Summary of Doctoral Thesis in Mathematics: On some problems of identifying unknown source term for parabolic equations

Research purposes: We consider some inverse source problems for parabolic equations, focus on three topics: First, we give stability estimates. Second, we propose regularization methods to solve these problems. Third, we set up algorithms and give numerical examples to illustrate the performance of the proposed regularization methods in this thesis.

VINH UNIVERSITY

LUONG DUY NHAT MINH

ON SOME PROBLEMS

OF IDENTIFYING UNKNOWN SOURCE TERM

FOR PARABOLIC EQUATIONS

CODE: 9 46 01 02

A SUMMARY OF DOCTORAL THESIS IN MATHEMATICS

Nghe An – 2021

1 Assoc Prof Dr Nguyen Van Duc

2 Dr Nguyen Trung Thanh

On the hour day month year

Thesis is stored in at:

1 National Library of Vietnam

2 Nguyen Thuc Hao Center of Information and Library, Vinh University

1 Rationale

The problems of identifying unknown source terms for parabolic equations hasbeen studied by many scientists from 1960s Mathematicians who have works onthis problem are Cannon, Dinh Nho Hao, Dang Duc Trong, Hasanov, Isakov, Li,Savateev, Prilepko, Yang, Fu,

The above problem is usually ill-posed in Hadamard’s sense A problem is calledwell-posed in the Hadamard’s sense if it satisfies all of the following conditions:i) The solution of the problem exists

ii) The solution of the problem is unique

iii) The solution depends continuously on the data of the problem

If at least one of the three above conditions is not satisfied, the problem is calledill-posed In ill-posed problems, a small error of data can also lead to large deviation

of the solution Therefore, ill-posed problems are often more difficult to solve thanwell-posed problems because the data used in ill-posed problems are often generated

by measurements, so there is inevitable error Furthermore, many calculations areonly performed approximately To solve an ill-posed problem, scientists often proposeregularization methods, that is to use the solution of a well-posed problem to makethe approximate solution to the original ill-posed problem

Research on the problems of identifying unknown source terms for parabolic tions often focuses on three main topics:

equa-i) The uniqueness of solutions

ii) Stability estimates

iii) Regularization methods and the numerical methods

There has been a lot of research on the problems of identifying unknown sourceterms of parabolic equations because they are the mathematical models of practicalproblems such as determining the heat source in the heat transfer equation, identify-ing the polluted water source, Currently, there are many open problems related tothe problem of determining the source terms in parabolic equations that need to bestudied, in which the results of stability estimates and regularization for parabolicequations with time-dependent coefficients has been fully studied, with only a few

results on the uniqueness of the solution of this type of problem.

The research direction of the problem of determining the source of the fractionalparabolic equations has received the attention of many scientists However, most

of the results listed above are for fraction parabolic equations by time or spatialvariable, with only a few results for fractional parabolic equations for both spatialand time variables Regarding the results of stability estimates and regularizationfor the problem of determining the source of parabolic equations in Banach space,according to our search, there are only a few relevant results

For the above reasons, we choose research topics for our thesis as: "On someproblems of identifying unknown source term for parabolic equations"

2 Research purposes

We consider some inverse source problems for parabolic equations, focus on threetopics: First, we give stability estimates Second, we propose regularization methods

to solve these problems Third, we set up algorithms and give numerical examples

to illustrate the performance of the proposed regularization methods in this thesis

3 Research subjects We focus on identifying unknown source terms of parabolicequations in three cases:

i) Parabolic equations with time dependent coefficients in Hilbert space L2(Rn);ii) Time-space fractional parabolic equations in Hilbert space L2(Rn);

iii) Parabolic equations in Banach spaces

4 Research scopes

We research stability estimates, regularization methods and numerical methods

to solve some problems of identifying unknown source terms for parabolic equations

5 Research methods

We used the logical reasoning method based on the previously known results Wealso use numerical methods to solve these inverse source problems

6 Scientific and practical meaning

The thesis contributes to enriching the research results in the field of inverseproblems The thesis has achieved some results on stability estimates, regularizationmethods and numerical methods to solve the inverse source problem of parabolicequations This thesis is the reference for students, graduated students

7 Overview and structure of the thesis

7.1 Overview of some issues related to the thesis

To facilitate the introduction of research results related to the inverse sourceproblems for parabolic equations, we take a concrete example of the linear parabolicequation in Hilbert space Let T be a positive real number, X is a Hilbert space

with the norm k · k, u : [0, T ] → X is a function from [0, T ] to X and F ∈ X Weconsider the initial value problem







u0(t) + Au(t) = F, t ∈ (0, T ),u(0) = 0,

(1)

where Ais the unbounded linear operator onX The problem (1) is forward problem,

in which we need to find uwhenF is known The problem of identifying an unknownsource for (1) is to find the source function F from the measurements of the function

u This is an inverse problem There are many different types of measurements inuse, for example: boundary measurements, at the final time measurements, or mea-surements at some discrete points, Therefore, there are many types of problems

of identifying an uknown source for parabolic equations Due to the ill-posedness,the solution of the problem does not always exist and if it exists, the solution maynot depend continuously on the data of the problem This makes the inverse sourceproplem difficult to solve Usually, mathematicians have to propose regularizationmethods to solve the ill-posed problems There are many regularization methods tosolve the inverse source problems for parabolic equations, such as: quasi-reversibilitymethod, Tikhonov method, finite element method, variational method, conjugategradient method, mollification method,

Here, we would like to summarize the types of inverse source problems for parabolicequations that we investigate and summarize the main results that we has achieved.i) First, we considered the inverse source problem for parabolic equations withtime dependent coefficients in Hilbert spaceL2(Rn) Find a pair of functions{u(x, t), f (x)}

space and time With domain Ω ⊂ Rn, they considered problem

We illustrated the above theoretical results by using numerical examples

ii) Second, we considered the inverse source problem for a time-space fractionalparabolic equation in Hilbert space L2(Rn): Find a pair functions {u(x, t), f (x)}

of finding the function u(t, x) and f (x) with t ∈ (0, T ) and x ∈ Ω = (−1, 1), whichsatisfies

(6)

here, ΩT := (0, T ) × Ω, r > 0 is a parameter, f (x) ∈ L2(Ω), h(t, x) is given,

β ∈ (0, 1), α ∈ (1, 2) are fractional order of the time and the space derivatives,respectively,T > 0is the final time and ∂β

∂tβ is the Caputo fractional derivative Withthe data at the final time T, the authors proved the existance and the uniqueness

of the solution In a research, Li and Wei considered the problem determining timedependent term p(t) of the source function f (x)p(t), which satisfies

here, ΩT := Ω × (0, T ], Ω ⊂ Rn; T > 0 is the final time; α ∈ (0, 1), β ∈ (1, 2)

are fractional order of the time and the space derivatives, respectively; ∂0+α is theCaputo fractional derivative They proved the uniqueness and proposed the sta-bility estimate In 2014, Tuan et al considered an inverse problem to identify anunknown source term in a space time fractional diffusion equation In addition tothe similar information of problem (6), they also replaced the first equation in (6) by

∂β

∂tβu(t, x) = −rβ(−∆)α/2u(t, x) + f (x)h(t) At this time, function h depends only

on the time variable t Moreover, in equation u(T, x) = ϕ(x), then x ∈ Ω instead of

x ∈ ¯Ω They proposed the Fourier truncation method to solve this problem Theyalso proposed a priori and a posteriori parameter choice rules and analyzed them.But no numerical examples were given

For the inverse source problem for a time-space fractional parabolic equation inHilbert space L2(Rn) in our thesis, we proved a H¨older-type stability estimate ofoptimal order (Theorem 3.2.3 and Remark 3.2.4) We regularized this problem bythe mollification method and we established the H¨older-type error estimates for theregularized solutions using both a priori and a posteriori parameter choice rules (The-

orem 3.3.2 and 3.3.6) We also demonstrated the above results by using numericalexamples.

iii) Third, we considered the inverse source problem for parabolic equation inBanach spaces SupposeX is the Banach space with norm k · k, A : D(A) ⊂ X → X

is the unbounded linear operator such that −A generates an analytic semigroup

{S(t)}t≥0 on X, with D(A) is the domain of A and assume that D(A) is dense in

X For t ∈ [0, T ], denote u(t) is a function from [0, T ] to X and F ∈ X, we identifysource function F from

u(T ) = g,

(8)

with g ∈ X is the data at the final time T To the best of our knowledge, theresults about this inverse source problem are not popular Some of the earliestworks on inverse source problems for parabolic equations in Banach spaces were due

to Iskenderov and Tagiev; Rundell The uniqueness of (8) is proven by Eidelman;Tikhonov In case of F is the time dependent function, the authors also proved theuniqueness In 2005, Tikhonov and Eidelman considered an inverse source problem

in Banach space E, with A is a closed linear operator, and the domain D(A) ⊂ E

(the domain may not dense in E) Let T > 0 and function ϕ 6= 0 is continous on

[0, T ] They found {u(t), p} satisfied

(9)

They proved the uniqueness of the solution of the inverse problem For a Banachspace X, k · k is the norm of X, in 1980, Rundell considered a problem find a pairfunctions {u(t), f } satisfy

(10)

here A is a linear operator in X and f ∈ X, additional information is of two values

of u at two fixed points (t = 0 and t = T > 0) With assumption u0, u1 ∈ D(A)

(D(A) is the domain of A, A−1 : X → D(A) exists and A generates a stronglycontinuous semigroup of operators {S(t)}t≥0 such that kS(t)k < 1, the authorsproved the existance and uniqueness of this problem and (u(t), f ) were described as

follow: u(t) = S(t)u0+

t

R

0

S(t − τ )f dτ and f = (I − S(T ))−1(Au1+ AS(T )u0), with

I is the identity operator in X In 1991, Eidelman used the theory of semigroups

of operators to prove the uniqueness of an inverse source problem in Banach space

E, the author considered the problem of finding a pair of functions {v(t), p} whichsatisfied ∂v

∂t = Av + f (t) + p, here A is an unbounded linear operator, f (t) is a

continous function on [0, t1] with values in E, p ∈ E is an unknown parameter andthey set the boundary conditions: v(0) = v0, v(t1) = v1 In 2007, Prilepko et alregularized an inverse source problem in Banach space, but they did not propose theconverate rates and numerical examples were not considered In 2013, Hasanov et alpresented a semigroup approach to propose a representation formula for a solution

of an inverse source problem for the heat equation ut = Au + F with measured data

at the final time uT(x) := u(x, T ), and they proved the uniquess of this problem

In our thesis, to regularize the inverse source problem for a parabolic equation inBanach spaces, base on the theory of semigroups of operators, we proposed a newregularization method and proved H¨older-type error estimates for the regularizedsolutions are proved for both a priori and a posteriori regularization parameter choicerules (Theorem 4.2.7 and 4.2.9) We also demonstrated the above theoretical results

by using numerical examples

7.2 Organization of the thesis

The main content of the thesis is presented in 04 chapters

In Chapter 1, we present the auxilary results, which are used in thesis

In Chapter 2, we present the new results of stability estimates and the obtainednew regularization for an inverse source problem for parabolic equation with timedependent coefficients in Hilbert space L2(Rn) by mollification method

In Chapter 3, we present the new result of stability estimate and the obtainednew regularization for an inverse source problem for time-space fraction parabolicequation in Hilbert space L2(Rn) by mollification method

In Chapter 4, we present the obtained new regularization results of the inversesource problem for parabolic equation in Banach spaces

The main results of the thesis were presented at the seminar of the AnalysisDepartment, Institute of Natural Pedagogy - Vinh University, Scientific seminar

"Researching and teaching Mathematics to meet the current education innovationrequirements" at Vinh University, Nghe An on September 21st, 2019 These resultshave published in 03 articles, including 01 article on Inverse Problems in Science andEngineering (SCIE, IF: 1.314), 01 article on Applicable Analysis (SCIE, IF: 1.107)and 01 article on Applied Numerical Mathematics (SCIE, IF: 1.979)

CHAPTER 1 AUXILIARY RESULTS

1.1 Some results in Analysis

Definition 1.1.1 For v :Rn → R is a measurable function, and p is a real numberwhich satisfies 1 ≤ p < ∞, Lp(Rn) is defined by

Z

Rn

k(x − y)f (y)dy = 1

(√2π)n

Z

Rn

eiξ·xv(ξ)dξ, x ∈ Rn,

here ξ · x is the inner product of two vectors ξ and x in Rn We call F(v)(ξ) and

F−1(v)(x) are Fourier transform and inverse Fourier transform of v

Definition 1.1.5 For p > 0, Hp(Rn) space is defined by

And we define norm ||| · |||q in Hp(Rn) space as follow.

Definition 1.1.6 Let q is a positive number For v : Rn → R is a measurable

Definition 1.1.9 For a positive vector η := (η1, , ηn) ∈ Rn, we define Mη,2(Rn)

as the collection of all entire functions of exponential type η which as functions of

is called the Dirichlet kernel

In the sequel, we use the following sets:

Mν := {x ∈ Rn : |xj| < ν, j = 1, 2, , n} and Qν := Rn \ Mν

Definition 1.1.14 We define the Gamma function by

Γ(z) =

Z ∞ 0

1.2 Some results about semigroup of bounded linear operators

Definition 1.2.2 LetAbe a linear operator with domainD(A)andD(A) is density

on Banach space X

i) Number λ ∈ C is said to be a regular value of A if the bounded linear operator

(λI − A)−1 exists on X The set of all regular values of the A operator is calledthe resolvent set and denoted by ρ(A) R(λ, A) = (λI − A)−1 is called resolvent

of A, with λ ∈ ρ(A)

ii) The set σ(A) = C\ ρ(A) is called the spectrum set of A.

Definition 1.2.4 Let Ω = {z : ϕ1 < arg z < ϕ2, ϕ1 < 0 < ϕ2} be a sector For

z ∈ Ω, let S(z) be a bounded linear operator The family {S(z)}z∈Ω is an analyticsemigroup in Ω if

i) For all z ∈ Ω, z → S(z)x is analytic in Ω;

ii) S(0) = I and for every x ∈ X, we have lim

z→0, z∈ΩS(z)x = x;iii) For z1, z2 ∈ Ω, S(z1 + z2) = S(z1)S(z2)

A semigroup {S(t)}t≥0 will be analytic if it is analytic in a sector Ω containing thenonnegative real axis

Definition 1.2.5 We will call the (possibly unbounded) operator, A, a tor if A generates a uniformly bounded strongly continous holomorphic semigroup

genera-{e−zA}Rez>0 By switching to the equivalent norm

|kxk| = sup

Re z>0

ke−zAk,

if necessary, we may assume that ke−zAk ≤ 1, whenever Re z > 0

Definition 1.2.6 If A is a generator and s > 0, then

where C is a path running in the resolvent set of A from ∞e−iv to ∞eiv, with

ω < v < π We also define Ab := A−b
−1 and A0 = I For positive integral values

of b the definition (1.2) coincides with the classical definition of (A−1)n

1.3 Ill-posed problem

1.3.1 Definition of ill-posed problem

In this section, we present the ill-posed problem

1.3.2 Regularization of ill-posed problem

In this section, we present the regularization for ill-posed problem and the larization operator

regu-1.3.3 Optimal oder

In this section, we present some content on optimal order

1.4 The mollification method

In order to solve the ill-posed problems, scientists must propose regularizationmethods In our thesis, we used the mollification method to solve two inverse sourceproblems for parabolic equations In this section, we summarize some content related

to mollification method