Inverse Coefficient Problem of the Parabolic Equation

Using Periodic Boundary and Integral Overdetermination Conditions

Ayaulym Zhaksylykova, Kamilla Zhamalbekova, Alpamys Tekebay,

Kazhymurat Berikbayev

Department of Mathematics

October 11, 2024

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 1 / 10
 Outline

1 Introduction

2 Problem Formulation

3 Theorems and Proofs

4 Numerical Methods

5 Examples

6 Conclusion

7 References

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 2 / 10
 Introduction

Introduction

We study an inverse problem for a parabolic equation with

time-dependent diffusion coefficient.
The equation has periodic boundary conditions and integral
overdetermination conditions.
We focus on the existence, uniqueness, and continuous dependence of
the solution.

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 3 / 10
 Problem Formulation

Problem Formulation
Consider the parabolic equation:

ut = a(t)uxx + F (x, t), (1)

with initial condition:

u(x, 0) = ϕ(x), 0 ≤ x ≤ 1, (2)

and periodic boundary conditions:

u(0, t) = u(1, t), (3)

ux (0, t) = ux (1, t), (4)

and the overdetermination condition:

Z 1
xu(x, t) dx = E (t). (5)
0

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 4 / 10
 Theorems and Proofs

Existence and Uniqueness Theorem

Theorem
Let assumptions on data hold:
E (t) ∈ C 1 [0, T ], ϕ(x) ∈ C 4 [0, 1], F (x, t) ∈ C (DT ).
Then, the inverse problem has a unique solution.

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 5 / 10
 Theorems and Proofs

Proof of Existence and Uniqueness

Using the Fourier method, we represent the solution as:

∞  Z t 
−(2πn)2 0t a(s) ds −(2πn)2 τt a(s) ds
X R R
u(x, t) = ϕn e + Fn (τ )e dτ sin(2πnx).
n=1 0
(6)
The uniqueness is established by applying the fixed-point theorem.

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 6 / 10
 Numerical Methods

Crank-Nicolson Method

We apply the Crank-Nicolson method to solve the equation

numerically.
This method is stable and has second-order accuracy.
Discretization of the equation:

uij+1 − uij a(t)  j+1 j+1 j+1

 F j+1 + F j
i i
= u i−1 − 2ui + u i+1 + . (7)
τ 2h2 2

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 7 / 10
 Examples

Example

Consider the inverse problem with the following data:

F (x, t) = (2π)2 sin(2πx)e t ,

ϕ(x) = sin(2πx),
1
E (t) = − e −t .
2π
The analytical solution is:
1
a(t) = + e 2t , u(x, t) = sin(2πx)e −t .
(2π)2

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 8 / 10
 Conclusion

Conclusion

We established the existence and uniqueness of the solution for the

inverse coefficient problem.
The numerical methods confirm the theoretical results with high
accuracy.
Future work could involve extending this method to higher
dimensions.

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 9 / 10
 References

References

F. Kanca, ”Inverse Coefficient Problem of the Parabolic Equation,”

Abstract and Applied Analysis, vol. 2013, Article ID 659804, 2013.

Ayaulym Zhaksylykova, Kamilla Zhamalbekova,

Inverse
Alpamys
Coefficient
Tekebay,Problem
Kazhymurat
of Parabolic
Berikbayev
Equation
(Department
October
of Mathematics)
11, 2024 10 / 10