1. Motivation of Fourier Series

1. Motivation of Fourier Series

The one-dimensional heat equation is a second order partial differential equation

(1.1) ∂u

∂t(x, t) = ∂²u

∂x²(x, t).

Assume the solution u=u(x, t) obeys the initial condition u(x,0) =f(x) and the boundary condition

u(0, t) =u(1, t) = 0.

Suppose that u(x, t) = X(x)T(t) satisfies (1.1). Then X(x)T⁰(t) = X⁰⁰(x)T(t). Dividing this equation byX(x)T(t),we obtain

T⁰(t)

T(t) = X⁰⁰(x) X(x) .

The left hand side of the above equation is a function oft while the right hand side of the equation is a function of x. This can only happen when they are constant functions, i.e.

there is a constant A so that

T⁰(t)

T(t) = X⁰⁰(x)

X(x) =−A.

Now, we obtain a system of ordinary differential equations:

X⁰⁰(x) +AX(x) = 0, T⁰(t) +AT(t) = 0.

We obtain T(t) = e^−AtT0 for some constant T0. The boundary condition implies X(0) = X(1) = 0.

Case 1. A is a positive number, i.e. A = k² for some real number k. In this case, the characteristic polynomial of the second order ordinary differential equation is

λ²+k² = 0.

Hence the characteristic polynomial has two conjugate complex roots λ = ±k. Then the general solution to the differential equation is given by

X(x) =C₁coskx+C₂sinkx.

Case 2. A= 0.Then X(x) =C1x+C2.

Case 3. A <0.Assume A=−k² for some real number k.The characteristic polynomial is given by λ²−k².Hence it has two distinct real roots ±k. The general solution to this equation is given by

X(x) =C1e^kx+C2e^−kx.

Using the boundary conditionsX(0) =X(1) = 0,we can check that in case 3,C₁=C₂ = 0.

In case 2,C1 =C2= 0.In case 1,X(x) =C2sinkx and thus X(1) =C₂sink= 0.

We obtain that k=nπ forn≥1.Hence if we denote X_n(x) = sinnπx,thenX_n(x) obeys X_n⁰⁰(x) +n²π²Xn(x) = 0.

1

2

In this caseTn(t) =e^−n2π2t obeys

T_n⁰(t) +n²π²Tn(t) = 0.

Hence if we denote u_n(x, t) = e^−n2π2tsinnπx, then u_n(x, t) is a solution to (1.1) obeying u_n(0, t) = u_n(1, t) = 0. Now, we also need to find a solution u(x, t) satisfies the initial condition u(x,0) =f(x).Let {a_n} be a sequence of real numbers so that

u(x, t) =

∞

X

n=1

a_ne^−n2t2sinnπx

obeys the initial conditionu(x,0) =f(x).(In fact, assuming the convergence of the infinite series, we knowu(0, t) =u(1, t) = 0.) Then we obtain

f(x) =

∞

X

n=1

a_nsinnπx.

Now, we want to express {a_n} in terms off(x).(The initial temperaturef(x) is given but a_n are unknown at this moment.) This motivates the study of Fourier series.