Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Volume 3, Number 1, March 2022, pp.1-11 https://doi.org/10.5206/mase/13511

ON APPROXIMATING INITIAL DATA IN SOME LINEAR EVOLUTION

EQUATIONS INVOLVING FRACTIONAL LAPLACIAN

RAMESH KARKI

Abstract. We study an inverse problem of recovering initial datum in a one-dimensional linear evolu-

tion equation with the Dirichlet boundary conditions when the solution to the equation is known only

at a suitably fixed space location and suitably chosen finitely many later time instances. To be more

explicit, we consider a one-dimensional linear evolutionary equation involving a Dirichlet fractional

Laplacian and an unknown initial datum f which is assumed to be in a suitable subset of a Sobolev

space, and then construct n future times so that from the known values of the solution at a suitably

fixed space location and at these n future times, we recover f with a desired accuracy.

1. Introduction

Consider the initial-boundary value problem

ut = −a(t)(−∆)1/2u, u(0, t) = u(π,t) = 0, u(x, 0) = f(x) (1.1)

where a is a positive continuous function of t > 0 such that
∫ t
0
a(s)ds exists, ∆ = ∂2/∂x2, and at the

moment, f is in L2[0,π]. The problem of finding a solution u(x,t) to (1.1) is quite common when

the initial datum f in a suitable function space is known. However, we are interested in studying an

inverse type problem, meaning a problem of recovering the initial datum f with a desired accuracy if

the solution u(x,t) is known only at a fixed point x0 in [0,π] and suitably selected n later time instances

tj, j = 1, 2, 3, . . . ,n.

This type of inverse problem is not well-posed in general. For the well-posed, we further assume that

f is in the closed subset Br of the Sobolev space Hr[0,π], r > 0, given by

Br :=
{
f ∈ Hr[0,π] : ||f||Hr[0,π] ≤ 1

}
. (1.2)

We are indeed motivated to study this problem from similar problems studied in [1, 9]. In [9], the authors

have considered the temperature distribution of a thin uniform one-dimensional body of finite length

represented by the one-dimensional heat equation with Dirichlet boundary conditions and an initial

condition. Then they have studied the recovery of the initial temperature measurement with a near

optimal rate when temperature measurements taken at a fixed point of the body and at finitely many

later times are known. Also, they have asked some questions, one of which is whether their method

can be extended to the case of involving a diffusion equation with a diffusion coefficient depending

continuously on time. This question has been addressed in [1]. Moreover, the authors in [1] have

also studied the problem of recovering initial data in an initial-boundary value problem involving a

parabolic PDE with constant coefficients and even order partial differential coefficients with respect to

Received by the editors 31 December 2020; revised 6 April 2021 and 5 January 2022; accepted 10 January 2022;

published online 19 January 2022.

2020 Mathematics Subject Classification. Primary 35S11.
Key words and phrases. Evolution equations, initial data, Fourier sine series, samples, later time instances, measure-

ment algorithm and optimal error.

1

https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/13511


2 R. KARKI

spacial variable. In this paper, we study the case, given in (1.1), that involves a fractional diffusion

equation with a diffusion coefficient depending continuously on time. It could be a base for generalizing

the method in other factional evolution equations, some of which are mentioned in Section 5. These

problems are of particular interest as they arise in different areas such as stochastic control theory and

mathematical finance, classical mechanics in the context of heat propagation, population dynamics, the

theory of water waves, quantum mechanics and phase transition problems [2, 5, 10, 13].

We know that the Dirichlet Laplacian −∆ on L2[0,π] has eigenvalues λn = n2, n = 1, 2, . . . with
the corresponding eigenfunctions en(x) := sin nx, n = 1, 2, . . . that form an orthonormal basis for

L2[0,π] when normalized with respect to the inner product 〈u,v〉L2 :=
2

π

∫ π
0

u(x)v(x) dx. Then each

u ∈ L2[0,π] has the following Fourier sine series representation

u =

∞∑
k=1

ûkek,

where the equality has to be understood in the L2-sense or in the sense of almost every x ∈ [0,π] and
ûk := 〈u,ek〉L2 , the kth Fourier sine coefficient of u. Referring to [3, 6, 12, 15], the fractional power
operator (−∆)s/2, s > 0 on the dense subset D((−∆)s/2) := {u ∈ L2[0,π] :

∑∞
k=1 k

2s|ûk|2 < ∞} =
Hr[0,π] of L2[0,π] is given by

(−∆)s/2u =
∞∑
k=1

ksûkek (1.3)

and has the eigenvalues λ
s/2
n = n

s, n = 1, 2, . . . with the corresponding eigenfunctions en, n = 1, 2, . . . .

Thus we can equip Hr[0,π] with the norm ||f||Hr =
∑∞
k=1 k

2r|f̂k|2.
The existence and uniqueness theory guarantees the existence of a (strong or L2-) solution u(x,t) to

(1.1), which then has the Fourier sine series representation

u(x,t) =

∞∑
k=1

ûk(t)ek(x) (1.4)

for almost every x ∈ [0,π]. However, solutions to more general evolutionary problems than (1.1) can
be expressed in the form of general Fourier series representations, which can be obtained by employing

tools from the semigroup theory and the spectral theory of self-adjoint operators on Hilbert spaces, and

have been discussed in [12, 14, 15, 16]. These representations can be used when extending our problem

to more general evolutionary equations.

Under the assumptions we made above, the regularity theory guarantees that the solution u(x,t) to

(1.1) is in D((−∆)1/2)). From (1.1), (1.3) and (1.4), we obtain that each time dependent Fourier sine
coefficient ûk(t) satisfies the initial value problem

d

dt
ûk(t) = −ka(t)ûk(t), ûk(0) = f̂k

whose solution is

ûk(t) = f̂ke
−kT(t), k = 1, 2, . . . , (1.5)

where T(t) =

∫ t
0

a(s)ds. From (1.4) and (1.5), we have

u(x,t) =

∞∑
k=1

f̂ke
−kT(t)ek(x) (1.6)

for t ≥ 0 and for almost every x ∈ [0,π]. The reason why the last equality holds in the a. e. sense is
because we have used the strong (not the classical) solution to (1.1). Since we are going to deal with



ON APPROXIMATING INITIAL DATA 3

the L2-error of approximation to f, it will be sufficient to have this type of solution. Throughout the

rest of the paper, we will have referred to this solution whenever we call the solution to (1.1).

Now we summarize what we are going to do in the rest of the paper. In Section 2, we will discuss

the selection of the space location x0 and prove that for an increasing sequence 0 < t1 < t2 < ... of

future times, the values u(x0, tj), j = 1, 2, . . . are enough to determine f uniquely. This consistency

result allows us to approximate f from the first n values u(x0, tj), j = 1, 2, . . . ,n, called n samples.

In Section 3, we will discuss the existence of a lower bound for an optimal error of approximation to

f. For this, we will use a measurement algorithm discussed in [1, 9] as an encoder or a continuous map

a from a subset B of L2[0,π] into Rn together with a decoder M or a continuous map from Rn into L2.
Using this measurement algorithm, we will find an approximation to f as M(a(f)) and also discuss the

existence of a lower bound for the optimal error of approximation to f in L2[0,π].

Like obtaining a lower bound for the optimal error of approximation to f, one may expect the

existence of an upper bound for the optimal error of the approximation. We will not address this in this

paper. However, in Section 4, we will prove that there exists a sequence of future times 0 < t1 < t2 < ...

such that from the first n samples u(x0, tj), j = 1, 2, . . . ,n, f can be approximated in L
2[0,π] with an

accuracy of order n−r.

In Section 5, we will discuss possibilities of extending this method to a few other evolutionary

equations and also possibilities of applying it to evolutionary equations with other boundary conditions.

2. Choice of x0 and consistency of recovery

We need to select x0 ∈ [0,π] in such a way that the samples u(x0, tj), j = 1, 2, . . . , determine f
uniquely provided the time sequence 0 < t1 < t2 < ... .

Observing (1.6), we see that the position x0 in [0,π] have to be chosen so that sin kx0 6= 0 for all
k = 1, 2, . . . . So, as in [9], we consider x0/π to be an algebraic number of second order, that is,

d

(
x0
π
,

{
0,

1

m
,

2

m
... ,

m

m
= 1

})
≥

c0
m2

, m = 1, 2, . . . (2.1)

where c0 is a constant. Then we have

d (kx0,{0,π, . . . ,kπ}) ≥ c0kπ−1, k = 1, 2, . . . ,

and hence

|sin kx0| ≥ d0k−1, k = 1, 2, . . . (2.2)
for some constant d0.

Now the following consistency result.

Lemma 2.1. For a sequence {tj}∞j=1 of later time instances satisfying t1 < t2 < t3 < ... and the choice
of x0 ∈ [0,π] described by (2.2), suppose u(x0, tj), j = 1, 2, 3, . . . are known. Then f can be determined
uniquely.

Proof: Consider the function

F0(z) :=

∞∑
k=1

ckz
k, (2.3)

where

ck := f̂kek(x0), k = 1, 2, 3, . . . (2.4)

Since the sequence {ck}∞k=1 is in l
2, F0 is holomorphic in the unit complex disk D := {z ∈ C : |z| < 1}.

Since the sequence of points zj = e
−T(tj), j = 1, 2, 3, . . . in D converges in D and F0(zj) = u(x0, tj), j =

1, 2, 3, . . . , the Identity Principle of one-complex variable implies that F0 can be determined uniquely.



4 R. KARKI

This together with (2.3) and (2.4) implies that ck, k = 1, 2, 3, . . . can be determined uniquely and hence

f̂k, k = 1, 2, 3, . . . . In this way, f can be determined uniquely.

3. Lower bound on optimal error

By following the techniques of [7, 9] (one may also see [8, 11]), we obtain a measurement algorithm

to determine a lower bound for the optimal error of recovery of f. First, we consider two continuous

mappings a and M, where a maps each f in a compact subspace B of L2 := L2[0,π] into a point in Rn
and M maps each point y ∈ Rn into a function M(y) in L2. We view the map a as an encoder or sensor,
whereas the map M as a decoder. The set {M(y) ∈ L2 : y ∈ Rn} can be viewed as an n-dimensional
manifold. An encoder a together with a decoder M forms our measurement algorithm. Using this

algorithm, we obtain point M(a(f)) =: f̄ in this manifold, which we consider as an approximation to

f, and define the manifold width δn(B,L2) as the best of optimal L2-errors

δa,M (B,L2) := sup
f∈B
||f − f̄||L2, (3.1)

or more precisely,

δn(B,L2) := inf
a,M

δa,M,n(B,L2) (3.2)

where n is fixed and the infimum is taken over all continuous maps a and M as described above.

In particular, for our approximation problem, we consider an encoder a as a map f 7→ (u1,u2, . . . ,un)
mapping Br into Rn, which extracts n samples uj := u(x0, tj), j = 1, 2, 3, . . . using the information
about f, and denote this map by an. This map is indeed continuous.

Lemma 3.1. In our measurement algorithm, the map an : f 7→ (u1,u2, . . . ,un) mapping Br into Rn is
continuous.

Proof: If f̄ ∈ Br with the Fourier sine coefficients ˆ̄fk, then for each f ∈ Br with the Fourier sine
coefficients f̂k and for each j = 1, 2, 3 . . . ,n, we have

|uj − ūj| =|
∞∑
k=1

f̂ke
−kT(tj)ek(x0) −

∞∑
k=1

ˆ̄fke
−kT(tj)ek(x0)|

≤
∞∑
k=1

|f̂k − ˆ̄fk|e−kT(tj)

≤

(
∞∑
k=1

|f̂k − ˆ̄f2k

)1/2 ( ∞∑
k=1

|e−kT(tj)|2
)1/2

≤||f − f̄||L2||{e−kT(t1)}∞k=1||l2,

from which the proof of the lemma follows.

In particular, for our approximation problem, we consider a decoder M as a map (u1,u2, . . . ,un) 7→ f̄n
mapping each n-tupple of n samples into an approximation f̄n ∈ L2 to f, and denote this map by Mn.
Thus δan,Mn,n(Br,L2) = supf∈Br ||f − f̄n||L2 , where f̄n = Mn(an(f)). Now we deduce the following.

Theorem 3.2. For a measurement algorithm with an encoder an : f 7→ (u1,u2, . . . ,un) and a decoder
Mn : (u1,u2, . . . ,un) 7→ f̄n, we have

δan,Mn,n(Br,L
2) ≥ Cn−r (3.3)

where C is a constant depending on r only.



ON APPROXIMATING INITIAL DATA 5

Proof: For Br, the idea discussed in Section 3 of [9] implies δn(Br,L2) ≥ C(r)n−r (or see [1, 7, 8]).
Therefore, for the measurement algorithm discussed above, we have

δan,Mn,n(Br,L
2) ≥ δn(Br,L2) ≥ C(r)n−r,

establishing (3.3).

Due to some technical challenges, we will not obtain an upper bound for the optimal error of approx-

imation to f. However, we will particularly prove in the next section that we can construct a sequence

of future times 0 < t1 < t2 < ... such that from the first n samples u(x0, tj), j = 1, 2, . . . ,n, f can be

approximated in L2[0,π] with an error that has an upper bound of order n−r. This is the main outcome

of this paper.

4. Optimal approximation to initial data

As we discussed at the end of the last section, our main goal is to select n future time instances

tj, j = 1, 2, 3, . . . ,n so that from the known n samples u(x0, tj), j = 1, 2, 3, . . . ,n, we can construct an

approximation to f in L2[0,π] with an accuracy of order n−r.

Theorem 4.1. Let Br be as described in (1.2), let f ∈ Br, r > 0, let a be as described in (1.1) and
let u(x,t) denote the solution to the problem (1.1). Fix x0 ∈ [0,π] such that (2.2) holds. Additionally,
fix t1 > 0 and ρ ≥ 2. There exists a sequence {tj}∞j=1 such that T(tj+1) = ρ

jT(t1), j = 1, 2, 3, . . . . If

u(x0, tj), j = 1, 2, . . . ,n are known, then there exists f̄n in L
2[0,π] such that

||f − f̄n||L2 ≤ Cn−r, (4.1)

where C is a constant that depends on d0, r, t1 and ρ.

We begin with considering an increasing sequence t1 < t2 < ... of later times. Set u(x0, tj) := U(tj),

j = 1, 2, 3, . . . . From (1.6), we have

U(tj) =

∞∑
k=1

cke
−kT(tj), j = 1, 2, 3, . . . , (4.2)

where ck = f̂kek(x0), k = 1, 2, 3, . . . . We use U(tn) to compute c1 and recursively, U(tn−k+1) to

compute each ck, k = 2, 3, 4, . . . . So, from (4.2) we obtain

c1 = e
T(tn)U(tn) −

∞∑
j=2

cje
(1−j)T(tn) (4.3)

and for each k = 2, 3, 4, . . .

ck = e
kT(tn−k+1)U(tn−k+1) −

k−1∑
j=1

cje
(k−j)T(tn−k+1) −

∞∑
j=k+1

cje
(k−j)T(tn−k+1). (4.4)

Suppose we have n samples U(tj), j = 1, 2, . . . ,n. From these n samples, we construct an approxi-

mation c̄1 to c1 as

c̄1 := e
T(tn)U(tn) (4.5)

and an approximation c̄k to each ck, k = 2, 3, 4, . . . ,n as

c̄k := e
kT(tn−k+1)U(tn−k+1) −

k−1∑
j=1

c̄je
(k−j)T(tn−k+1). (4.6)

These ck’s and c̄k’s satisfy an important estimate which is described in the next lemma.



6 R. KARKI

Lemma 4.2. Let f, a, x0 and t1 and ρ be as in Theorem 4.1. There exists a sequence {tk}∞k=1 such
that T(tk+1) = ρ

kT(t1), k = 1, 2, 3, . . . . For this sequence,

|ck − c̄k| ≤ S(t1)2ke−T(tn−k+1), k = 1, 2, 3, . . . . (4.7)

Proof: Notice that T is a strictly increasing positive continuous function of t > 0. For each

k = 1, 2, 3, . . . , we have T(t1) < ρ
kT(t1) and thus ρ

kT(t1) is in the range of T . Therefore, for each

k = 1, 2, 3, . . . , we can choose tk+1 > 0 such that T(tk+1) = ρ
kT(t1).

We use the method of induction to prove the second part of the lemma. Since f ∈ Br, we have
|cj|2 ≤ |f̂(j)|2 ≤ j−2r

∑∞
k=j k

2r|f̂(k)|2 ≤ j−2r for all j = 1, 2, 3, . . . . Then from 4.4 and 4.6,

|c1 − c̄1| ≤
∞∑
j=2

j−re(1−j)T(tn) ≤ e−T(tn)
∞∑
j=0

e−jT(t1) ≤ 2S(t1)e−T(tn), (4.8)

where S(t1) :=
∑∞
j=0 e

−jT(t1) =
1

1 −e−T(t1)
. Thus we have proved that (4.7) holds true for k = 1.

Assume that (4.7) holds true for each j ∈ {1, 2, . . . ,k − 1}, where k ≥ 2. For each k ≥ 2, we obtain
from (4.4) and (4.6) that

|ck − c̄k| ≤
k−1∑
j=1

e(k−j)T(tn−k+1)|cj − c̄j| +
∞∑

j=k+1

j−re(k−j)T(tn−k+1). (4.9)

Using the induction hypothesis and the formula for T(tj), j = 1, 2, 3, . . . , we have

k−1∑
j=1

e(k−j)T(tn−k+1)|cj − c̄j| ≤S(t1)
k−1∑
j=1

2je(k−j)T(tn−k+1)−T(tn−j+1)

=S(t1)

k−1∑
j=1

2je(k−j)T(tn−k+1)−ρ
k−jT(tn−k+1)

=S(t1)

k−1∑
j=1

2je(k−j−ρ
k−j)T(tn−k+1). (4.10)

Also, we have

∞∑
j=k+1

j−re(k−j)T(tn−k+1) ≤(k + 1)−r
∞∑

j=k+1

e(k−j)T(tn−k+1)

=(k + 1)−r
∞∑
j=0

e(−j−1)T(tn−k+1)

=(k + 1)−re−T(tn−k+1)
∞∑
j=0

e−jT(tn−k+1)

≤(k + 1)−re−T(tn−k+1)
∞∑
j=0

e−jT(t1)

≤S(t1)(k + 1)−re−T(tn−k+1). (4.11)



ON APPROXIMATING INITIAL DATA 7

From (4.9), (4.10) and (4.11), we have

|ck − c̄k| ≤S(t1)
k−1∑
j=1

2je(k−j−ρ
k−j)T(tn−k+1) + S(t1)(k + 1)

−re−T(tn−k+1)

=S(t1)e
−T(tn−k+1)


k−1∑
j=1

2je(k−j−ρ
k−j+1)T(tn−k+1) + (k + 1)−r




Since ρ ≥ 2 and x + 1 ≤ 2x for x ≥ 1, we have k − j + 1 ≤ 2k−j ≤ ρk−j for all j = 1, 2 . . . ,k − 1. So,

|ck − c̄k| ≤S(t1)e−T(tn−k+1)

k−1∑
j=1

2j + (k + 1)−r




≤S(t1)2ke−T(tn−k+1),

proving that (4.7) holds true for k ≥ 2. This completes the proof of the lemma.
Now we are ready to prove Theorem 4.1.

Under the assumptions of Theorem 4.1 and in the view of ck = f̂kek(x0), we use the relation

c̄k =
ˆ̄fkek(x0) to determine approximate Fourier sine coefficients

ˆ̄fk to f̂k, k = 1, 2, . . . ,n. So, we define
ˆ̄fk :=

c̄k
ek(x0)

, k = 1, 2, . . . ,n and then define an approximation to f as

f̄n(x) :=

m∑
k=1

ˆ̄fkek(x), where m =
⌈n

2

⌉
. (4.12)

Then the L2-error of approximation to f satisfies

||f − f̄n||2L2 ≤
m∑
k=1

|f̂k − ˆ̄fk|2 +
∞∑

k=m+1

|f̂k|2

≤
m∑
k=1

|f̂k − ˆ̄fk|2 +
∞∑

k=m+1

(
k

m

)2r
|f̂k|2

≤
m∑
k=1

|f̂k − ˆ̄fk|2 + m−2r
∞∑

k=m+1

k2r|f̂k|2

≤
m∑
k=1

|f̂k − ˆ̄fk|2 + m−2r||f||2Hr

≤
m∑
k=1

|f̂k − ˆ̄fk|2 + m−2r (4.13)

Using (2.2) and Lemma 4.2, we have

|f̂k − ˆ̄fk| =
|ck − c̄k|
|ek(x0)|

≤ C0k2ke−T(tn−k+1), k = 1, 2, . . . ,n, (4.14)

where C0 := S(t1)/d0. Since 0 ≤ m −
n

2
< 1, we have n − m ≤

n

2
< n − m + 1 and, therefore,

T(tn−k+1) = ρ
n−kT(t1) ≥ ρn/2−1T(t1) for all k = 1, 2, 3, . . . ,m. Also ln k ≤ k for all k = 1, 2, 3, . . . ,m.



8 R. KARKI

With these inequalities, we obtain from (4.13) and (4.14) that

||f − f̄n||2L2 ≤
m∑
k=1

C20k
222ke−2T(tn−k+1) + m−2r

≤C20e
−2T(tn−m+1)

m∑
k=1

k222k + m−2r

≤C20e
−2T(t1)ρn/2−1m ·m222m + m−2r

≤C20e
−2T(t1)ρn/2−1n322n + 22rn−2r. (4.15)

Notice that
j2r+322j

e2T(t1)ρ
j/2−1 ≤

e(2r+3+2 ln 2)j

e2T(t1)ρ
j/2−1 → 0 as j →∞. So, for each set of choice of r > 0, t1 > 0 and

ρ ≥ 2, there exists a constant C1 depending on r, t1 and ρ such that
j2r+322j

e2T(t1)ρ
j/2−1 ≤ C1

for all j = 1, 2, 3, . . . . In particular, we have

n2r+322n

e2T(t1)ρ
n/2−1 ≤ C1. (4.16)

Form (4.15) and (4.16), we obtain

||f − f̄n||L2 ≤ Cn−r,
where C is a constant depending on d0, t1, r and ρ. In this way, we have established (4.1) and completed

the proof of Theorem 4.1.

Finally, we consider the initial-boundary value problem

ut = −a(−∆)1/2u, u(0, t) = u(π,t) = 0, u(x, 0) = f(x), (4.17)

where a is positive real number and f ∈Br. As a special case of Theorem 4.1, we obtain the following.

Corollary 4.3. Let Br be as described in (1.2), let f ∈Br, r > 0 and let u(x,t) denote the solution to
the problem (4.17). Fix x0 ∈ [0,π] such that it satisfies (2.2). Also, fix t1 > 0 and let ρ ≥ 2. Consider
a sequence {tj}∞j=1 with tj+1 := ρ

jt1, j = 1, 2, 3, . . . . If u(x0, tj), j = 1, 2, . . . ,n are known, then there

exists f̄n in L
2[0,π] such that

||f − f̄n||L2 ≤ Cn−r, (4.18)
where C is a constant that depends on d0, r, t1 and ρ.

Proof: Set T(t) = at. We can see that all the assumptions of Theorem 4.1 are satisfied. Therefore,

the proof of the corollary follows from Theorem 4.1.

In the next example, we will particularly choose a function f and illustrate the accuracy of an

approximation to f versus tk, ρ and n of Theorem 4.1.

Example 4.4. Consider r = 2 and define f : [0,π] → R by

f(x) =
1

4
x(π −x).

Then a straightforward calculation gives us the kth Fourier since coefficient

f̂k =
2

π

∫ π
0

f(x) sin kx dx =
1

πk3
(
1 + (−1)k+1

)
and also

||f||2Hr =
∞∑
k=1

k2r|f̂k|2 ≤
∞∑
k=1

k2r ·
4

π2k6
=

4

π2

∞∑
k=1

1

k2
=

4

π2
·
π2

6
≤ 1.



ON APPROXIMATING INITIAL DATA 9

Thus f ∈ Hr([0,π]). Consider the following problem

ut = −2t(−∆)
1
2 u, u(x, 0) = u(x,π) = 0, u(x, 0) = f(x) (4.19)

Then T(t) = t2, t ≥ 0. Due to (1.6), the solution to this problem is

u(x,t) =

∞∑
k=1

e−kt
2

f̂kek(x). (4.20)

Fix x0, t1 and ρ as in Theorem 4.1 and consider the time sequence {tj}∞j=1 as t
2
j+1 = ρ

jt21. Consider

n values u(x0, tj), j = 1, 2, 3, . . . ,n. Using these n values, we will determine an approximation f̄n to f

and will demonstrate the desired accuracy of f̄n versus tk, ρ and n.

As in the proof of Theorem 4.1, we will use u(x0, tn) to obtain an approximation c̄1 to c1 and

u(x0, tn−k+1) to obtain an approximation c̄k to ck for k = 2, 3, . . . ,n. More precisely, from u(x0, tn) =∑∞
k=1 cke

−kt2n where ck = f̂kek(x0), we have

c1 = e
t2nu(x0, tn) −

∞∑
j=2

cje
(1−j)t2j

and from u(x0, tn−k+1) =
∑∞
k=1 cke

−kt2n−k+1 , k = 2, 3, . . . ,n, we have

ck = e
kt2n−k+1u(x0, tn−k+1) −

k−1∑
j=1

cje
(k−j)t2n−k+1 −

∞∑
j=k+1

cje
(k−j)t2n−k+1.

Set c̄1 = e
t2nu(x0, tn) and

c̄k = e
kt2n−k+1u(x0, tn−k+1) −

k−1∑
j=1

c̄je
(k−j)t2n−k+1, k = 2, 3 . . . ,n.

Using the method of the proof of Lemma 4.2, we get

|ck − c̄k| ≤ S(t1)2ke−t
2
n−k+1, k = 1, 2, 3, . . . ,n

where S(t1) =
∑∞
j=1 e

−jt21 . Thus the kth Fourier coefficient f̂k and its approximation
ˆ̄fk = c̄k/ek(x0)

give

|f̂k − ˆ̄fk| ≤
|ck − c̄k|
|ek(x0)|

≤
S(t1)

d0
k2ke−t

2
n−k+1

where d0 as in (2.2). Then an approximation to f defined by f̄n =
∑m
k=1

ˆ̄fkek where m = dn2e satisfies

||f − f̄n||2L2 ≤
m∑
k=1

|f̂k − ˆ̄fk|2 + m−2r

≤
m∑
k=1

S(t1)
2

d20
k222ke−2t

2
n−k+1 + m−2r

≤
S(t1)

2

d20
e−2t

2
n−m+1

m∑
k=1

k222k + m−2r

≤
S(t1)

2

d20
e−2t

2
n−m+1n222n ·n +

(n
2

)−2r
=n−4

(
S(t1)

2

d20
e−2ρ

n/2−1t21n722n + 4

)
.



10 R. KARKI

But e−2ρ
j/2−1t21j722j = e−2ρ

j/2−1t21+7 ln j+2j ln 2 ≤ C0 for some constant C0 depending on t1 and ρ because{
e−2ρ

j/2−1t21+7 ln j+2j ln 2
}∞
j=1

is a convergent sequence. Using this into the last inequality, we get

||f − f̄n||L2 ≤ Cn−2

for some constant C depending on (r = 2,) t1 and ρ, thereby verifying (4.1).

5. Remarks

We may ask whether our approximation method studied in the preceding sections work for more

general problems like the following

ut = −a(−∆)ηu, u(0, t) = u(π,t) = 0, u(x, 0) = f(x), (5.1)

where a is a positive real number, η ∈ (0, 1] and f ∈Br, and

ut = −a(t)(−∆)ηu, u(0, t) = u(π,t) = 0, u(x, 0) = f(x), (5.2)

where a is a positive continuous function of t > 0, η ∈ (0, 1] and f ∈Br. Even the method has worked
for various special cases of these problems such as when a = 1 and η = 1 (see [9]); when η = 1 (see [1])

and when η = 1
2
, the method may require more advanced analytical tools related to spectral properties

of unbounded self-adjoint operators on Hilbert spaces (see [12, 15, 16]) to address these general cases.

We may also ask whether it is possible for the current method to be applied for evolutionary equations

with other boundary conditions such as the Neumann and Robin boundary conditions.

It would be worth answering any of the above questions.

References

[1] R. Aceska, A. Arsie, R. Karki, On near-optimal time samplings for initial data best approximation, Matematiche

(Catania) 74 (2019), no. 1, 173-190.

[2] G. Alberti, G. Bellettini, A nonlocal anisotropic phase transitions I. The optimal profile problem, Math. Ann. 310

(1998), no. 3, 527- 560

[3] X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math.

224 (2010), no. 5, 2052-2093.

[4] L. Caffarelli, L. Silvetre An Extention Probelm Related to the Fractional Laplacian, Communications on Partial

Differential Equations, 32 (2007), 1245-1260.

[5] W. Craig, M.D. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion 19(1994),

no. 4, 367- 389.

[6] P. Constantin, M. Ignatova, Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications,

Int. Math. Res. Not. 2017 (2017), no. 6, 1653–1673.

[7] R. DeVore, R. Howard, C. Micchelli, Optimal nonlinear approximation, Manuscipta Math. 63 (1989), 469-478.

[8] R. DeVore, G. Kyriazis, D. Leviatan, V. Tikhomirov, Wavelet compression and nonlinear n-widths, Adv. Comput.

Math. 1 (1993), 197-214.

[9] R. DeVore, E. Zuazua, Recovery of an initial temperature from discrete sampling, Math. Models Methods Appl. Sci.

24 (2014), 2487.

[10] A. Garroni, G. Palatucci, A singular perturbation result with a fraction norm, in: Variational Problems in Material

Sience, in: Progr. Nonlinear Differential Equations Appl. vol 68, Birkhauser, Basel, 2006, pp. 111-126.

[11] D. S. Gulliam, B. A. Mair, C. F. Martin, Determination of initial states of parabolic system from discrete data,

Inverse Problems 6 (1990), 737-747.

[12] R. Karki, Sobolev gradient & application to nonlinear pseudo-differential equations, Neural Parallel Sci. Comput. 22

(2014), no. 3, 359-373.

[13] P.I. Naumkin, I.A. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical

Society, Providence, RI, 1994.

[14] J. Pöschel, E. Trubowitz, Inverse Spectral Theory, Pure and Applied Mathematics, Vol. 130, Academic Press, 1987.



ON APPROXIMATING INITIAL DATA 11

[15] K. Schmudgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Text in Mathematics 265, Springer-

Verlag, NY, 2012.

[16] G. Sell, Y. You Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Vol. 143, Springer-Verlag,

2002.

Corresponding author, Natural Science and Mathematics, Indiana University East, Richomond, IN 47374,

U.S.A.

Email address: rkarki@iue.edu


	1. Introduction
	2. Choice of x0 and consistency of recovery
	3. Lower bound on optimal error
	4. Optimal approximation to initial data
	5. Remarks
	References