Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 211–225. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

A wavelet Galerkin method applied to partial differential equations with variable coefficients ^{∗}

José Roberto Linhares de Mattos & Ernesto Prado Lopes

Abstract We consider the problem K(x)u_{xx} = u_{t} , 0 < x < 1, t ≥ 0, where K(x) is bounded below by a positive constant. The solution on the boundary x = 0 is a known function g and u_{x}(0, t) = 0. This is an ill-posed problem in the sense that a small disturbance on the boundary specification g, can produce a big alteration on its solution, if it exists. We consider the existence of a solution u(x, ·) ∈ L^{2}(R) and we use a wavelet Galerkin method with the Meyer multi-resolution analysis, to filter away the highfrequencies and to obtain well-posed approximating problems in the scaling spaces V_{j}. We also derive an estimate for the difference between the exact solution of the problem and the orthogonal projection, onto V_{j}, of the solution of the approximating problem defined in V_{j−1}.

1 Introduction

In this paper, we consider the following problem, for 0 < α ≤ K(x) < +∞,

\begin{matrix}K(x)u_{xx}(x, t) = u_{t}(x, t), t ≥ 0, 0 < x < 1 \\ u(0, ·) = g, u_{x}(0, ·) = 0 \end{matrix}  (1.1)

We assume that this problem has a solution u(x, ·) ∈ L^{2}(R), for K continuous, and we extend u(x, t) and g to R assuming that both vanish for t < 0.

Problem (1.1) is ill-posed in the sense that a small disturbance on the boundary specification g, can produce a big alteration on its solution, if it exists. This means that if the solution exists, it does not depend continuously on g (see note 1 below).

We consider the Meyer multi-resolution analysis. The advantage of using this method is that it has good localization in the frequency domain, since its Fourier transform has compact support. The orthogonal projection onto Meyer scaling spaces, can be considered as a low pass filter, cutting off the high frequencies. We get a version of the Gronwall inequality that we use to obtain an estimate for the frequency of the solution of the problem (1.1).

∗Mathematics Subject Classifications: 65T60.

Key words: Wavelet, multi-resolution analysis.

\begin{matrix}c \\ ◯ \end{matrix}2003 Southwest Texas State University.

Published February 28, 2003.

¿From the variational formulation of the approximating problem on the scaling space V_{j}, we get an infinite-dimensional system of second order ordinary differential equations with variable coefficients. An estimate obtained for the solution of this evolution problem, is used to get the stability of the wavelet Galerkin method. Using an estimate obtained for the difference between the exact solution of the problem (1.1) and its orthogonal projection onto V_{j}, we get an estimate for the difference between the exact solution of the problem (1.1) and the orthogonal projection, onto V_{j}, of the solution of the approximating problem defined on the scaling space V_{j−1}.

Our approach is similar to the one used in [4] for the sideway heat equation. The problem considered in [4] is an inverse problem for the heat equation with constant coefficient. There the variational formulation, on the scaling space V_{j}, of the approximating problem, produces an infinite-dimensional system of second order ordinary differential equations with constant coefficients, for which the solution is known. Stability and convergence of the method follows from form of this solution.

In section 2, we construct the Meyer multi-resolution analysis. In section 3, we get the estimates of the numerical stability and the convergence of the wavelet Galerkin method.

For a function h ∈ L^{1}(R) ⋂ L^{2}(R) its Fourier Transform is given by ĥ(ξ) := ∫_{ℝ} h(x)e^{−ixξ}dx.

2 Meyer multi-resolution analysis

To construct a wavelet basis from a mother wavelet, we need an structure in L^{2}(ℝ) which allows us to decompose L^{2}(ℝ) in a direct sum of mutually orthogonal spaces.

Definition A multi-resolution analysis is a sequence of closed subspaces V_{j} in L^{2}(ℝ), called scaling spaces, satisfying:

(M1) V_{j} ⊆ V_{j−1} for all j ∈ ℤ

(M2) ⋃_{j∈ℤ} V_{j} is dense in L^{2}(ℝ)

(M3) ⋂_{j∈ℤ} V_{j} = {0}

(M4) f ∈ V_{j} if and only if f(2^{j}·) ∈ V_{0}

(M5) f ∈ V_{0} if and only if f(· − k) ∈ V_{0} for all k ∈ ℤ

(M6) There exists ϕ ∈ V_{0} such that {ϕ_{0,k} : k ∈ ℤ} is an orthonormal basis in V_{0}, where ϕ_{j,k}(x) = 2^{−j/2}ϕ(2^{−j}x − k) for all j, k ∈ ℤ. The function ϕ is called the scaling function of the multi-resolution analysis.

Remarks 1) M4 and M6 imply {ϕ_{j,k} : k ∈ ℤ} being an orthonormal basis for the space V_{j}.

2) Let ϕ ∈ L^{2}(ℝ) and V_{j} = \overline{span\{ϕ_{jk}\}}_{k∈ℤ} where ϕ_{jk}(t) := 2^{−j/2}ϕ(2^{−j}t − k) and j ∈ ℤ. Thus, V_{0} = \overline{span\{ϕ(· − k)\}}_{k∈ℤ}. We have that V_{j} satisfy M1 if only if ϕ ∈ V_{−1}, that is, if only if there exists a 2π-periodic square integrable function m_{0}, such that

ϕ̂(ξ) = m_{0}(\frac{ξ}{2})ϕ̂(\frac{ξ}{2}).

The Meyer multi-resolution analysis is constructed in the following way: Let φ be the scaling function defined by its Fourier transform by

φ̂(ξ) = \begin{cases}1, & if |ξ| ≤ 2π/3 \\ cos[\frac{π}{2} ν(\frac{3}{2π} |ξ| − 1)] & if 2π/3 ≤ |ξ| ≤ 4π/3 \\ 0, & otherwise, \end{cases}

where ν is a differentiable function satisfying

ν(x) = \begin{cases}0 & if x ≤ 0 \\ 1 & if x ≥ 1 \end{cases}  (2.1)
ν(x) + ν(1 − x) = 1  (2.2)

¿From (2.2), it follows that ∑_{k∈ℤ} |φ̂(ξ + 2kπ)|^{2} = 1, which is equivalent to the orthonormality of φ(· − k), k ∈ ℤ. Then M6 is satisfied. Here m_{0} can be constructed on [0, 2π], from φ̂, by

m_{0}(ξ) = ∑_{l∈ℤ} φ̂(2(ξ + 2πl))

This function is 2π-periodic, square integrable, and, for ξ ∈ [0, 2π],

m_{0}(\frac{ξ}{2})φ̂(\frac{ξ}{2}) = ∑_{l∈ℤ} φ̂(ξ + 4πl)φ̂(\frac{ξ}{2}) = φ̂(ξ)φ̂(\frac{ξ}{2}) = φ̂(ξ)

The second equality above follows from

φ̂(ξ + 4πl)φ̂(\frac{ξ}{2}) = 0, ∀l ≠ 0

and the third equality follows from φ̂(ξ/2) = 1 for all ξ ∈ supp φ̂. Then M1 is satisfied and the other conditions of the definition can also be proved. The associated mother wavelet is given by (see [2])

ψ̂(ξ) = e^{iξ/2}\overline{m_{0}(ξ/2 + π)}φ̂(ξ/2)
= e^{iξ/2} ∑_{l∈ℤ} φ̂(ξ + 2π(2l + 1))φ̂(ξ/2)
= e^{iξ/2}[φ̂(ξ + 2π) + φ̂(ξ − 2π)]φ̂(ξ/2)

or equivalently,

ψ̂(ξ) = \begin{cases}e^{iξ/2} sin[\frac{π}{2} ν(\frac{3}{2π} |ξ| − 1)], & if \frac{2π}{3} ≤ |ξ| ≤ \frac{4π}{3} \\ e^{iξ/2} cos[\frac{π}{2} ν(\frac{3}{4π} |ξ| − 1)], & if \frac{4π}{3} ≤ |ξ| ≤ \frac{8π}{3} \\ 0, & otherwise. \end{cases}

The function ψ is the Meyer wavelet.

Now, we consider the Meyer multi-resolution analysis. We have

ψ̂_{jk}(ξ) = ∫_{ℝ} ψ_{jk}(x)e^{−ixξ}dx
= ∫_{ℝ} 2^{− \frac{j}{2}} ψ(2^{−j}x − k)e^{−ixξ}dx
= ∫_{ℝ} 2^{j/2}ψ(y − k)e^{−i2^{j}yξ}dy
= 2^{j/2} ∫_{ℝ} ψ(t)e^{−i2^{j}(t+k)ξ}dt
= 2^{j/2} ∫_{ℝ} ψ(t)e^{−i2^{j}tξ−i2^{j}kξ}dt = 2^{j/2}e^{−i2^{j}kξ}ψ̂(2^{j}ξ)

Since supp(ψ̂) = \{ξ : \frac{2}{3}π ≤ |ξ| ≤ \frac{8}{3}π\} we have that

supp(ψ̂_{jk}) = \{ξ; \frac{2}{3}π2^{−j} ≤ |ξ| ≤ \frac{8}{3}π2^{−j}\} ∀k ∈ ℤ  (2.3)

Furthermore,

supp(φ̂_{jk}) = \{ξ; |ξ| ≤ \frac{4}{3}π2^{−j}\} ∀k ∈ ℤ  (2.4)

Now we consider the orthogonal projection onto V_{j}, P_{j} : L^{2}(R) → V_{j},

P_{j}f(t) = ∑_{k∈ℤ}⟨f, φ_{jk}⟩φ_{jk}(t)

The hypothesis M1 and M2 imply that lim_{j→−∞} P_{j}f = f, for all f ∈ L^{2}(R). This means that from a representation of f in a given scale, we can get f by adding details which are given at higher frequencies. ¿From (2.4), we see that P_{j} filters away the frequencies higher than \frac{4}{3}π2^{−j} (low pass filter).

We have, for all f ∈ L^{2}(R),

f = P_{j}f − P_{j}f + f
= P_{j}f + (I − P_{j})f
= ∑_{k∈ℤ}⟨f, φ_{jk}⟩φ_{jk} + ∑_{l≤j} ∑_{k∈ℤ}⟨f, ψ_{lk}⟩ψ_{lk}

This implies

P̂_{j}f(ξ) = f̂(ξ) for |ξ| ≤ \frac{2}{3}π2^{−j}  (2.5)

since, by (2.3), ψ̂_{lk}(ξ) = 0 for all l ≤ j and |ξ| ≤ \frac{2}{3}π2^{−j}.

Considering the corresponding orthogonal projections in the frequency space, P̂_{j} : L^{2}(R) → V̂_{j} = \overline{span\{φ̂_{jk}\}}_{k∈ℤ},

P̂_{j}f = ∑_{k∈ℤ} \frac{1}{2π} ⟨f, φ̂_{jk}⟩φ̂_{jk}

we have

P̂_{j}f̂ = ∑_{k∈ℤ} \frac{1}{2π} ⟨f̂, φ̂_{jk}⟩φ̂_{jk} = ∑_{k∈ℤ}⟨f, φ_{jk}⟩φ̂_{jk} = P̂_{j}f

Then (2.5) implies that

\begin{aligned}∥(I − P_{j})f∥ & = \frac{1}{\sqrt{2π}} ∥[(I − P_{j})f]^{∧}∥ = \frac{1}{\sqrt{2π}} ∥(I − P̂_{j})f̂∥ \\ & = \frac{1}{\sqrt{2π}} ∥(I − P̂_{j})χ_{j}f̂∥ ≤ ∥χ_{j}f̂∥ \end{aligned}  (2.6)

where χ_{j} is the characteristic function in (−∞, −\frac{2}{3}π2^{−j}] ∪ [\frac{2}{3}π2^{−j}, +∞).

3 Results of Stability and Convergence

Hereafter, the multi-resolution analysis considered corresponds to the Meyer multi-resolution analysis with scaling function φ. The next lemma is a version of the Gronwall inequality.

Lemma 3.1 Let u and v be positive continuous functions, x ≥ a and c > 0. If u(x) ≤ c + ∫_{a}^{x} ∫_{a}^{s} v(τ)u(τ) dτds then

u(x) ≤ c exp ( ∫_{a}^{x} ∫_{a}^{s} v(τ) dτds) .

Proof. Let w(x) = c + ∫_{a}^{x} ∫_{a}^{s} v(τ)u(τ) dτds. Then w′(x) = ∫_{a}^{x} v(τ)u(τ) dτ. Therefore,

w′′(x) = v(x)u(x) ≤ v(x)w(x) and \frac{w′′(x)}{w(x)} ≤ v(x)

Now

\frac{w′′(x)}{w(x)} = (\frac{w′}{w})′(x) + (\frac{w′(x)}{w(x)})^{2}

Thus (\frac{w′}{w} )′(x) ≤ v(x) which implies

\frac{w′(x)}{w(x)} ≤ ∫_{a}^{x} v(τ) dτ and (ln w(x))′ ≤ ∫_{a}^{x} v(τ) dτ;

Therefore,

ln w(x) − ln w(a) ≤ ∫_{a}^{x} ∫_{a}^{s} v(τ) dτds

Since w(a) = c, ln w(x) − ln c ≤ ∫_{a}^{x} ∫_{a}^{s} v(τ) dτds, which implies

ln \frac{w(x)}{c} ≤ ∫_{a}^{x} ∫_{a}^{s} v(τ) dτds and w(x) ≤ c exp ( ∫_{a}^{x} ∫_{a}^{s} v(τ) dτds).

Since, by hypothesis, u(x) ≤ w(x), we have

u(x) ≤ c exp ( ∫_{a}^{x} ∫_{a}^{s} v(τ) dτds)

which completes the proof. □

Applying the Fourier Transform with respect to time in Problem (1.1), we obtain the following problem in the frequency space:

û_{xx}(x, ξ) = \frac{iξ}{K(x)}û(x, ξ), 0 < x < 1, ξ ∈ R

û(0, ξ) = ĝ(ξ), û_{x}(0, ·) = 0

whose solution satisfies

û(x, ξ) = ĝ(ξ) + ∫_{0}^{x} ∫_{0}^{s} \frac{iξ}{K(τ)}û(τ, ξ) dτds

Then, from lemma 3.1, for ĝ(ξ) ≠ 0, we have

|û(x, ξ)| ≤ |ĝ(ξ)| exp [|ξ| ∫_{0}^{x} ∫_{0}^{s} \frac{1}{K(τ)} dτds]  (3.1)

The next lemma corresponds to proposition 3.1 in [4], when K(x) is constant.

Lemma 3.2 The operator D_{j}(x) defined by

[(D_{j})_{lk}(x)]_{l∈ℤ, k∈ℤ} = [\frac{1}{K(x)}⟨φ′_{jl}, φ_{jk}⟩]_{l∈ℤ,k∈ℤ}

satisfies the following three conditions: 1) (D_{j})_{lk}(x) = −(D_{j})_{kl}(x)

2) (D_{j})_{lk}(x) = (D_{j})_{(l−k)0}(x). Hence, (D_{j})_{lk}(x) are equal along diagonals.

3) ∥D_{j}(x)∥ ≤ \frac{π2^{−j}}{K(x)}

Proof. This proof follows quite closely the proof in [4] for D_{j}(x) independent of x.

1) As we already know, φ̂_{j0}(ξ) = 2^{j/2}φ̂(2^{j}ξ), φ̂_{jk}(ξ) = 2^{j/2}e^{−ikξ2^{j}} φ̂(2^{j}ξ) = e^{−ikξ2^{j}} φ̂_{j0}(ξ), and the Fourier transform of the scaling function φ is even. Then φ̂_{j0}(ξ) = φ̂_{j0}(−ξ) and

(D_{j})_{lk}(x) = \frac{1}{K(x)}⟨φ′_{jl}, φ_{jk}⟩ = \frac{1}{K(x)} \frac{1}{2π} ⟨φ̂′_{jl}, φ̂_{jk}⟩
= \frac{1}{K(x)} \frac{1}{2π} ∫_{ℝ} φ̂′_{jl}(ξ)\overline{φ̂_{jk}}(ξ) dξ
= \frac{i}{K(x)} \frac{1}{2π} ∫_{ℝ} ξφ̂_{jl}(ξ)\overline{φ̂_{jk}}(ξ) dξ
= \frac{i}{K(x)} \frac{1}{2π} ∫_{ℝ} ξe^{−i(l−k)ξ2^{j}}|φ̂_{j0}(ξ)|^{2}dξ

Then

(D_{j})_{lk}(x) = \frac{i}{K(x)} \frac{1}{2π} ∫_{ℝ}(−ω)e^{−i(k−l)ω2^{j}}|φ̂_{j0}(−ω)|^{2}dω
= −\frac{i}{K(x)} \frac{1}{2π} ∫_{ℝ} ωe^{−i(k−l)ω2^{j}}|φ̂_{j0}(ω)|^{2}dω
= −\frac{i}{K(x)} \frac{1}{2π} ∫_{ℝ} ωφ̂_{jk}(ω)\overline{φ̂_{jl}}(ω) dω
= −\frac{1}{K(x)} \frac{1}{2π} ∫_{ℝ} φ̂′_{jk}(ω)\overline{φ̂_{jl}}(ω) dω.

Thus

(D_{j})_{lk}(x) = −\frac{1}{K(x)} \frac{1}{2π} ⟨φ̂′_{jk}, φ̂_{jl}⟩ = −\frac{1}{K(x)}⟨φ′_{jk}, φ_{jl}⟩ = −(D_{j})_{kl}(x)

2) As proved above

(D_{j})_{lk}(x) = \frac{i}{K(x)} \frac{1}{2π} ∫_{ℝ} ξe^{−i(l−k)ξ2^{j}}|φ̂_{j0}(ξ)|^{2}dξ = (D_{j})_{(l−k)0}(x)

Then (D_{j})_{lk}(x) is equal along diagonals.

3) We have

∥D_{j}(x)∥ = ∥\frac{1}{K(x)}B_{j}∥ = \frac{1}{K(x)}∥B_{j}∥

where (B_{j})_{lk} = ⟨φ′_{jl}, φ_{jk}⟩. ¿From results 1) and 2), we have (B_{j})_{lk} = −(B_{j})_{kl}, (B_{j})_{lk} = \frac{1}{2π} ∫_{ℝ} ξe^{−i(l−k)ξ2^{j}}|φ̂_{j0}(ξ)|^{2}dξ = (B_{j})_{(l−k)0} and (B_{j})_{lk} is constant along diagonals. We will show that ∥B_{j}∥ ≤ π2^{−j}. Thus, we will have

∥D_{j}(x)∥ ≤ \frac{π}{K(x)}2^{−j}

For |t| ≤ π2^{−j},

Γ_{j}(t) =i2^{−j}[(t − 2^{−j+1}π)|φ̂_{j0}(t − 2^{−j+1}π)|^{2} + t|φ̂_{j0}(t)|^{2}
+ (t + 2^{−j+1}π)|φ̂_{j0}(t + 2^{−j+1}π)|^{2}]

Extend Γ_{j} periodically to ℝ and expand it in Fourier series as

Γ_{j}(t) = ∑_{k∈ℤ} γ_{k}e^{ikt2^{j}}

We have γ_{k} = b_{k} for all k, where b_{k} is the element in diagonal k of B_{j}. In fact, since φ̂_{j0}(t) = 0 for |t| ≥ \frac{4}{3}π2^{−j}, it follows that

γ_{k} = \frac{1}{2^{−j+1}π} ∫_{−π2^{−j}}^{π2^{−j}} Γ_{j}(t)e^{−ikt2^{j}}dt

= \frac{i}{2π} ∫_{−π2^{−j}}^{π2^{−j}}(t − 2^{−j+1}π)|φ̂_{j0}(t − 2^{−j+1}π)|^{2}e^{−ikt2^{j}}dt

+\frac{i}{2π} ∫_{−π2^{−j}}^{π2^{−j}} t|φ̂_{j0}(t)|^{2}e^{−ikt2^{j}}dt

+\frac{i}{2π} ∫_{−π2^{−j}}^{π2^{−j}}(t + 2^{−j+1}π)|φ̂_{j0}(t + 2^{−j+1}π)|^{2}e^{−ikt2^{j}}dt

Making a change of variable, we obtain:

γ_{k} = \frac{i}{2π} ∫_{−3π2^{−j}}^{−π2^{−j}} t|φ̂_{j0}(t)|^{2}e^{−ikt2^{j}}dt + \frac{i}{2π} ∫_{−π2^{−j}}^{π2^{−j}} t|φ̂_{j0}(t)|^{2}e^{−ikt2^{j}}dt

+\frac{i}{2π} ∫_{π2^{−j}}^{3π2^{−j}} t|φ̂_{j0}(t)|^{2}e^{−ikt2^{j}}dt

= \frac{i}{2π} ∫_{−3π2^{−j}}^{3π2^{−j}} t|φ̂_{j0}(t)|^{2}e^{−ikt2^{j}}dt
= \frac{i}{2π} ∫_{ℝ} t|φ̂_{j0}(t)|^{2}e^{−ikt2^{j}}dt = b_{k}

Now, ∥B_{j}∥ = sup_{∥f∥=1} ∥B_{j}f∥ where ∥f∥^{2} = ∑_{k∈ℤ} |f_{k}|^{2}. Let F (t) = ∑_{k∈ℤ} f_{k}e^{ikt2^{j}} and define W (t) = Γ_{j}(t)F (t). We have

W (t) = ∑_{k∈ℤ} ω_{k}e^{ikt2^{j}} and ω_{k} = ∑_{l∈ℤ} b_{k−l}f_{l} = (B_{j}f)_{k}

Hence

∥ω∥^{2} = ∑_{k∈ℤ} |ω_{k}|^{2} = \frac{1}{2π2^{−j}} ∫_{−π2^{−j}}^{π2^{−j}} |W (t)|^{2}dt

= \frac{1}{2π2^{−j}} ∫_{−π2^{−j}}^{π2^{−j}} |Γ_{j}(t)F (t)|^{2}dt

≤ sup_{|t|≤π2^{−j}} |Γ_{j}(t)|^{2} \frac{1}{2π2^{−j}} ∫_{−π2^{−j}}^{π2^{−j}} |F (t)|^{2}dt

= sup_{|t|≤π2^{−j}} |Γ_{j}(t)|^{2}∥f∥^{2}

Then

∥B_{j}∥ ≤ sup_{|t|≤π2^{−j}} |Γ_{j}(t)|^{2}

On the other hand, Γ_{j} is an odd function. Hence

sup_{|t|≤π2^{−j}} |Γ_{j}(t)|^{2} = sup_{0≤t≤π2^{−j}} |Γ_{j}(t)|^{2}

But, for 0 ≤ t ≤ π2^{−j}, we have t + π2^{−j+1} ≥ π2^{−j+1} and t − π2^{−j+1} ≤ 0. Hence

φ̂_{j0}(t + π2^{−j+1}) = 0 and (t − π2^{−j+1})|φ̂_{j0}(t − π2^{−j+1})|^{2} ≤ 0

for t ∈ [0, π2^{−j}]. Thus

sup_{0≤t≤π2^{−j}} |Γ_{j}(t)|^{2} ≤ π2^{−j+1} sup_{0≤t≤π2^{−j}} t|φ̂_{j0}(t)|^{2}
= π2^{−j+1} sup_{0≤t≤π2^{−j}}(t2^{j})|φ̂(2^{j}t)|^{2}
= π2^{−j+1} sup_{0≤s≤π} s|φ̂(s)|^{2}

By definition of φ̂ we have |φ̂(s)|^{2} ≤ \frac{1}{2π} and therefore s|φ̂(s)|^{2} ≤ \frac{π}{2π} = \frac{1}{2} for 0 ≤ s ≤ π. Then

sup_{0≤t≤π2^{−j}} |Γ_{j}(t)|^{2} ≤ sup_{0≤s≤π} s|φ̂(s)|^{2} ≤ \frac{π2^{−j+1}}{2} = π2^{−j}

Thus

∥D_{j}(x)∥ = \frac{1}{K(x)}∥B_{j}∥ ≤ \frac{1}{K(x)} sup_{|t|≤π2^{−j}} |Γ_{j}(t)|^{2} ≤ \frac{π2^{−j}}{K(x)}

which completes the proof of lemma 3.2. □

Let us now consider the following approximating problem in V_{j}, where the projection in the first equation of (3.2) is due to the fact that we can have φ ∈ V_{j} with φ′ ∉ V_{j} (see note 2 below),

\begin{matrix}K(x)u_{xx}(x, t) = P_{j}u_{t}(x, t), t ≥ 0, 0 < x < 1 \\ u(0, ·) = P_{j}g u_{x}(0, ·) = 0 \\ u(x, t) ∈ V_{j} \end{matrix}  (3.2)

Its variational formulation is

⟨K(x)u_{xx} − u_{t}, φ_{jk}⟩ = 0
⟨u(0, ·), φ_{jk}⟩ = ⟨P_{j}g, φ_{jk}⟩, ⟨u_{x}(0, ·), φ_{jk}⟩ = ⟨0, φ_{jk}⟩, k ∈ ℤ

where φ_{jk} is the orthonormal basis of V_{j} given by the scaling function φ. Consider u_{j} a solution of the approximating problem (3.2), given by u_{j}(x, t) = ∑_{l∈ℤ} w_{l}(x)φ_{jl}(t). Then, we have (u_{j})_{t}(x, t) = ∑_{l∈ℤ} w_{l}(x)φ′_{jl}(t) and (u_{j})_{xx}(x, t) = ∑_{l∈ℤ} w′′_{l}(x)φ_{jl}(t). Therefore,

K(x)(u_{j})_{xx}(x, t) − (u_{j})_{t}(x, t) = K(x) ∑_{l∈ℤ} w′′_{l}(x)φ_{jl}(t) − ∑_{l∈ℤ} w_{l}(x)φ′_{jl}(t)

Hence

⟨K(x)(u_{j})_{xx} − (u_{j})_{t}, φ_{jk}⟩ = 0

⟺ ⟨∑_{l∈ℤ} K(x)w′′_{l}φ_{jl} − ∑_{l∈ℤ} w_{l}φ′_{jl}, φ_{jk}⟩ = 0

⟺ ∑_{l∈ℤ} K(x)w′′_{l}⟨φ_{jl}, φ_{jk}⟩ = ∑_{l∈ℤ} w_{l}⟨φ′_{jl}, φ_{jk}⟩

⟺ K(x)w′′_{k} = ∑_{l∈ℤ} w_{l}⟨φ′_{jl}, φ_{jk}⟩ k ∈ ℤ.

Therefore,

\frac{d^{2}}{dx^{2}} w_{k} = ∑_{l∈ℤ} w_{l} \frac{1}{K(x)}⟨φ′_{jl}, φ_{jk}⟩ and \frac{d^{2}}{dx^{2}} w_{k} = ∑_{l∈ℤ} w_{l}(D_{j})_{lk}(x)

where, as defined before, (D_{j})_{lk}(x) = \frac{1}{K(x)}⟨φ′_{jl}, φ_{jk}⟩. Thus, we get an infinitedimensional system of ordinary differential equations

\begin{matrix}\frac{d^{2}}{dx^{2}} w = −D_{j}(x)w \\ w(0) = γ, w′(0) = 0 \end{matrix}  (3.3)

where γ is given by

P_{j}g = ∑_{z∈ℤ} γ_{z}φ_{jz} = ∑_{z∈ℤ}⟨g, φ_{jz}⟩φ_{jz}

Lemma 3.3 If w is a solution of the evolution problem of second order (3.3), then

∥w(x)∥ ≤ ∥γ∥ exp (2^{−j}π ∫_{0}^{x} ∫_{0}^{s} \frac{1}{K(τ)} dτ ds)

Proof Since w(x) = γ + ∫_{0}^{x} ∫_{0}^{s}(−D_{j})(τ)w(τ) dτds,

∥w(x)∥ ≤ ∥γ∥ + ∫_{0}^{x} ∫_{0}^{s} ∥D_{j}(τ)∥∥w(τ)∥ dτ ds

By lemma 3.2 this implies

∥w(x)∥ ≤ ∥γ∥ + ∫_{0}^{x} ∫_{0}^{s} \frac{2^{−j}π}{K(x)}∥w(τ)∥ dτ ds.

Then by lemma 3.1 we have

∥w(x)∥ ≤ ∥γ∥ exp (2^{−j}π ∫_{0}^{x} ∫_{0}^{s} \frac{1}{K(τ)} dτds)

which completes the proof. □

Theorem 3.4 (Stability of the wavelet Galerkin method) Let u_{j} and v_{j} be solutions in V_{j} of the approximating problems (3.2) for the boundary specifications g and g̃, respectively. If ∥g − g̃∥ ≤ ϵ then

∥u_{j}(x, ·) − v_{j}(x, ·)∥ ≤ ϵ exp (\frac{2^{−j−1}π}{α} x^{2})

where α satisfies 0 < α ≤ K(x) < +∞ as in the definition of the problem (1.1). For j such that 2^{−j} ≤ \frac{2α}{π} log ϵ^{−1} we have

∥u_{j}(x, ·) − v_{j}(x, ·)∥ ≤ ϵ^{1−x^{2}}

Proof. u_{j}(x, t) = ∑_{l∈ℤ} w_{l}(x)φ_{jl}(t), v_{j}(x, t) = ∑_{l∈ℤ} w̃_{l}(x)φ_{jl}(t) where w and w̃ are solutions of the Galerkin problem (3.3) with conditions w(0) = γ and w̃(0) = γ̃, respectively. So, by lemma 3.3 and linearity of (3.3) we have

∥u_{j}(x, ·) − v_{j}(x, ·)∥ = ∥w(x) − w̃(x)∥
≤ ∥γ − γ̃∥ exp(2^{−j}π ∫_{0}^{x} ∫_{0}^{s} \frac{1}{K(τ)} dτds)
≤ ϵ exp(2^{−j}π ∫_{0}^{x} ∫_{0}^{s} \frac{1}{α} dτds)
= ϵ exp(2^{−j−1} \frac{π}{α}x^{2})

For j = j(ϵ) such that 2^{−j} ≤ \frac{2α}{π} log ϵ^{−1}, we have

∥u_{j}(x, ·) − v_{j}(x, ·)∥ ≤ ϵ exp(x^{2} log ϵ^{−1}) = ϵ^{1−x^{2}}

which completes the proof. □

Now, we are interested in the solutions u(x, ·) ∈ L^{2}(R) of problem (1.1), for the functions g ∈ L^{2}(R) such that ĝ(·) exp(| · |/(2α)) ∈ L^{2}(R), where ĝ is the Fourier Transform of g. The Inverse Fourier Transform of exp(−\frac{ξ^{2}+|ξ|}{2α}), for instance, satisfies this condition. Define

f := ĝ(·) exp (\frac{| · |}{2α}) ∈ L^{2}(R)  (3.4)

Proposition 3.5 If u(x, t) is a solution of problem (1.1), then

∥u(x, ·) − P_{j}u(x, ·)∥ ≤ ∥f∥_{L^{2}(ℝ)} exp(−\frac{1}{3} \frac{π}{α}2^{−j}(1 − x^{2}))

where f is given by (3.4).

Proof ¿From (2.6) and (3.1), we have

∥(I − P_{j})u(x, ·)∥ ≤ ∥χ_{j}û(x, ·)∥
= [∫_{|ξ|> \frac{2}{3} π2^{−j}} |û(x, ξ)|^{2} dξ]^{1/2}
≤ [∫_{|ξ|> \frac{2}{3} π2^{−j}} |ĝ(ξ)|^{2} exp[2|ξ| ∫_{0}^{x} ∫_{0}^{s} \frac{1}{K(τ)} dτ ds] dξ]^{1/2}

Then

∥(I − P_{j})u(x, ·)∥ ≤ [∫_{|ξ|> \frac{2}{3} π2^{−j}} |ĝ(ξ)|^{2} exp(|ξ|\frac{x^{2}}{α}) dξ]^{1/2}
≤ [∫_{|ξ|> \frac{2}{3} π2^{−j}} |f(ξ)|^{2} exp(−\frac{|ξ|}{α}) exp(\frac{|ξ|}{α}x^{2})dξ]^{1/2}
= [∫_{|ξ|> \frac{2}{3} π2^{−j}} |f(ξ)|^{2} exp(−\frac{|ξ|}{α}(1 − x^{2})) dξ]^{1/2}

For |x| < 1,

∥(I − P_{j})u(x, ·)∥ ≤ [∫_{ℝ} |f(ξ)|^{2} dξ]^{1/2} exp(−\frac{(2/3)π2^{−j}}{2α} (1 − x^{2}))
≤ ∥f∥_{L^{2}(ℝ)} exp(−\frac{1}{3} \frac{π}{α}2^{−j}(1 − x^{2}))

which completes the proof. □

Proposition 3.6 If u is a solution of problem (1.1) and u_{j−1} is a solution of the approximating problem in V_{j−1} then

û(x, ξ) = û_{j−1}(x, ξ) for |ξ| ≤ \frac{4}{3}π2^{−j}  (3.5)

Consequently,

P_{j}u(x, ·) = P_{j}u_{j−1}(x, ·)  (3.6)

Proof Let Λ(x, ξ) = û(x, ξ) − û_{j−1}(x, ξ). We will show that Λ(x, ξ) = 0 for |ξ| ≤ \frac{4}{3}π2^{−j}. Consider the approximating problem in V_{j−1}:

K(x)(u_{j−1})_{xx} = P_{j−1}(u_{j−1})_{t} t ∈ ℝ, 0 < x < 1
u_{j−1}(0, ·) = P_{j−1}g, (u_{j−1})_{x}(0, ·) = 0
u_{j−1}(x, ·) ∈ V_{j−1}

Applying the Fourier transform with respect to time, we have

K(x)(û_{j−1})_{xx}(x, ξ) = P̂_{j−1}[(u_{j−1})_{t}] (x, ξ) = P̂_{j−1}(iξû_{j−1}(x, ξ))

for 0 ≤ x < 1, ξ ∈ ℝ, with the conditions: û_{j−1}(0, ξ) = P̂_{j−1}ĝ(ξ) and (û_{j−1})_{x}(0, ·) = 0. Now, by (2.5),

P̂_{j−1}(iξû_{j−1}(x, ξ)) = iξû_{j−1}(x, ξ) and P̂_{j−1}û(0, ξ) = û(0, ξ)

for |ξ| ≤ \frac{4}{3}π2^{−j}. Thus, for |ξ| ≤ \frac{4}{3}π2^{−j}, we have

K(x)Λ_{xx}(x, ξ) − iξΛ(x, ξ)
= K(x)û_{xx}(x, ξ) − K(x)(û_{j−1})_{xx}(x, ξ) − iξ[û(x, ξ) − û_{j−1}(x, ξ)] = 0

Λ(0, ξ) = û(0, ξ) − û_{j−1}(0, ξ) = û(0, ξ) − P̂_{j−1}ĝ(ξ) = û(0, ξ) − P̂_{j−1}û(0, ξ) = 0

Λ_{x}(0, ξ) = û_{x}(0, ξ) − (û_{j−1})_{x}(0, ξ) = 0

Hence, for |ξ| ≤ \frac{4}{3}π2^{−j}, fixed, Λ(x, ξ) is solution on 0 ≤ x < 1 of the problem

K(x)Λ_{xx}(x, ξ) − iξΛ(x, ξ) = 0, 0 < x < 1
Λ(0, ξ) = 0, Λ_{x}(0, ξ) = 0

This problem has an unique solution Λ(x, ξ) = 0, for all x ∈ [0, 1). Thus,

û(x, ξ) = û_{j−1}(x, ξ) for |ξ| ≤ \frac{4}{3}π2^{−j}

Now, (3.6) is consequence of (3.5) and the definition of P̂_{j}. □

Theorem 3.7 Let u be a solution of (1.1) with the condition u(0, ·) = g, and let f be given by (3.4). Let v_{j−1} be a solution of (3.2) in V_{j−1} for the boundary specification g̃ such that ∥g − g̃∥ ≤ ϵ. If j = j(ϵ) is such that 2^{−j} = \frac{α}{π} log ϵ^{−1}, then

∥P_{j}v_{j−1}(x, ·) − u(x, ·)∥ ≤ ϵ^{1−x^{2}} + ∥f∥_{L^{2}(R)} · ϵ^{\frac{1}{3} (1−x^{2})}

Proof Note that

∥P_{j}v_{j−1}(x, ·) − u(x, ·)∥ ≤ ∥P_{j}v_{j−1}(x, ·) − P_{j}u(x, ·) + P_{j}u(x, ·) − u(x, ·)∥
≤ ∥P_{j}v_{j−1}(x, ·) − P_{j}u(x, ·)∥ + ∥P_{j}u(x, ·) − u(x, ·)∥ .

Let u_{j−1} be a solution of (3.2) in V_{j−1} for the boundary specification g. By (3.6), P_{j}u(x, ·) = P_{j}u_{j−1}(x, ·). Thus, by theorem 3.4, we have

∥P_{j}v_{j−1}(x, ·) − P_{j}u(x, ·)∥ = ∥P_{j}v_{j−1}(x, ·) − P_{j}u_{j−1}(x, ·)∥
≤ ∥v_{j−1}(x, ·) − u_{j−1}(x, ·)∥ ≤ ϵ^{1−x^{2}}

Now, by proposition 3.5,

∥P_{j}u(x, ·) − u(x, ·)∥ ≤ ∥f∥_{L^{2}(ℝ)} exp(−\frac{1}{3} \frac{π}{α}2^{−j}(1 − x^{2})) ≤ ∥f∥_{L^{2}(ℝ)} · ϵ^{\frac{1}{3} (1−x^{2})}

Then ∥P_{j}v_{j−1}(x, ·) − u(x, ·)∥ ≤ ϵ^{1−x^{2}} + ∥f∥_{L^{2}(R)}ϵ^{\frac{1}{3} (1−x^{2})} □

Conclusion

We had considered solutions u(x, ·) ∈ L^{2}(R) of the problem K(x)u_{xx} = u_{t}, 0 < x < 1 , t ≥ 0 , with boundary specification g and u_{x}(0, ·) = 0, where K(x) is bounded below by a positive constant. The inequality (3.1) implies that a solution of the problem above will be in L^{2}(ℝ) if ĝ has a rapid decay at high frequencies. Since the Meyer wavelet has compact support in the frequency domain, it cuts the high frequencies. Utilizing a wavelet Galerkin method with the Meyer multi-resolution analysis, we regularize the ill-posedness of the problem, approaching it by well-posed problems in the scaling spaces, as shown by theorem 3.4. We had shown the convergence of the wavelet Galerkin method applied to our problem, with an estimate error, in theorem 3.7. A more direct result would be to have a similar estimate for the difference between the exact solution of the problem and the solution of the approximating problem defined on the scaling space V_{j}. We are working towards this goal at the moment.

Notes: 1) Consider the problem

u_{xx}(x, t) = u_{t}(x, t), t ≥ 0, 0 < x < 1
u(0, ·) = g_{n}, u_{x}(0, ·) = 0 ,

where

g_{n}(t) = \begin{cases}n^{−2} cos 2n^{2}t, & if 0 ≤ t ≤ t_{0} \\ 0, & if t > t_{0} . \end{cases}

The solution of this problem is

u_{n}(x, t) = \begin{cases}∑_{j=0}^{∞} n^{−2} cos(2n^{2}t + j \frac{π}{2} )^{\frac{(\sqrt{2}nx)^{2j}}{(2j)!}} , & if 0 ≤ t ≤ t_{0} \\ 0, & if t > t_{0} . \end{cases}

Note that g_{n}(t) converges uniformly to zero as n tends to infinity, while for x > 0, the solution u_{n}(x, t) does not tend to zero. This example was inspired by [1].

2) Note that (φ_{jl})′ ∉ V_{j}. In fact, if (φ_{jl})′ ∈ V_{j} then (φ_{jl})′ = ∑_{k∈Z} α_{k}φ_{jk}. Hence

{ \atop (φ_{jl})′} = ∑_{k∈Z} α_{k}φ̂_{jk}

So, we would have

i2^{j/2}e^{−i2^{j}lξ}ξφ̂(2^{j}ξ) = ∑_{k∈Z} α_{k}2^{j/2}e^{−i2^{j/2}ξ}φ̂(2^{j}ξ)

This equality implies ξ = ∑_{k∈Z} α_{k}e^{−i[2^{j}(k−l)ξ+ \frac{π}{2} ]}.

References

[1] J. R Cannon, The one dimensional heat equation, (Reading) MA: Addison-Wesley, 1984.

[2] I. Daubechies, I., Ten Lectures on Wavelets, CBMS - NSF 61 SIAM, Regional Conferences Series in Applied Mathematics, 1992.

[3] T. Reginska, Sideways heat equation and wavelets, J. Comput. Appl. Math. 63 (1995) 209-214.

[4] T. Reginska and L. Eldén, Solving the sideways heat equation by a wavelet Galerkin method, Inverse Problems 13 (1997) 1093-1106.

[5] T. Reginska, Stability and Convergence of a wavelet Galerkin method for the Sideways Heat Equation, J. Inverse Ill Posed Probl., 8, no 1, 2000.

José Roberto Linhares de Mattos Rural Federal University of Rio de Janeiro, Exact Sciences Institute, Department of Mathematics, BR465 Km 7, Seropédica RJ, CEP 23890-000, Brazil email: linhares@cos.ufrj.br

Ernesto Prado Lopes Federal University of Rio de Janeiro, COPPE, Systems and Computing Engineering Program, Tecnology Center, Bloco H and Institute of Mathematics, Tecnology Center, Bloco C, Ilha do Fundão, Rio de Janeiro RJ, CEP 21945-970, Brazil email: lopes@cos.ufrj.br