Question

Define the formal solution of $$ \begin{array}{c} u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}=0, \quad \rho_{0}

Short answer

Question: Find the formal solution of the partial differential equation (PDE) in polar coordinates with boundary conditions given by: $$u_{rr}+\frac{1}{r}{u}_r+\frac{1}{r^2}{u}_{\theta \theta }=0,u({\rho }_0,\theta )=f(\theta ),u(\rho ,\theta )=0,-\pi \le \theta <\pi$$ Answer: The formal solution of the given PDE with the provided boundary conditions is given by: $$u(r,\theta )=\sum _{n=1}^{\mathrm{\infty }}[C_n{J}_n(\sqrt{{\lambda }_n}r)][A_n\cos (n\theta )+B_n\sin (n\theta )]$$ where $J_n$ are Bessel functions of the first kind, $n$ is an integer, and the coefficients $A_n$, $B_n$, and $C_n$ are determined by expanding $f(\theta )$ as a Fourier series and matching the coefficients.

Step-by-step solution

Separate the variables

Begin by assuming that the function $u(r,\theta )$ can be separated into a product of two functions, $R(r)$ and $T(\theta )$: $$u(r,\theta )=R(r)T(\theta )$$ Substitute this into the given PDE: $$R^{″}(r)T(\theta )+\frac{1}{r}{R}^{\prime }(r)T(\theta )+\frac{1}{r^2}R(r)T^{″}(\theta )=0$$

Rearrange the equation

Divide both sides of the equation by $R(r)T(\theta )$, and rearrange: $$\frac{R^{″}(r)}{R(r)}+\frac{1}{r}\frac{R^{\prime }(r)}{R(r)}=-\frac{T^{″}(\theta )}{T(\theta )r^2}$$ Since the left side of the equation depends only on $r$, and the right side depends only on $\theta$, each side must equal a constant, which we'll call $\lambda$: $$\frac{R^{″}(r)}{R(r)}+\frac{1}{r}\frac{R^{\prime }(r)}{R(r)}=\lambda =-\frac{T^{″}(\theta )}{T(\theta )r^2}$$

Solve the $T(\theta )$ ODE

Now we have two separated ODEs. Let's start with the $\theta$-dependent equation: $$T^{″}(\theta )+\lambda r^{2}T(\theta )=0$$ The general solution of this ODE will be: $$T(\theta )=A\cos (\sqrt{\lambda }r\theta )+B\sin (\sqrt{\lambda }r\theta )$$

Solve the $R(r)$ ODE

Now let's move onto the $r$-dependent equation: $$R^{″}(r)+\frac{1}{r}{R}^{\prime }(r)-\lambda R(r)=0$$ This equation is a Bessel's equation, and its general solution is: $$R(r)=C{J}_n(\sqrt{\lambda }r)+D{Y}_n(\sqrt{\lambda }r)$$ Here, $J_n$ and $Y_n$ are Bessel functions of the first and second kind, respectively, and $n$ is an integer.

Apply the boundary conditions

We can now apply the boundary conditions to the general solution of the separated ODEs: $$u(r,\theta )=R(r)T(\theta )=[C{J}_n(\sqrt{\lambda }r)+D{Y}_n(\sqrt{\lambda }r)][A\cos (\sqrt{\lambda }r\theta )+B\sin (\sqrt{\lambda }r\theta )]$$ Apply the boundary condition $u(\rho ,\theta )=0$: $$u(\rho ,\theta )=[C{J}_n(\sqrt{\lambda }\rho )+D{Y}_n(\sqrt{\lambda }\rho )][A\cos (\sqrt{\lambda }\rho \theta )+B\sin (\sqrt{\lambda }\rho \theta )]=0$$ To satisfy this condition, we require $J_n(\sqrt{\lambda }\rho )=0$. As $J_n$ has an infinite number of zeros, we will have an infinite series of separated solutions. Finally, apply the boundary condition $u({\rho }_0,\theta )=f(\theta )$: $$u({\rho }_0,\theta )=\sum _{n=1}^{\mathrm{\infty }}[C_n{J}_n(\sqrt{{\lambda }_n}{\rho }_0)][A_n\cos (n\theta )+B_n\sin (n\theta )]=f(\theta )$$

Determine coefficients

To determine the coefficients $A_n$, $B_n$, and $C_n$, we need to expand $f(\theta )$ as a Fourier series and match the coefficients. This will ultimately provide us with the formal solution of the given PDE with the given boundary conditions. The final solution will be in the form: $$u(r,\theta )=\sum _{n=1}^{\mathrm{\infty }}[C_n{J}_n(\sqrt{{\lambda }_n}r)][A_n\cos (n\theta )+B_n\sin (n\theta )]$$

Key concepts

Understanding Partial Differential Equations

Partial differential equations (PDEs) are fundamental to modeling various physical phenomena, from heat conduction and fluid flow to sound and elasticity. A PDE involves derivatives with respect to multiple variables, and solving a PDE means finding a function that satisfies the equation across a specified domain.

PDEs can be classified based on order, linearity, and homogeneity. The order is the highest derivative present, while linearity pertains to whether the equation is a linear combination of the function and its derivatives. Homogeneity involves whether the PDE equals zero (homogeneous) or a non-zero function (non-homogeneous).

In the exercise provided, you came across a homogeneous, linear second-order PDE with derivatives with respect to two variables, radius $r$ and angle $\theta$. The solution approach employed 'separation of variables', a technique where one assumes that the solution can be expressed as a product of functions, each depending on a single variable. This simplification facilitates solving PDEs by breaking them down into ordinary differential equations (ODEs).

Grasping the Essence of Bessel Functions

Bessel functions, denoted by $J_n$ and $Y_n$, are canonical solutions to Bessel's differential equation. This equation arises naturally in problems with cylindrical or spherical symmetry, such as in heat conduction, wave propagation, and in our exercise, the vibrational modes of a circular membrane.

Bessel functions of the first kind $J_n$ are used when dealing with finite solutions at the origin, which is crucial for physical problems where the solution must be bounded. On the other hand, Bessel functions of the second kind $Y_n$ are often ruled out in physical applications due to their singularity at the origin. These functions form a complete set, allowing us to express solutions to PDEs in terms of an infinite series involving $J_n$.

In our problem, a Bessel function solution emerges when applying the separation of variables to the radial part of the PDE. The boundary conditions lead us to seek zeros of these functions since they must satisfy certain conditions at the boundaries of the defined domain, ${\rho }_0<r<\rho$.

Decomposing Functions with Fourier Series

Fourier series decompose periodic functions into an infinite sum of sine and cosine functions, each multiplied by a coefficient. This powerful tool enables us to express complex, periodic functions in terms of simple, harmonic components. These series play a pivotal role in solving PDEs by assisting in boundary condition applications and within the process of separation of variables.

In our solution, after separating the variables, the angular part of the solution is represented by a Fourier series. When applying the given boundary conditions, the Fourier series helps in determining the coefficients involved in the solution. By matching coefficients from the series with those of the boundary condition $f(\theta )$, we ensure that the solution to the PDE adheres to the specified constraints at the domain edges.

The method of Fourier series helps to encapsulate the initial condition $f(\theta )$ into an analytical form that aligns with the separated solution, resulting in a complete and comprehensive description of the wave profile over the domain, neatly incorporating the boundary data.