Separation of Variables and Fourier Series

1Introduction

This lecture explains separation of variables as a method that decomposes a PDE into spatial modes compatible with its boundary conditions.

2Terminology and Definitions

Separation of variables assumes a product form such as $u(x,t)=X\left(x\right)T\left(t\right)$u(x,t)=X(x)T(t) and decomposes a PDE into ordinary differential equations. A Fourier series represents a function as a series of mutually orthogonal trigonometric functions. We use sine series adapted to homogeneous Dirichlet conditions on $0<x<L$0<x<L.

3Strategy

The boundary conditions determine a spatial eigenvalue problem. We expand the initial data in its eigenfunctions and solve the corresponding time-dependent ODE for each coefficient.

4Why a Separation Constant Appears

After substituting a nonzero product $u(x,t)=X\left(x\right)T\left(t\right)$u(x,t)=X(x)T(t) and dividing by $XT$XT where it is nonzero, one side depends only on $x$x and the other only on $t$t. Because $x$x and $t$t vary independently, both sides must equal one constant, called the separation constant. On intervals containing zeros, uniqueness for the resulting linear ODEs extends the solutions while preserving the same constant obtained on the nonzero regions.

5Typical Models

For the heat equation on $0<x<L$0<x<L with $u(0,t)=u(L,t)=0$u(0,t)=u(L,t)=0, the spatial modes are $\sin (n\pi x/L)$\sin(n\pi x/L). The initial distribution is expanded in a sine series, and every mode decays with time.

For the wave equation under the same boundary conditions, the spatial modes are identical, but the time factors oscillate as $\cos (cn\pi t/L)$\cos(cn\pi t/L) and $\sin (cn\pi t/L)$\sin(cn\pi t/L) rather than decaying exponentially. The difference between heat and waves therefore appears in the time equation associated with the same spatial eigenvalue.

6Example 1: The Heat Equation

Let $\kappa >0$\kappa>0 and consider

Substitution of $u=XT$u=XT gives

The spatial problem is

For real-valued $X$X, multiplication by $X$X and integration by parts show that every nonzero mode satisfies

Thus neither $\lambda =0$\lambda=0 nor $\lambda <0$\lambda<0 yields a nonzero fixed-end mode. Since every nonzero scalar multiple spans the same eigenspace, representatives of the eigenfunctions and their eigenvalues may be chosen as

with $T_n\left(t\right)=e^{-\kappa {\lambda }_{n}t}$T_n(t)=e^{-\kappa\lambda_n t}. To expand the initial condition, we invoke the completeness theorem stating that the sine system is complete in $L^2(0,L)$L^2(0,L). It follows by extending $f$f oddly to $(-L,L)$(-L,L) and applying $L^2$L^2 completeness of the Fourier system. Thus

This formula uses

where ${\delta }_{nm}$\delta_{nm} is the Kronecker delta. Formally,

If $f\in L^2(0,L)$f\in L^2(0,L), meaning ${\int }_0^L\left|f\right(x){|}^{2}dx<\mathrm{\infty }$\int_0^L|f(x)|^2dx<\infty, its sine series represents $f$f in the $L^2$L^2 sense: the mean-square error tends to zero. For every $t>0$t>0, exponential suppression of high frequencies permits termwise spatial and temporal differentiation and produces a smooth classical solution. Extension to a pointwise classical solution at $t=0$t=0 requires additional regularity and compatibility such as $f\left(0\right)=f\left(L\right)=0$f(0)=f(L)=0.

7Example 2: The Wave Equation

Let $c>0$c>0 and consider

The spatial modes are the same as for the heat equation, while the time equation is

Thus, formally,

where

The denominator of $d_n$d_n follows because the sine coefficient of $u_t(x,0)$u_t(x,0) is $(cn\pi /L)d_n$(cn\pi/L)d_n. Unlike the heat equation, the wave equation has no exponential high-frequency decay. Twice differentiating the series termwise as a classical solution therefore requires decay of the Fourier coefficients sufficient for the differentiated series to converge, together with all endpoint compatibility conditions required by those derivatives. Conditions such as $f\left(0\right)=f\left(L\right)=g\left(0\right)=g\left(L\right)=0$f(0)=f(L)=g(0)=g(L)=0 are necessary low-order compatibility conditions but are not by themselves a complete classical-solvability theorem. For weaker data, one must specify both the convergence and the sense in which the equation is satisfied.

8Eigenvalue Problems and Orthogonality

Boundary conditions determine the domain of the spatial operator and hence its eigenvalue problem. With fixed endpoints, the eigenfunctions of $-d^2/d{x}^2$-d^2/dx^2 are sine functions. Their orthogonality decomposes the initial data into coefficients. Neumann conditions instead produce cosine functions and a constant mode, while periodic conditions naturally produce complex exponentials. Separation of variables therefore constructs an eigenfunction system compatible with the boundary conditions and uses it as a basis when the relevant completeness theorem applies.

9Limitations

Complicated domains or coefficients may prevent a product decomposition $X\left(x\right)T\left(t\right)$X(x)T(t). For nonlinear PDEs, superposition also fails, so a simple Fourier-series expansion may not apply.

10Common Errors

• Choosing a Fourier expansion without checking the boundary conditions.
• Separating the eigenvalue problem from the reason the Fourier series appears.
• Ignoring convergence of the series solution or its behavior at the boundary.

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