-inverse) expecting a matrix

> Y:=map(simplify,Y):
> Digits:=5;

Digits := 5

> for i from 1 to N do x[i](t):=evalf((Y[i,1]));od:
Error, invalid subscript selector

> for i from 0 to N+1 do x[i](t):=eval(x[i](t));od;


> setcolors(["Red", "Blue", "LimeGreen", "Goldenrod", "maroon",
> "DarkTurquoise", "coral", "aquamarine", "magenta", "khaki", "sienna",
> "orange", "yellow", "gray"]);

["Red", "LimeGreen", "Goldenrod", "Blue", "MediumOrchid",


> pp:=plot([seq(x[i](t),i=0..N+1)],t=0..10,thickness=4);pt:=textplot([[0.28,0.15,typeset("Follow the arrow: ",x[0],"(t), ..., ",
> x[N+1],"(t).")]]):
Warning, unable to evaluate the functions to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct


VIEW(0. .. 10., DEFAULT))

> display([pp,pt,arw],title="Figure 5.13",axes=boxed,labels=[t,"x"]);
Error, (in plots:-display) expecting plot structures but received: [arw]


In advance, thanks for the time of reading it!



I have been struggling (reading Ore/Weyl Algebra documentation) to understand how to input a PDE system with polynomial coeff. in Weyl algebra notation so I can compute a Groebner basis for it. I would be very grateful if someone could  show, using the simple example below, which differential operators in Ore_algebra[diff_algebra] should one declare to express the system in Weyl algebra notation. The systems I'm working are more complicated but all have many dependent variables, f and g functions in this example:

pdesys:= [ x*diff( f(x,y,z),x)- z*diff( g(x,y,z),y) = 0, (x^2-y)*diff( f(x,y,z),z)- y*diff( g(x,y,z),z) = 0 ]

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