ODE: Fourier Methods for PDEs

Partial Differential Equations (PDEs) govern phenomena from heat diffusion and wave propagation to electrostatics and quantum mechanics. While daunting, many linear PDEs with classic boundary conditions yield to a powerful and elegant technique: the method of separation of variables coupled with Fourier analysis. This method transforms a complex PDE into simpler, solvable Ordinary Differential Equations (ODEs), whose solutions are then synthesized using Fourier series or integrals. Mastering this approach is fundamental for engineers and physicists, providing a systematic toolbox for solving the core equations of mathematical physics.

The Foundation: Separation of Variables

Separation of variables is a technique that seeks solutions to a PDE in the form of a product of functions, each depending on only one of the independent variables. The core idea is to assume a solution structure that allows the PDE to be "separated" into a set of ODEs.

Consider a prototypical problem: the one-dimensional heat equation on a finite interval $0<x<L$ with homogeneous Dirichlet boundary conditions. $$u_t={\alpha }^2{u}_{xx},u(0,t)=0,u(L,t)=0,u(x,0)=f(x)$$ We assume a product solution of the form $u(x,t)=X(x)T(t)$. Substituting into the PDE gives: $$X(x)T^{\prime }(t)={\alpha }^2{X}^{\prime \prime }(x)T(t)$$ Dividing both sides by ${\alpha }^{2}X(x)T(t)$ separates the variables: $$\frac{T^{\prime }(t)}{{\alpha }^{2}T(t)}=\frac{X^{\prime \prime }(x)}{X(x)}$$ The left side is a function of $t$ only, and the right side is a function of $x$ only. For this equality to hold for all $x$ and $t$, both sides must be equal to a constant, which we denote $-\lambda$. This yields two ODEs:

1. Temporal ODE: $T^{\prime }(t)+{\alpha }^2\lambda T(t)=0$
2. Spatial ODE (Boundary Value Problem): $X^{\prime \prime }(x)+\lambda X(x)=0$, with $X(0)=0$ and $X(L)=0$.

The power of the method is now clear: the original PDE with its boundary conditions has been reduced to solving an ODE initial value problem and an ODE boundary value problem (BVP).

Sturm-Liouville Theory and Eigenfunction Expansions

The spatial BVP, $X^{\prime \prime }+\lambda X=0$ with $X(0)=X(L)=0$, is not just any ODE problem; it is a Sturm-Liouville problem. In general, a regular Sturm-Liouville problem has the form: $$\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+[q(x)+\lambda w(x)]y=0,a<x<b$$ with prescribed homogeneous boundary conditions (e.g., Dirichlet, Neumann, or Robin). For our simple case, $p(x)=1$, $q(x)=0$, and $w(x)=1$.

These problems have a profound theory with critical results:

• They possess an infinite set of eigenvalues ${\lambda }_n$ and corresponding non-zero eigenfunctions $X_n(x)$.
• The eigenvalues are real, countable, and can be ordered ${\lambda }_1<{\lambda }_2<{\lambda }_3<...$.
• The eigenfunctions are orthogonal with respect to the weight function $w(x)$. In our case, orthogonality means ${\int }_0^L{X}_n(x)X_m(x)dx=0$ for $n\ne m$.

Solving our simple BVP, we find the eigenvalues are ${\lambda }_n=(n\pi /L{)}^2$ and the corresponding eigenfunctions are $X_n(x)=\sin (n\pi x/L)$ for $n=1,2,3,...$. These sine functions form a complete orthogonal set on the interval $[0,L]$.

The concept of eigenfunction expansion states that any sufficiently nice function $f(x)$ defined on $[0,L]$ satisfying the boundary conditions can be expanded as an infinite series (a Fourier sine series) in these eigenfunctions: $$f(x)=\sum _{n=1}^{\infty }{b}_n\sin \left(\frac{n\pi x}{L}\right),\text{where }{b}_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \left(\frac{n\pi x}{L}\right)dx.$$ This expansion is the key to satisfying the initial condition of our PDE.

Constructing Fourier Series Solutions

We now synthesize the product solutions to solve the complete initial-boundary value problem. For each eigenvalue ${\lambda }_n$, the temporal ODE $T^{\prime }+{\alpha }^2{\lambda }_{n}T=0$ has the solution $T_n(t)=C_n{e}^{-{\alpha }^2{\lambda }_{n}t}=C_n{e}^{-{\alpha }^2(n\pi /L{)}^{2}t}$.

Thus, for each $n$, we have a product solution satisfying the PDE and boundary conditions: $$u_n(x,t)=X_n(x)T_n(t)=b_n\sin \left(\frac{n\pi x}{L}\right)e^{-{\alpha }^2(n\pi /L{)}^{2}t}$$ where we have absorbed the constant $C_n$ into the coefficient $b_n$. By the principle of superposition (valid for linear PDEs), the general solution is the sum of all such product solutions: $$u(x,t)=\sum _{n=1}^{\infty }{b}_n\sin \left(\frac{n\pi x}{L}\right)e^{-{\alpha }^2(n\pi /L{)}^{2}t}$$ Finally, we impose the initial condition $u(x,0)=f(x)$: $$u(x,0)=\sum _{n=1}^{\infty }{b}_n\sin \left(\frac{n\pi x}{L}\right)=f(x)$$ This is precisely the eigenfunction expansion of $f(x)$. Therefore, the coefficients $b_n$ are the Fourier sine coefficients of the initial temperature distribution $f(x)$. The series solution is now complete, describing how any initial heat profile evolves over time, decaying exponentially while maintaining zero temperature at the ends.

Applying the Method to Wave and Potential Equations

The same logical framework applies to other fundamental PDEs, changing only the details of the ODEs and the resulting series.

For the Wave Equation $u_{tt}=c^2{u}_{xx}$ with the same boundaries $u(0,t)=u(L,t)=0$, separation $u(x,t)=X(x)T(t)$ leads to: $$X^{\prime \prime }+\lambda X=0,T^{\prime \prime }+c^2\lambda T=0.$$ The spatial BVP is identical, yielding the same eigenvalues and eigenfunctions $\sin (n\pi x/L)$. The temporal ODE is a harmonic oscillator, giving solutions of the form $\cos (c\sqrt{{\lambda }_n}t)$ and $\sin (c\sqrt{{\lambda }_n}t)$. The full solution becomes: $$u(x,t)=\sum _{n=1}^{\infty }\left[A_n\cos \left(\frac{n\pi ct}{L}\right)+B_n\sin \left(\frac{n\pi ct}{L}\right)\right]\sin \left(\frac{n\pi x}{L}\right)$$ The coefficients $A_n$ and $B_n$ are determined by expanding the initial position $u(x,0)$ and initial velocity $u_t(x,0)$ as Fourier sine series.

For Laplace's Equation (the potential equation) in a rectangle, ${\nabla }^{2}u=u_{xx}+u_{yy}=0$, with boundary conditions like $u(0,y)=u(a,y)=0$, $u(x,0)=0$, and $u(x,b)=f(x)$, separation $u(x,y)=X(x)Y(y)$ yields: $$X^{\prime \prime }+\lambda X=0,Y^{\prime \prime }-\lambda Y=0.$$ The homogeneous conditions on $x$ dictate the eigenvalues ${\lambda }_n=(n\pi /a{)}^2$ and eigenfunctions $X_n(x)=\sin (n\pi x/a)$. The ODE for $Y$ becomes $Y^{\prime \prime }-(n\pi /a{)}^{2}Y=0$, with solution $Y_n(y)=C_n\sinh (n\pi y/a)+D_n\cosh (n\pi y/a)$. Applying the homogeneous condition at $y=0$ simplifies this. The final solution is a Fourier sine series in $x$, with coefficients that are hyperbolic functions of $y$, tailored to match the non-homogeneous condition at $y=b$.

Common Pitfalls

1. Misapplying Boundary Conditions: The most critical step is correctly applying the boundary conditions to the separated spatial ODE, not the product solution. For example, if $u(0,t)=0$, you apply $X(0)=0$. Forgetting this leads to an incorrect eigenvalue problem and invalid eigenfunctions.

• Correction: Always substitute the product form into the boundary conditions immediately after separation to derive the conditions for $X(x)$ (or the relevant spatial function).

1. Incorrect Coefficient Formulas: Using the wrong formula for Fourier coefficients is a common algebraic error. The coefficient formula depends on the eigenfunction, the interval, and the weight function from the Sturm-Liouville problem.

• Correction: Derive the coefficient formula using orthogonality: Multiply both sides of $f(x)=\sum a_n{\varphi }_n(x)$ by ${\varphi }_m(x)w(x)$ and integrate over the domain. For the standard sine series on $[0,L]$, $w(x)=1$, and the orthogonality integral gives $b_n=\frac{{\int }_0^{L}f(x)\sin (n\pi x/L)dx}{{\int }_0^L{\sin }^2(n\pi x/L)dx}$.

1. Ignoring the Superposition Principle: Attempting to find a single product solution that satisfies both the PDE and the initial condition is impossible. The initial condition is only satisfied by the infinite sum.

• Correction: Remember the workflow: Find all product solutions (the eigenfunctions $X_n(x)$ times their corresponding $T_n(t)$), sum them to form the general solution, then use the initial condition to determine the coefficients in that sum.

1. Confusion Between Fourier Series and Integrals: This method on a finite domain with homogeneous boundary conditions always leads to a Fourier series (sine, cosine, or mixed). On infinite or semi-infinite domains, the eigenvalue spectrum becomes continuous, leading to Fourier integral (transform) solutions.

• Correction: Identify the domain first. Finite intervals with S-L problems lead to series. Infinite domains require transform techniques.

Summary

• The separation of variables technique reduces linear PDEs with compatible boundary conditions to a set of simpler ODEs: a temporal ODE and a spatial Sturm-Liouville boundary value problem.
• Sturm-Liouville theory guarantees the existence of a complete, orthogonal set of eigenfunctions $X_n(x)$. Any admissible function (like an initial condition) can be represented as an eigenfunction expansion (a Fourier series).
• The general PDE solution is constructed via superposition of product solutions $X_n(x)T_n(t)$. The series coefficients are determined by projecting the initial condition onto the orthogonal eigenfunctions.
• This framework systematically solves the classic PDEs: The heat equation solutions involve exponential decay in time, the wave equation solutions involve oscillatory modes in time, and Laplace's equation solutions involve hyperbolic functions.
• Success depends on meticulous application of boundary conditions to the spatial ODE to derive the correct eigenvalue problem and on using the proper orthogonal coefficient formulas to satisfy the initial condition.