Solving partial differential equation of second in two variables

how to solve this partial differential equation? here u is a function of r and theta, I know the boundary conditions but how to apply them and solve this equation to get u as a function of r and theta solution.

[Torsten]

Just to clarify: all gamma's in your equation are r's ?

So it reads:

d^2u/dr^2 + 1/r * du/dr + 1/r^2 * d^2u/dtheta^2 = 0 ?

[Apurva Suman]

Yes they are r, the equation you have written is right ,that only i need to solve.

[Apurva Suman]

Kindly given me the solution.

[Apurva Suman]

The boundary conditions are as follows : At r = 0 , u = 0 And at theta = 0 , du/d(theta) = 0

[Torsten]

Is this PDE an artifial one or does it have a physical background ? For the second case: Can you give a reference ?

[Apurva Suman]

Equation comes from axial equation of the momentum conservation for the liquid flow , and I've removed some terms from that Another boundary layer condition is also there that may be helpful : At r = R , du/dr = 0 ; R is a constant

[Apurva Suman]

[Torsten]

And at theta = 2*pi periodic boundary condition ?

[Apurva Suman]

At theta = pi/4 , du/dtheta = 0

[Torsten]

Then I think the only solution you can get with a numerical solver is

u(r,theta) == 0

for all r and theta.

[Apurva Suman]

Can't i solve to get u as a function of r and theta , and yes how to solve with numerical solver?

[Torsten]

Why spend the effort to use a numerical solver of the result is obvious with the boundary conditions you gave ?

And no: there is no ready-to-use MATLAB solver for 2-dimensional partial differential equations.

You will have to discretize the PDE and use "linsolve" to solve for the values of u in the grid points.

Maybe you could search in the file exchange for "general elliptic partial differential equation on rectangle" or see whether MATLAB's PDE toolbox supports the form of your equation.

Answer 1

[jessupj]

Context for this question is insufficient. Using matlab, are you attempting a numerical or symbolic answer?

[Apurva Suman]

Symbolic. I need u as a function of r and theta as a equation.