- #1
NapoleonZ
- 6
- 0
Urr+(1/r)*Ur+(1/r^2)*Uθθ=0
a<r<b, 0<θ<w
with the conditions
U(r,0)=U1
U(r,w)=U2
U(a,θ)=0
U(b,θ)=f(θ)
a<r<b, 0<θ<w
with the conditions
U(r,0)=U1
U(r,w)=U2
U(a,θ)=0
U(b,θ)=f(θ)
NapoleonZ said:Urr+(1/r)*Ur+(1/r2)*Uθθ=0
a<r<b, 0<θ<w
with the conditions
U(r,0)=U1
U(r,w)=U2
U(a,θ)=0
U(b,θ)=f(θ)
NapoleonZ said:I have got two sets of solutions
U(r,θ)=A*ln(r)+B*
U(r,θ)=(C*r^λ+D/r^λ)*(E*sinλθ+F*cosλθ)
My problem is the boundary conditions are nonhomogeneous, with which I cannot work out the coefficients.
tiny-tim said:Hi NapoleonZ!
i] I'm a little confused about the conditions … U(a,θ)=0 seems incompatible with U(r,0)=U1
and U(r,w)=U2, if the conditions are continuous
ii] Doesn't the condition U(a,θ)=0 make it fairly clear what E and F are (unless λ = 0)?
The Laplace equation in polar coordinate is a mathematical equation that describes the relationship between the second derivatives of a function in polar coordinates. It is used to solve problems in physics and engineering, such as heat conduction and electrostatics.
The Laplace equation in polar coordinate has various applications in physics and engineering, including heat conduction, electrostatics, and fluid dynamics. It is also used in image and signal processing, as well as in finding the equilibrium state of a system.
To solve the Laplace equation in polar coordinate, you first need to express the equation in terms of polar coordinates. Then, you can use separation of variables, the method of images, or other techniques to find the solution. Boundary conditions must also be considered to find a unique solution.
The boundary conditions for the Laplace equation in polar coordinate depend on the specific problem being solved. In general, these conditions specify the values or derivatives of the function at the boundaries of the region of interest. They are essential in finding a unique solution to the equation.
The Laplace equation in polar coordinate is used in physics to describe the behavior of physical systems in a variety of fields, such as heat conduction, electrostatics, and fluid dynamics. It is also used to model the behavior of electromagnetic fields and to analyze the equilibrium state of a system.