# Solving Laplace Equation with Boundary Conditions: Is it Wrong?

• Logarythmic
In summary, the problem involves finding solutions for the equation \nabla ^2 u(r,\theta) = 0 with the boundary conditions of u(1,\theta) = u(2,\theta) = \sin^2 \theta. The general solutions can be found in both spherical and plane polar coordinates, with different forms for each. By manipulating the coefficients and using trigonometric identities, the boundary conditions can be met in both forms. For simpler cases, the coefficients can be easily determined through inspection, but for more complex cases, integration may be necessary.
Logarythmic
Does

$$\nabla ^2 u(r,\theta) = 0$$

with the boundary conditions

$$u(1,\theta) = u(2,\theta) = \sin^2 \theta$$

have any solutions?

This was a problem on my exam but someone must have written the conditions wrong, or am I stupid?

Why not? It's a well-defined, well-behaved boundary condition over an enclosed space (the region between two concentric spheres).

The general solution in spherical coordinates is sum (Ai r^i + Bi / r^(i+1))Pi(cos theta). Plug in r=1 and r=2, and rewrite the sin^2 theta as 1 - cos^2 theta... then you only need the first few Legendre polynomials and you can solve for the coefficients.

But the problem is to be done in plane polar coordinates...?

Ah, right, sorry, I'm used to looking at these things in spherical coordinates... oh well, same wine, different bottle.

The general solutions look like A0 ln r + B0 + sum (Ai r^i + Bi / r^i)(Ci cos (i theta) + Di sin (i theta)). Use the double angle formula to get your cos^2 term.

How am I ever going to get $$u(2,\theta)$$, which contains an $$ln 2$$, to equal $$\sin^2 \theta$$? And how can you possibly write $$\sin^2 \theta$$ as a linear combination of sin and cos?

Last edited:
$$\sin^{2}(\theta)=\frac{1}{2}-\frac{\cos(2\theta)}{2}$$

Logarythmic said:
How am I ever going to get $$u(2,\theta)$$, which contains an $$ln 2$$, to equal $$\sin^2 \theta$$?

Basically, you are free to choose any combination of the coefficients you need to match the boundary conditions. There are some integrals you can do to solve this analytically, but in most cases that you are likely to see in homework or on a test, you can just pick out the coefficients by inspection.

Damn it. I'm not worth those 4 points. ;)

## 1. What is the Laplace equation and what does it represent?

The Laplace equation is a partial differential equation that describes the behavior of a scalar field in space. It represents the equilibrium state of a system, where the rate of change of the field is equal to zero.

## 2. Why is it important to solve the Laplace equation with boundary conditions?

Boundary conditions provide additional information about the system, which is necessary to uniquely determine the solution to the Laplace equation. Without boundary conditions, the solution would be a general solution that does not represent the specific system being studied.

## 3. Can the Laplace equation be solved analytically or does it require numerical methods?

The Laplace equation can be solved analytically for simple boundary conditions and geometries. However, for more complex systems, numerical methods such as finite difference, finite element, or boundary element methods are often used.

## 4. What are some common mistakes or errors when solving the Laplace equation with boundary conditions?

Some common mistakes include not specifying all necessary boundary conditions, using incorrect boundary conditions, or making mathematical errors in the solution process. It is important to carefully check the solution for consistency and accuracy.

## 5. How is the solution to the Laplace equation with boundary conditions validated?

The solution can be validated by comparing it to analytical solutions for simple cases, or by comparing it to experimental data for more complex cases. It is also important to check for convergence and accuracy of the numerical method used to solve the equation.

Replies
12
Views
625
Replies
1
Views
903
Replies
11
Views
2K
Replies
7
Views
1K
Replies
1
Views
681
Replies
4
Views
854
Replies
2
Views
532
Replies
1
Views
864