How to solve this partial differential equation which is a Laplace equation

[dimension10]

I was trying to solve this partial differential equation which arose because I wanted to find a general solution to the Laplace equation in the case f=f(x,y).

$$\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}=0$$

Thanks in advance.

 

[romsofia]

The general solution would be a harmonic function, if there was a general solution for an unbounded Laplace equation. The reason why I said "would be" is because without bounds, the Laplace equation has an infinite number of solutions.

You really have to give the boundary conditions when dealing with Laplace equation, and in general for PDEs, boundary and initial conditions.

 

[jackmell]

The general solution is:

$$f(x,y)=g(z)+h(\overline{z}),\quad z=x+iy$$

where g and h are arbitrary $C^2$ functions of a single variable z=x+iy. We can show this by factoring the equation:

$$\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right) f=\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)=\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)u=0$$

where:

$$u=\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}$$

So first solve:

$$u_t-iu_x=0$$

which is an easy first-order equation which turns out to be u=g(x+it) then substitute that into:

$$f_x+if_y=g(x+it)$$

another easy one to obtain so that the general solution is:

$$f(x,y)=g(x+it)+h(x-it)$$

Note that means the general solution is not analytic since it contains $\overline{z}$ but analyticity is not a requirement for the solution but only differentiability.

 

[dimension10]

The general solution is:

$$f(x,y)=g(z)+h(\overline{z}),\quad z=x+iy$$

where g and h are arbitrary $C^2$ functions of a single variable z=x+iy. We can show this by factoring the equation:

$$\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right) f=\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)=\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)u=0$$

where:

$$u=\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}$$

So first solve:

$$u_t-iu_x=0$$

which is an easy first-order equation which turns out to be u=g(x+it) then substitute that into:

$$f_x+if_y=g(x+it)$$

another easy one to obtain so that the general solution is:

$$f(x,y)=g(x+it)+h(x-it)$$

Note that means the general solution is not analytic since it contains $\overline{z}$ but analyticity is not a requirement for the solution but only differentiability.

Thanks.

 

[blue_raver22]

To solve this partial differential equation, you can use various methods such as separation of variables, Fourier series, or Green's function. However, since the equation is a Laplace equation, it is a special case of the more general Poisson equation, which has well-established methods for solving it.

One approach is to use the method of separation of variables, where you assume that the solution can be written as a product of two functions, one depending only on x and the other depending only on y. You can then substitute this into the equation and rearrange to get two ordinary differential equations, which can be solved separately. The solutions can then be combined to form the general solution.

Another approach is to use Fourier series, where you can express the solution as a sum of sine and cosine functions with unknown coefficients. These coefficients can be determined by applying the boundary conditions of the problem.

Finally, you can also use Green's function, which is a powerful tool for solving linear differential equations. It involves finding a function that satisfies the equation and certain boundary conditions, and then using it to construct the solution to the original problem.

In summary, there are various methods available for solving the Laplace equation, and the choice of method will depend on the specific problem and your personal preference. I would recommend consulting a textbook or seeking guidance from a mathematical expert for further assistance.