Items tagged with partial_differential_equations

Feed

will give me

which is indeed a solution of the PDE1

will give me

which is not a solution of the PDE2

However, both differential equations are equal, only the arguments are swapped around. Am I doing something wrong, or is this a bug?

Thanks

PDE := diff(u(x, t), t$2) = (1/16)*(diff(u(x, t), x$2))-(1/5)*(diff(u(x, t), t)); IBCs := u(x, 0) = x*(1-(1/2)*x), (D[2](u))(x, 0) = x*(1-(1/2)*x), u(0, t) = 0, (D[1](u))(1, t) = 0; Sol := pdsolve({IBCs, PDE}, HINT = f(x)*g(t)); Sol := subs(op([2, 2, 1], Sol) = n, Sol)

I thought that the addition of HINT wolud help but no.

The numeric solution gives the right answer but the analytical gives way too small numbers and wrong shape.

 

Hi all,

 

I have a partial differential equation similar to the following:

Equation: f_x(x,y) + f_y(x,y) = f(x,y) + f(x,0),
Boundary value conditions: f(x,10) = f(10,y) = 0.

The solution is that f is identically equal to 0.

 

However, I am having trouble solving this equation in Maple. I type the following:

pde := diff(f(x, y), x)+diff(f(x, y), y) = f(x, y)+f(x, 0);

bv1 := f(x, 10) = 0;

bv2 := f(10, y) = 0;

solution := pdsolve(pde, {bv1, bv2}, numeric, time = x, range = 0 .. 10);

 

When Maple tries to evaluate the last expression, I get the error

Error, (in pdsolve/numeric/process_PDEs) PDEs can only contain dependent variables with direct dependence on the independent variables of the problem, got {f(x, 0)}

 

It seems to have difficulties with the expression "f(x,0)". Is there some trick to typing this in a way that makes Maple interpret it correctly?

 

Edit: I encounter the same problem, when I try to solve the ODE f'(x) = f(x) + f(0), where f(10) = 0.

 

Best regards.

I am trying to solve the wave equation in polar coordinates.  The initial condition on u is given by f(r,theta) and the initial condition on u_t is zero.  The weight function is w(r).  I am not sure why it will not evaluate this as I know the solution remains finite on the domain (the unit disk).  Here is the code: 
 

Wave Equation in Polar Coordinates

restart; with(plots); addcoords(u_cylindrical, [u, r, theta], [r*cos(theta), r*sin(theta), u])

Example:

rho := 1; 1; c := 1; 1; w := proc (r) options operator, arrow; r end proc

1

 

1

 

proc (r) options operator, arrow; r end proc

(1)

f := proc (r, theta) options operator, arrow; 2.5*(1-r^2)*r*sin(theta) end proc

proc (r, theta) options operator, arrow; 2.5*(1-r^2)*r*sin(theta) end proc

(2)

assume('n', integer); 1; assume('m', integer)

lambda := proc (n, m) options operator, arrow; BesselJZeros(n, m)^2/rho^2 end proc;

proc (n, m) options operator, arrow; BesselJZeros(n, m)^2/rho^2 end proc

(3)

c0 := proc (m) options operator, arrow; (int(int(f(r, theta)*BesselJ(0, sqrt(lambda(0, m))*r)*w(r), theta = -Pi .. Pi), r = 0 .. rho))/(int(int(BesselJ(0, sqrt(lambda(0, m))*r)^2*w(r), theta = -Pi .. Pi), r = 0 .. rho)) end proc; 1; a := proc (n, m) options operator, arrow; (int(int(f(r, theta)*BesselJ(n, sqrt(lambda(n, m))*r)*cos(n*theta)*w(r), theta = -Pi .. Pi), r = 0 .. rho))/(int(int(BesselJ(n, sqrt(lambda(n, m))*r)^2*cos(n*theta)^2*w(r), theta = -Pi .. Pi), r = 0 .. rho)) end proc; 1; b := proc (n, m) options operator, arrow; (int(int(f(r, theta)*BesselJ(n, sqrt(lambda(n, m))*r)*sin(n*theta)*w(r), theta = -Pi .. Pi), r = 0 .. rho))/(int(int(BesselJ(n, sqrt(lambda(n, m))*r)^2*sin(n*theta)^2*w(r), theta = -Pi .. Pi), r = 0 .. rho)) end proc

proc (m) options operator, arrow; (int(int(f(r, theta)*BesselJ(0, sqrt(lambda(0, m))*r)*w(r), theta = -Pi .. Pi), r = 0 .. rho))/(int(int(BesselJ(0, sqrt(lambda(0, m))*r)^2*w(r), theta = -Pi .. Pi), r = 0 .. rho)) end proc

 

proc (n, m) options operator, arrow; (int(int(f(r, theta)*BesselJ(n, sqrt(lambda(n, m))*r)*cos(n*theta)*w(r), theta = -Pi .. Pi), r = 0 .. rho))/(int(int(BesselJ(n, sqrt(lambda(n, m))*r)^2*cos(n*theta)^2*w(r), theta = -Pi .. Pi), r = 0 .. rho)) end proc

 

proc (n, m) options operator, arrow; (int(int(f(r, theta)*BesselJ(n, sqrt(lambda(n, m))*r)*sin(n*theta)*w(r), theta = -Pi .. Pi), r = 0 .. rho))/(int(int(BesselJ(n, sqrt(lambda(n, m))*r)^2*sin(n*theta)^2*w(r), theta = -Pi .. Pi), r = 0 .. rho)) end proc

(4)

u := proc (n, m, r, theta, t) options operator, arrow; sum(BesselJ(0, sqrt(lambda(0, j))*r)*c0(j)*cos(sqrt(lambda(0, j))*c*t), j = 1 .. m)+sum(sum(BesselJ(i, sqrt(lambda(i, j))*r)*(a(i, j)*cos(i*theta)+b(i, j)*sin(i*theta))*cos(sqrt(lambda(i, j))*c*t), j = 1 .. m), i = 1 .. n) end proc

proc (n, m, r, theta, t) options operator, arrow; sum(BesselJ(0, sqrt(lambda(0, j))*r)*c0(j)*cos(sqrt(lambda(0, j))*c*t), j = 1 .. m)+sum(sum(BesselJ(i, sqrt(lambda(i, j))*r)*(a(i, j)*cos(i*theta)+b(i, j)*sin(i*theta))*cos(sqrt(lambda(i, j))*c*t), j = 1 .. m), i = 1 .. n) end proc

(5)

soln := evalf(u(3, 3, r, theta, t));

(Float(infinity)+Float(infinity)*I)*BesselJ(1., 3.831705970*r)*sin(theta)*cos(3.831705970*t)+(Float(infinity)+Float(infinity)*I)*BesselJ(1., 7.015586670*r)*sin(theta)*cos(7.015586670*t)+(Float(infinity)+Float(infinity)*I)*BesselJ(1., 10.17346814*r)*sin(theta)*cos(10.17346814*t)-0.3676566232e-9*BesselJ(2., 5.135622302*r)*sin(2.*theta)*cos(5.135622302*t)-0.1879633956e-10*BesselJ(2., 8.417244140*r)*sin(2.*theta)*cos(8.417244140*t)-0.5146823927e-10*BesselJ(2., 11.61984117*r)*sin(2.*theta)*cos(11.61984117*t)+(Float(infinity)+Float(infinity)*I)*BesselJ(3., 6.380161896*r)*sin(3.*theta)*cos(6.380161896*t)+(Float(infinity)+Float(infinity)*I)*BesselJ(3., 9.761023130*r)*sin(3.*theta)*cos(9.761023130*t)+(Float(infinity)+Float(infinity)*I)*BesselJ(3., 13.01520072*r)*sin(3.*theta)*cos(13.01520072*t)

(6)

plot3d(soln, r = 0 .. 1, theta = 0 .. 2*Pi, coords = u_cylindrical, axes = boxed)

NULL

NULL


 

Download Section_6.3.mw

Any assistance would be greatly appreciated. 

f2 := (diff(y(a, b), a)-(-(1/2)*x-1/2+(1/2)*sqrt(-3*x^2-2*x-3))/x^2)*(diff(y(a, b), b)-(-(1/2)*x-1/2-(1/2)*sqrt(-3*x^2-2*x-3))/x^2);
f := collect(expand(f2), [diff(y(a,b),a),diff(y(a,b),b),diff(y(a,b),a)*diff(y(a,b),b)]);

Hello everyone!

I'm trying to understand how to work with PDEs in Maple by solving a basic PDE such as the wave equation, along with conditions at t = 0 in terms of arbitrary functions. I stumbled upon the very same example, included in the Maple Online help at this page under the part "Three Textbook Examples":

I tried to copy and paste the code and execute it, but to no avail: it seems like pdsolve doesn't give me any other result than a blank space, unlike in the example reported in the Online help. Can anyone tell me if (and where) I am doing wrong? Is it possible that my version of Maple (Maple 13) may not be compatible with the procedures required by this example?

 A warm thank you to whoever will help me!

Page 1 of 1