465 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 

 
Timewise–dependent Coefficients Identification Problems for Third-Order 

Pseudo-Parabolic Equations from Nonlocal Extra Conditions 

 

Sayl Gani1 , Mohammed S. Hussein2*  , and Taysir E. Dyhoum3  

1Department of Computer Techniques Engineering, Imam Al-Kadhum College, Baghdad, Iraq. 
2Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. 

3Department of Computing and Mathematics, Faculty of Science and Engineering, Manchester Metropolitan 

University, Manchester, UK.  

 

*Corresponding Author.  

 

 Received:10 September 2023                 Accepted:13 November 2023           Published:20 January 2025  

 doi.org/10.30526/38.1.3671 

 
Abstract  

     This study aims to find the time-dependent potential terms in the two inverse problems of the 

third-order pseudo-parabolic with initial and various boundary conditions supplemented by the 

overdetermination data. The nonlinear inverse problems have significant applications in physics 

and engineering fields. We proved the existence and uniqueness of the solution of the two 

problems are being proved, but they still need to be proposed (since tiny perturbations in input 

data cause considerable errors in the output potential term). Consequently, the regularized 

methods should be employed. A finite difference schema is used for solving direct problems. In 

contrast, the inverse problems were reformulated as nonlinear least-square minimization and 

solved efficiently by optimizing MATLAB routine lsqnonlin. Tikhonov's regularization method 

was applied to get stable results. The numerical results were explained by presenting a test 

example for each problem. In addition, the stability was discussed by utilizing the Von Neumann 

stability analysis. The results showed that the time-dependent potential terms were reconstructed 

successfully and were stable and accurate. 

Keywords: Von Neumann stability analysis, Finite difference method, Tikhonov regularization 

method, Pseudoparabolic inverse problem, Inverse problem. 

 

1. Introduction 

     For the inverse problems, identifying the unknown coefficients of the parabolic problem has 

many applications in engineering and science. Many researchers have identified the unknown 

coefficients of the parabolic inverse problem recently. For example, Hussian et al. investigated 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://doi.org/10.30526/38.1.3501
https://orcid.org/0000-0003-4595-2251
mailto:saylgani@alkadhum-col.edu.iq
https://orcid.org/0000-0002-9456-4303
mailto:mmmsh@sc.uobaghdad.edu.iq
https://orcid.org/0000-0002-1761-5904
mailto:T.Dyhoum@mmu.ac.uk


IHJPAS. 2025, 38 (1) 
 

466 
 

the parabolic inverse problem to identify the multiple time-dependent coefficients of thermal 

problems with unknown free boundary conditions in (1). Hussein and Lesnic investigated the 

one-dimensional parabolic inverse problem for determining the two time-dependent conductivity 

with Cauchy data and heat capacity storage in (2). While in (3), the authors presented two 

parabolic inverse problems for identifying the space and time-dependent coefficients from the 

overdetermination conditions. In (4), the authors presented the one-dimensional parabolic inverse 

problem for recovering the heat source and time-dependent thermal conductivity with the heat 

flux overdetermination condition for the other related work see (5-8). 

The pseudo-parabolic equations of a higher order play an essential role in the mathematical 

modelling of moisture transfer, fluid filtration and heat propagation (9) and (10). The pseudo-

parabolic inverse problems have been utilized in modelling various phenomena such as the wave 

processes, chemical, engineering, diffusion, plasma physics and heat conduction (11). In 

addition, they have many applications in real-life phenomena, such as the theory of small 

oscillation of a rotating fluid (12) and the infiltration of homogeneous fluids in strata (13). 

Moreover, Lyubonova and Tani (14) discussed the stabilization of a multi-dimensional pseudo-

parabolic inverse problem with a coefficient of piezo conductivity and the regularity of the 

solution. In (15), the authors analyzed the uniqueness and existence of the solution of the third-

order pseudo-parabolic inverse problem with periodic and integral conditions. Abylkairov and 

Khompysh (16) studied the existence and uniqueness of a solution for the right side of the pseudo-

parabolic inverse problem, which was described as the motion of Kelvin - Voight fluids. Antotsev 

et al. (17) proved the unique solvability for the pseudo-parabolic inverse problem with a P-

Laplacian and under a nonlocal integral overdetermination condition by using the Galerkin 

method. Many other researchers have examined the pseudo-parabolic inverse problem to identify 

the unknown time-dependent coefficients. In studies (18, 19), the pseudo-parabolic inverse 

problem was presented to determine the unknown coefficient of filtration and diffusion. In 

addition, the asymptotic behaviour of the pseudo-parabolic inverse problem to determine 

unknown source terms with integral conditions has been considered by Yaman and G¨ozükizil 

(20).  

A few years ago, the numerical solution containing unknown coefficients was investigated by 

Beshtokv for the pseudo-parabolic equations from a class of the third-order (21). The third-order 

pseudo-parabolic equations result from the problem with heat and moisture transmission and 

fluid filtration (22,23). An inverse problem of reformulating an unknown potential element has 

been studied (24). Moreover, both Huntual et al.  (25,26) and Ramazanova et al. (27) examined 

the fourth-order inverse problem for identifying the time-dependent potential coefficient from 

additional conditions and the cubic-spline method as a direct method. 

In this work, two pseudo-parabolic inverse problems were presented from the third-order 

equation to recover the potential time-dependent coefficient numerically with different boundary 

conditions. Since the periodic conditions were used for both problems, the Neuman and non-

local integral conditions were used with the first and second problems, respectively. For both 

problems, the over-specification data was utilized for recovering the unique potential terms. The 

uniqueness and existence were proved in (28) for the first problem and in (29) for the second 



IHJPAS. 2025, 38 (1) 
 

467 
 

one. 

The formation of this study is as follows: Section 2 presents the mathematical form of the Inverse 

Problem I and II, which are contained in Subsection 1. The FDM is used to discretize the direct 

problems I and II, subsection 2, and the stability analysis is provided in subsection 3, Examples 

of tests for direct problems I and II. Section 3 presents the numerical technique of functional 

minimization and numerical results of the inverse problems I and II. Finally, in Section 4, the 

conclusions are highlighted. 

 

2. Mathematical formulation of the inverse problem I and II 

     In rectangle domain 𝑄𝑇 ≔ {0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑡 ≤ 𝑇}, consider the inverse problems of 

identifying a pair of functions (𝑢(𝑥, 𝑡),  𝑝(𝑡)),  which satisfy the 1D pseudo-parabolic equation  

𝜕𝑢(𝑥, 𝑡)

𝜕𝑡
= 𝑏

𝜕3𝑢(𝑥, 𝑡)

𝜕𝑥2𝜕𝑡
+ 𝑎(𝑡)

𝜕2𝑢(𝑥, 𝑡)

𝜕𝑥2
+ 𝑝(𝑡) 𝑢(𝑥, 𝑡)

+ 𝑓(𝑥, 𝑡)          (𝑥, 𝑡) 𝜖 𝑄𝑇                                                                                              (1) 

the initial condition  

𝑢(𝑥, 0) = 𝜑(𝑥),                      0 ≤ 𝑥

≤ 1,                                                                                                                                    (2) 

the periodic condition 

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

≤ 𝑇,                                                                                                                                    (3) 

the Neumann condition 

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

𝑇                                                                                                                                                                    (4)       

the nonlocal integral condition  

∫𝑢(𝑥, 𝑡) = 0,

1

0

                           0 ≤ 𝑡

≤ 𝑇                                                                                                                                    (5) 

and the additional conditions are  

𝑢 (
1

2
, 𝑡) + ∫𝑢(𝑥, 𝑡)𝑑𝑥

1

0

= ℎ1(𝑡),              0 ≤ 𝑡

≤ 𝑇                                                                                                                                  (6) 

𝑢(𝑥0, 𝑡) = ℎ2(𝑡),                                0 ≤ 𝑡

≤ 𝑇,                                                                                                                                (7)  

 

where 𝑥0 ∈ (0,1),  and 𝑏 > 0 is a given number. 

We call Equations (1) - (4), (6) as the inverse problem I (IP-I) where 𝑎 is constant and the 

Equations (1)-(3), (5), (7) as the inverse problem II (IP-II), where 𝑎 is time-dependent function. 

In particular, if we put 𝑏 = 0 in Equation (1), then the resulting is a heat equation which has been 

investigated previously by many authors such as(7, 8, 30). The functions 𝑓, 𝜑 and ℎ1 and ℎ2 are 



IHJPAS. 2025, 38 (1) 
 

468 
 

given functions. In these problems, 𝑝(𝑡) is a potential term and (𝑈𝑋, 𝑡) is the temperature 

distribution, and these functions are unknown. These problems have been utilized in the 

modelling of various phenomena such as wave processes, chemical, engineering, diffusion, 

plasma physics and heat conduction (11). The unique solvability of IP-I has been established in 

(28), whilst for IP-II in (29) and the following their unique solvability theorems:  

 

Definition: The classical solution of the IP-I and IP-II is the pair {𝑢(𝑥, 𝑡), 𝑝(𝑡)} satisfies the 

following properties: 

i- 𝑢(𝑥, 𝑡) is continuous in 𝑄𝑇 with all its derivatives.      

ii- 𝑝(𝑡) is continuous on [0, 𝑇]. 

iii- All Equation (1)– (5) conditions are satisfied in the ordinary sense. 

Lemma 1 for IP-I: Suppose that 𝑎 > 0,    𝑏 > 0,    𝜑(𝑥) ∈  𝐶[0,1], 𝑓(𝑥, 𝑡) ∈ 𝐶(𝑄𝑇), ℎ1(𝑡)   ∈

 𝐶1[0, 𝑇], ℎ1(𝑡) ≠ 0 for (0 ≤ 𝑡 ≤ 𝑇) and 𝜑 (
1

2
) + ∫ 𝜑(𝑥)

1

0
𝑑𝑥 = ℎ1(0). Then the problem of 

defining the functions 𝑢(𝑥, 𝑡) and 𝑝(𝑡) is equivalent to the problem of finding the solution of 

problem Equations (1)– (4), (6) possessing the properties (i) and (ii) of the solution of problem 

Equations (1)–(4), (6) from relations Equations (1)–(4), and  

ℎ1
′(𝑡) + 𝑏 (𝑢𝑡𝑥(0, 𝑡) + 𝑢𝑡𝑥𝑥 (

1

2
, 𝑡)) + 𝑎 (𝑢𝑥(0, 𝑡) + 𝑢𝑥𝑥 (

1

2
, 𝑡))

= 𝑝(𝑡)ℎ1(𝑡) + 𝑓 (
1

2
, 𝑡)

+ ∫𝑓(𝑥, 𝑡)

1

0

𝑑𝑥,      (0 ≤ 𝑡 ≤ 𝑇)                                                                                   (8) 

Theorem 1 for IP-I. Let the problem Equations (1) – (4), (8) satisfy the following conditions: 

1. 𝜑(𝑥) ∈ 𝐶2[0,1], 𝜑′′′(𝑥)  ∈  𝐿2(0,1),

𝜑(0) =  𝜑(1),    𝜑′(1) = 0, 𝜑′′(0)

=  𝜑′′(1);                                                                                                                                                        (9) 

 

2. 𝑓(𝑥, 𝑡)  ∈  𝐶(𝑄𝑇),   𝑓𝑥(𝑥, 𝑡)  ∈  𝐿2(𝑄𝑇),

𝑓(0, 𝑡)

= 𝑓(1, 𝑡),   (0 ≤ 𝑡

≤ 𝑇).                                                                                                                                                              (10) 

 

3. 𝑎 > 0, 𝑏 > 0,    ℎ1(𝑡) ∈ 𝐶
1[0, 𝑇],   ℎ1(𝑡)

≠ 0  (0 ≤ 𝑡

≤ 𝑇).                                                                                                                                                             (11) 

 

4. 𝜑 (
1

2
) + ∫ 𝜑(𝑥)𝑑𝑥

1

0

= ℎ1(0).                                                                                                                            (12) 



IHJPAS. 2025, 38 (1) 
 

469 
 

Then IP-I has a unique solution in the ball K = 𝐾𝑅‖𝑧‖𝐸𝑇3 ≤  R =
( A(T) +  2) of the space 𝐸𝑇

3 

in Banach space. 

 

where,  

𝐴(𝑇) =
1

3
‖𝜑(𝑥)‖𝐿2(0,1) +

1

3
√𝑇‖𝑓(𝑥, 𝑡)‖𝐿2(𝑄𝑇)

+ ‖ℎ1
−1(𝑡)‖

𝐶[0,𝑇]
‖ℎ1

′(𝑡) − 𝑓 (
1

2
, 𝑡) − ∫ 𝑓(𝑥, 𝑡)𝑑𝑥

1

0

‖
𝐶[0,𝑇]

+ (
√6

2
+
4𝑎

𝑏2
𝑇 +

√3𝑎

3𝑏
‖ℎ1

−1(𝑡)‖
𝐶[0,𝑇)

)‖𝜑′′′(𝑥)‖𝐿2[0,1] 

+(
√6𝑇

2𝑏
+ 4 (

1

𝑏
+
𝑎

𝑏3
𝑇)√𝑇 +

√3𝑇𝑎

3𝑏2
‖ℎ1

−1(𝑡)‖
𝐶[0,𝑇]

) × ‖𝑓𝑥(𝑥, 𝑡)‖𝐿2(𝑄𝑇)

+ 2‖𝑥𝜑′′′(𝑥) + 3𝜑′′(𝑥)‖𝐿2(0,1)

+
√3

3
‖ℎ1

−1(𝑡)‖
𝐶[0,𝑇]

× ‖‖𝑓𝑥(𝑥, 𝑡)‖𝐶[0,𝑇]‖𝐿2(0,1)
,                                                                                                                  (13) 

𝐵(𝑇) =  (1 +
1

𝑏
(2√2 + √3) +

4√2

𝑏
(1 +

𝑎

𝑏2
𝑇))𝑇

+ (
𝑎

𝑏
𝑇 + 𝑏) ‖ℎ1

−1(𝑡)‖
𝐶[0,𝑇]

,                                                                                  (14) 

 

Lemma 2 for IP-II: Suppose that 𝑎(𝑡) > 0,    𝑏 > 0, 𝑎(𝑡) ∈  𝐶[0, 𝑇], 𝜑(𝑥) ∈ [0,1], 𝑓(𝑥, 𝑡) ∈

𝐶(𝑄𝑇), ℎ2(𝑡)   ∈  𝐶
1[0, 𝑇], ℎ2(𝑡) ≠ 0, ∫ 𝑓(𝑥, 𝑡)𝑑𝑥 = 0

1

0
 for (0 ≤ 𝑡 ≤ 𝑇),   ∫ 𝜑(𝑥)

1

0
𝑑𝑥 = 0, 

𝜑′(0) = 𝜑′(1), 𝜑(𝑥0) = ℎ2(0). Then, the problem of defining the functions 𝑢(𝑥, 𝑡) and 𝑝(𝑡) is 

equivalent to the problem of finding the solution of IP-II, possessing the properties (i) and (ii) of 

the solution of IP-II, from relations (i)–(iii), and  

 

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

≤ 𝑇,                                                                                                                                 (15) 

ℎ2
′(𝑡) − 𝑏𝑢𝑡𝑥𝑥(𝑥0, 𝑡) − 𝑎(𝑡)𝑢𝑥𝑥(𝑥0, 𝑡)

= 𝑝(𝑡)ℎ2(𝑡) + 𝑓(𝑥0, 𝑡)     (0 ≤ 𝑡 ≤ 𝑇)                                                                  (16) 

 

Theorem 2 for IP-II. Let the problem Equations (1) – (3), (5), (16) satisfy the following: 

1. 𝜑(𝑥) ∈ 𝐶2[0,1], 𝜑′′′(𝑥)  ∈  𝐿2(0,1), 𝜑(0) =  𝜑(1),      𝜑
′(0) =  𝜑′(1), 𝜑′′(0) =

 𝜑′′(1).                                                                                                                                                       (17) 

 

2. 𝑓(𝑥, 𝑡)  ∈  𝐶(𝑄𝑇),   𝑓𝑥(𝑥, 𝑡)  ∈  𝐿2(𝑄𝑇), 𝑓(0, 𝑡) = 𝑓(1, 𝑡),   (0 ≤ 𝑡 ≤

𝑇).                                     (18) 



IHJPAS. 2025, 38 (1) 
 

470 
 

 

3.𝑎(𝑡) > 0, 𝑏 > 0, 𝐶  [0, 𝑇],   ℎ2(𝑡) ∈ 𝐶
1[0, 𝑇],   ℎ2(𝑡) ≠ 0  (0 ≤ 𝑡 ≤

𝑇).                                                                                                                                                                       (19) 

 

4. 𝜑(𝑥0)

=   ℎ2(0).                                                                                                                                                            (20) 

Then IP- II has a unique solution in the ball K = 𝐾𝑅 (‖𝑧‖𝐸𝑇3 ≤  R =  A
(T) +  2) of the space 𝐸𝑇

3 

in Banach space. 

 

Where  

𝐴1(𝑇)

= ‖𝜑(𝑥)‖𝐿2(0,1) + √𝑇‖𝑓(𝑥, 𝑡)‖𝐿2(𝑄𝑇) + 2√3‖𝜑
′′′(𝑥)‖𝐿2(0,1)

+
2√3

𝑏
√𝑇‖𝑓𝑥(𝑥, 𝑡)‖𝐿2(𝑄𝑇),                                                                                                                          (21) 

 

𝐴2(𝑇) = ‖[ℎ2(𝑡)]
−1‖𝐶[0,𝑇] {‖ℎ2

′(𝑡) − 𝑓(𝑥0, 𝑡)‖𝐶[0,𝑇] +
2

√6
‖‖𝑓𝑥(𝑥, 𝑡)‖𝐶[0,𝑇]‖𝐿2(0,1)

+
2

√6
‖𝜑′(𝑥)‖𝐿2(0,1)

+
2

𝑏√6
‖𝑎(𝑡)‖𝐶[0,𝑇](‖𝜑

′(𝑥)‖𝐿2(0,1)

+ ‖𝑓(𝑥, 𝑡)‖𝐿2(𝑄𝑇))}                                                                                                     (22) 

 

𝐵1(𝑇)

= (1 +
2√3

𝑏
)𝑇,                                                                                                                                       (23) 

𝐵2(𝑇)

=
2

√6
 (1 +

1

𝑏2
𝑇‖𝑎(𝑡)‖𝐶[0,𝑇]) ‖[ℎ2(𝑡)]

−1‖𝐶[0,𝑇],                                                                            (24) 

where 

𝐴(𝑇) = 𝐴1(𝑇) + 𝐴2(𝑇), 𝐵(𝑇) = 𝐵1(𝑇) +

𝐵2(𝑇).                                                                                                                                                        (25) 

 

2.1 Discretization of the direct solver 

 Consider the direct solver for IP-I contains the Equations (1)- (4) and required data Equation (6). 

Also, the direct solver for IP-II contains the Equations (1)-(3), (5) and the required data Equation 

(7). In these direct problems, the only unknown quantity that should be determined is 𝑢(𝑥, 𝑡), 

that is, all other components are known. Discretising Equation (1) by a form of (FDM) as follows: 



IHJPAS. 2025, 38 (1) 
 

471 
 

 Denote for 𝑢(𝑥𝑖 , 𝑡𝑗) = 𝑢𝑖,𝑗, and 𝑓(𝑥𝑖 , 𝑡𝑗) = 𝑓𝑖,𝑗 where space node 𝑥𝑖 = 𝑖∆𝑥, time node 𝑡𝑗 =

𝑗∆𝑡, the space step length ∆𝑥 =
1

𝑀
 and time step length ∆𝑡 =

𝑇

𝑁
 for 𝑖 = 0,1, … ,𝑀, 𝑗 =

0,1,2,… , 𝑁 where 𝑀,𝑁 are positive integers. Based on the finite difference method, Equation (1) 

can be expressed as: 

𝑢𝑖,𝑗+1 − 𝑢𝑖,𝑗

∆𝑡
= 𝑎𝑗 (

𝑢𝑖+1,𝑗 − 2𝑢𝑖,𝑗 + 𝑢𝑖−1,𝑗
(∆𝑥)2

) +
𝑏

∆𝑡
(
𝑢𝑖+1,𝑗+1 − 2𝑢𝑖,𝑗+1 + 𝑢𝑖−1,𝑗+1

(∆𝑥)2
) 

−
𝑏

∆𝑡
(
𝑢𝑖+1,𝑗−2𝑢𝑖,𝑗+𝑢𝑖−1,𝑗

(∆𝑥)2
) + 𝑝𝑗𝑢𝑖𝑗 + 𝑓𝑖,𝑗                    (26)  

 

𝑢(𝑥, 0) = 𝜑(𝑥𝑖),          𝑖

= 0, 1, … ,𝑀,                                                                                                                  (27) 

 

𝑢(0, 𝑡𝑗) = 𝑢(1, 𝑡𝑗),         𝑗

= 0,1, … ,𝑁,                                                                                                                   (28) 

the Neumann condition  

𝑢𝑥(1, 𝑡) = 0  gives 𝑢𝑀+1,𝑗 = 𝑢𝑀−1,𝑗 ,         𝑗 =

0,1, … , 𝑁                                                                                                                                                    (29)  

via central difference formula.  

Using the trapezoidal rule approximation to the integral in Equation (5), we get the following 

formula 

 ∑𝑢𝑖𝑗

𝑀

𝑖=1

= 0,

𝑗 = 0,1, … , 𝑁.                                                                                                                                           (30) 

Also, the approximate formula for overdetermination condition Equation (6) via trapezoidal rule 

is given as: 

ℎ1(𝑡𝑗)

= 𝑢 (
𝑀

2
, 𝑡𝑗) +

1

𝑀
∑𝑢𝑖𝑗

𝑀

𝑖=1

,                                                                                                                     (31) 

and the overdetermination condition Equation (7) is given as: 

ℎ2(𝑡𝑗) = 𝑢(𝑥0, 𝑗),    𝑗

= 0,1, … ,𝑁.                                                                                                                 (32) 

 

Then, the discrete difference equation governing Equation (26) using the FDM scheme, we obtain 

the following difference equation. 

−𝛼𝑢𝑖−1,𝑗+1 + (1 + 2𝛼)𝑢𝑖,𝑗+1 − 𝛼𝑢𝑖+1,𝑗+1 = 𝛾𝑗𝑢𝑖−1,𝑗 + (1 − 2𝛾𝑗 + 𝑐𝑗)𝑢𝑖,𝑗 

+𝛾𝑗𝑢𝑖+1,𝑗 + ∆𝑡𝑓𝑖,𝑗 ,    𝑖 = 1,2, … ,𝑀 ,       𝑗 = 0,1,2, …𝑁,                                              (33)         

where  



IHJPAS. 2025, 38 (1) 
 

472 
 

𝛼 =
𝑏

(∆𝑥)2
 ,        𝛾𝑗 =

𝑎𝑗∆𝑡

(∆𝑥)2
−

𝑏

(∆𝑥)2
  ,   𝑐𝑗

= ∆𝑡 𝑝𝑗                                                                                                                            (34) 

The last difference Equation (33) can be encoded by the following linear system for Equations 

(1)-(4)  

𝑫1𝑣
𝑗+1 = 𝑬1𝑣

𝑗 + 𝑍1,           𝑗

= 0,1,2, … , 𝑁,                                                                                                                (35) 

and for Equations (1)-(3) and (5) 

𝑫2𝑣
𝑗+1 = 𝑬2𝑣

𝑗 + 𝑍2,             𝑗

= 0,1,2, … , 𝑁,                                                                                                                (36) 

where the matrices have the form  

𝐷1 =

(

 
 

       
1 + 2𝛼 −𝛼 0 ⋯ 0 0 −𝛼
−𝛼 1 + 2𝛼 −𝛼 ⋯ 0 0 0
⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0 0 0 ⋯ −𝛼 1 + 2𝛼 −𝛼
0 0 0 ⋯ 0 −2𝛼 1 + 2𝛼)

 
 

𝑀×𝑀

 

  

𝐷2 =

(

 
 
 

       
2 2 2 2 2 2 0
−𝛼 1 + 2𝛼 −𝛼 0 0 0 0
0 −𝛼 1 + 2𝛼 −𝛼 0 0 0
⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0 0 0 ⋯ −𝛼 1 + 2𝛼 −𝛼
0 2 2 ⋯ 2 2 2 )

 
 
 

𝑀+1×𝑀+1

 

 

 

𝐸1 =

(

 
 
 

        
𝛾𝑗 1 + 2𝛾𝑗 + 𝑐𝑗 𝛾𝑗 0 ⋯ 0 0 0

0 𝛾𝑗 1 + 2𝛾𝑗 + 𝑐𝑗 𝛾𝑗 ⋯ 0 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0 0 0 0 ⋯ 𝛾𝑗 1 + 2𝛾𝑗 + 𝑐𝑗 𝛾𝑗
0 0 0 0 ⋯ 0 2𝛾𝑗 1 + 2𝛾𝑗 + 𝑐𝑗)

 
 
 

𝑀×𝑀

 

 

𝐸2 =

(

 
 
 
 

       
0 0 0 0 0 0 0
𝛾𝑗 1 + 2𝛾𝑗 + 𝑐𝑗 𝛾𝑗 0 0 0 0

0 𝛾𝑗 1 + 2𝛾𝑗 + 𝑐𝑗 𝛾𝑗 0 0 0

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0 0 0 ⋯ 𝛾𝑗 1 + 2𝛾𝑗 + 𝑐𝑗 𝛾𝑗
0 0 0 ⋯ 0 0 0)

 
 
 
 

𝑀+1×𝑀+1

 

 



IHJPAS. 2025, 38 (1) 
 

473 
 

𝑍1 =

(

  
 

∆𝑡𝑓1,𝑗
∆𝑡𝑓2,𝑗
⋮

∆𝑡𝑓𝑀−1,𝑗 

∆𝑡𝑓𝑀,𝑗 )

  
 
,                   𝑍2 =

(

 
 
 

0
∆𝑡𝑓1,𝑗
∆𝑡𝑓2,𝑗
⋮

∆𝑡𝑓𝑀−1,𝑗 

0 )

 
 
 

,  

where,  𝑣𝑗+1 = (𝑢1,𝑗+1, 𝑢2,𝑗+1, … , 𝑢𝑀,𝑗+1)   and    𝑣𝑗 = (𝑢1,𝑗, 𝑢2,𝑗 , … , 𝑢𝑀,𝑗). 

2.2 Stability analysis  

In this section, we apply the Von Neumann stability analysis (24, 31) for direct problems I and 

II. Assume that 𝑓(𝑥, 𝑡) = 0,  for simplicity, and local constant  𝑝𝑗 = �̂� for known time level in 

Equation (33) where �̂� = max
𝑡=[0,𝑇]

|𝑝(𝑡)|, and �̂� = max
𝑡=[0,𝑇]

|𝑎(𝑡)|, then we obtain:  

−𝛼𝑢𝑖−1,𝑗+1 + (1 + 2𝛼)𝑢𝑖,𝑗+1 − 𝛼𝑢𝑖+1,𝑗+1

= 𝛾𝑢𝑖−1,𝑗 + (1 − 2𝛾 + ∆𝑡 ∗ �̂�)𝑢𝑖,𝑗 + 𝛾𝑢𝑖+1,𝑗                                                        (37) 

where, 

𝛼 =
𝑏

(∆𝑥)2
 ,        𝛾

=
�̂�∆𝑡

(∆𝑥)2

−
𝑏

(∆𝑥)2
  ,                                                                                                                       (38) 

 

Apply decomposition of the numerical solution into a Fourier sum as  
𝑢𝑖,𝑗

= 𝑆𝑗𝑒𝑤𝑖𝜃,                                                                                                                                                      (39) 

where S is the amplification factor, the phase angle 𝜃 = ∅ℎ where ∅ =
2𝜋

𝑁
  and 𝑤 = √−1. The 

amplification factor S is said to satisfy the von Neumann condition if |S|< 1. To find S, plug 

Equation (39)  into Equation (37) as follows:  

 

−𝛼𝑆𝑗+1𝑒𝑤𝜃(𝑖−1) + (1 + 2𝛼)𝑆𝑗+1𝑒𝑤𝑖𝜃 − 𝛼𝑆𝑗+1𝑒𝑤𝜃(𝑖+1)

= 𝛾𝑆𝑗𝑒𝑤𝜃(𝑖−1) + (1 − 2𝛾)𝑆𝑗𝑒𝑤𝜃(𝑖)

+ 𝛾𝑆𝑗𝑒𝑤𝜃(𝑖+1),                                                                                                            (40) 

 

after simplifying above equation, we get: 

 

−2𝛼 𝑆 (
𝑒−𝑤𝜃 + 𝑒𝑤𝜃

2
) + (1 + 2𝛼)𝑆

= 2𝛾 (
𝑒−𝑤𝜃 + 𝑒𝑤𝜃

2
) + (1 − 2𝛾),                                                                            (41) 

 



IHJPAS. 2025, 38 (1) 
 

474 
 

This equation gives the following: 

 

((1 + 2𝛼) − 2𝛼 cos 𝜃)𝑆

= (1 − 2𝛾) + 2𝛾 cos 𝜃                                                                                                (42) 

 

Equation (42) which can be written as, 

 

𝑆

=
(1 − 2𝛾) + 2𝛾 𝑐𝑜𝑠 𝜃

(1 + 2𝛼) − 2𝛼 𝑐𝑜𝑠 𝜃
.                                                                                                                              (43) 

 

In order to ensure the stability, the last quantity should be less than one in the sense of absolute 

value, that is 

|S| = |
(1 − 2𝛾) + 2𝛾 𝑐𝑜𝑠 𝜃

(1 + 2𝛼) − 2𝛼 𝑐𝑜𝑠 𝜃
|

< 1                                                                                                                                                             (44) 

This gives 

|(1 + 2𝛼) − 2𝛼 𝑐𝑜𝑠 𝜃| ≤ |1 + 2𝛼| + 2𝛼|cos 𝜃|   

≤ |1 + 2𝛼|

+ 2𝛼,                                                                                                                                                         (45) 

since 𝑀 > 0, 𝛼 =
𝑏

(∆𝑥)2
= 𝑏𝑀2 , applying in Equation (45) this gives 

 

≤ |1 + 2𝑏𝑀2| + 2𝑏𝑀2 = 1 + 2𝑏𝑀2 + 𝑏𝑀2 = 1 + 4𝑏𝑀2

> 1                                                                                                                                   (46) 

 

since b> 0 we guarantee that |S|< 1, therefor method is unconditionally stable. 

 

The convergence of the proposed scheme is obtained from the Lax-Richtmyer equivalence 

theorem, which states that "a consistent finite-difference scheme for a linear non-fractional 

partial differential equation for which the initial-value problem is well posed is convergent if and 

only if it is stable”. For a proof, see (32). 

2.3 Examples of direct problems  

2.3.1 Example for problem I 

We consider the direct problem I Equations (1)-(4) with T=1 with 𝑎 = 𝑏 = 0.01 and the 

following input data: 

𝑢(𝑥, 0) =
cos(2𝜋𝑥)

𝑒1
,       𝑥

∈ [0,1]                                                                                                                         (47) 

𝑝(𝑡) = cos(2 𝜋 𝑡) ,        𝑡

∈ [0, 𝑇]                                                                                                                       (48) 



IHJPAS. 2025, 38 (1) 
 

475 
 

𝑓(𝑥, 𝑡) = 𝑒−𝑡(−0.367879 − 0.367879 cos(2𝜋 𝑡)) cos(2𝜋𝑥),     (𝑥, 𝑡)

∈ 𝑄𝑇                                                                                                                            (49) 

 

The analytic solution is given by 

 

𝑢(𝑥, 𝑡) = 𝑒−1−𝑡 cos(2𝜋𝑥),     (𝑥, 𝑡)

∈ 𝑄𝑇 ,                                                                                                                              (50) 

and overdetermination condition  

ℎ1(𝑡) = −𝑒
−1−𝑡,        𝑡

∈ [0, 𝑇].                                                                                                                          (51) 

The numerical and exact solution of 𝑢(𝑥, 𝑡)  and the absolute error is plotted in Figure 1. 

when  𝑀 = 𝑁 = 40. This figure shows the excellent matching with the error magnitude of order 

O(10−3 ). Figure 2. presents the comparison between the exact solution and numerical for 

desired outputs ℎ1(𝑡). Also, excellent agreements were obtained. 

 

 
Figure 1. The exact and numerical solutions with an absolute error when M = N = 40, for Example, of the inverse 

problem I. 



IHJPAS. 2025, 38 (1) 
 

476 
 

 
Figure 2. The required output ℎ1(𝑡), with 𝑁 =  𝑀 =  40, for Example of inverse problem I. 

 

2.3.2 Example of Direct Problem II 

Consider the direct problem II Equations (1)-(3) and (5) with T=1, 𝑥0 =
1

2
 , 𝑎 = 𝑏 = 0.01  and 

the following input data 

𝑢(𝑥, 0) = −
cos(2𝜋𝑥)

𝑒1
,       𝑥

∈ [0,1],                                                                                                                          (52) 

𝑝(𝑡) = sin(2 𝜋 𝑡) ,        𝑡

∈ [0, 𝑇]                                                                                                                         (53) 

 

𝑓(𝑥, 𝑡) =
1

√1 + 𝑡
𝑒−√1+𝑡(0.697392 − 0.394784√1 + 𝑡) + √1 + 𝑡 sin(2𝜋𝑡) cos(2𝜋𝑥) , (𝑥, 𝑡)

∈ 𝑄𝑇                                                                                                                                                          (54) 

 

the analytic solution is given as 

𝑢(𝑥, 𝑡) = −𝑒−√1+𝑡 cos(2𝜋𝑥),     (𝑥, 𝑡)

∈ 𝑄𝑇                                                                                                                               (55) 

and overdetermination condition  

ℎ2(𝑡) = 𝑒
−√1+𝑡,        𝑡

∈ [0, 𝑇]                                                                                                                           (56) 

that can be checked by direct substitution. 

 

Figure 3. presents the numerical solution, exact solution and the absolute error between them for 

the temperature 𝑢(𝑥, 𝑡) when 𝑀 = 𝑁 = 40. The figure shows an excellent agreement between 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

h
(t

)

exact and numerical values of h(t) 

exact

numerical



IHJPAS. 2025, 38 (1) 
 

477 
 

the exact and numerical solutions with the error magnitude of order 𝑂(10−4). Figure 4. shows 

the comparison between the exact and numerical solutions of ℎ2(𝑡). 

 

 
Figure 3. The exact and numerical solutions with the corresponding absolute error when M = N = 40, for 

Example, direct problem II. 

 

 

 
Figure 4. The required output ℎ2(𝑡), with 𝑁 =  𝑀 =  40, for Example of direct problem II. 

 

3. Computational approach for inverse problems  

      Our goal in this section is devoted to solving IP- I and IP-II. To find stable reconstructions 

for unknown coefficient 𝑝(𝑡), in addition to heat distribution 𝑢(𝑥, 𝑡) which satisfy Equation (1)- 

(4), (6) for IP-I and Equations (1)-(3),(5), (7) for IP-II. These problems are reformulated as 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

h
(t

)

exact and numerical values of h(t) 

exact

numerical



IHJPAS. 2025, 38 (1) 
 

478 
 

nonlinear optimization problems and solved numerically by minimizing the gap between extra 

measurement data Equations (6) or (7) and the associated computed solution. To gain reliable 

results we apply the Tikhonov’s regularization method due to the ill-possedness of the under 

investigation problems. The cost functional can be constructed as (33-40). 

 

𝐾𝐼(𝑝) = ‖𝑢 (
1

2
, 𝑡) + ∫𝑢(𝑥, 𝑡)𝑑𝑥 − ℎ1(𝑡)

1

0

‖

2

+ 𝛽‖𝑝(𝑡)‖2,                                                                                                                  (57) 

for IP-I and the functional  

𝐾𝐼𝐼(𝑝) = ‖𝑢(𝑥0, 𝑡) − ℎ2(𝑡)‖
2

+ 𝛽‖𝑝(𝑡)‖2,                                                                                                                  (58) 

 

For IP-II, where  β ≥ 0 the regularization parameter, should be selected according to some 

selection strategy such as L-Curve (41), Mozorov discriperey principle (42), or trial and error as 

in (43, 44). The approximate form of the above functionals are: 

 

𝐾𝐼 (𝑝) =∑(𝑢 (
1

2
, 𝑡𝑗) + ∫𝑢(𝑥, 𝑡𝑗)𝑑𝑥 − ℎ1(𝑡𝑗)

1

0

)

2
𝑁

𝑗=1

+ 𝛽∑𝑝𝑗
2

𝑁

𝑗=1

,                                                                                                                    (59) 

 

𝐾𝐼𝐼 (𝑝) =∑(𝑢(𝑥0, 𝑡𝑗) − ℎ2(𝑡𝑗))
2

𝑁

𝑗=1

+ 𝛽∑𝑝𝑗
2

𝑁

𝑗=1

,                                                                                                                     (60) 

For IP-I and IP-II, respectively.  

 

The objective functions Equations (25) and (26), it is minimized via the subroutine lsqnonlin 

from the MATLAB optimization toolbox. This routine tries to solve nonlinear least-squares curve 

fitting problems starting from the initial guess for unknown coefficient 𝑝. The upper and lower 

bounds on the variable 𝑝 are specified as 10−2 ≤ 𝑝 ≤ 102. Also, in this routine, there is no need 

to calculate the gradient separately; this is something impeded inside the routine package. 

The following parameters are essential to initiate the optimization process of Equations (57) or 

(58); the minimization process will terminate when the following prescribed parameters are 

reached: 

 



IHJPAS. 2025, 38 (1) 
 

479 
 

 • Allowed number of iterations = 6000. 

 • Specified solution and objective function Tolerance = 10−20. 

The inverse problems I or II are solved concerning noisy/ exact measurement data in Equations 

(6) or (7). The additive noise type as presented in (45- 50): 

 

ℎ𝑙
𝜖(𝑡𝑗) = ℎ𝑙(𝑡𝑗) + 𝜖𝑗,    𝑗 = 1,2, … ,𝑁,     𝑙

= 1,2,                                                                                                                              (61) 

 

Where 𝜖 is a Gaussian random vector, and standard deviation 𝜇 is: 

 

𝜇𝑙 = 𝑞 × max
𝑡∈[0,𝑇]

|ℎ𝑙(𝑡)| ,                  𝑙

= 1,2,                                                                                                                              (62) 

 

Where 𝑞 represents the percentage of noise. Here, we use the normrnd built-in function to 

generate the random variables 𝜖 = (𝜖𝑗)   𝑗 = 1,2, … ,𝑁 as follows: 

𝜖 =  𝑛𝑜𝑟𝑚𝑟𝑛𝑑(0, 𝜇𝑙 , 𝑁)                      𝑙

= 1,2                                                                                                                               (63) 

 

3.1 Results and Discussion 

We introduce a test example for each inverse problem. To explain the stability and accuracy of 

the computational procedure that is based on the finite difference method combined with the 

depreciation of Tikhonov's functional Equations (59) and (60). 

To assess the reconstruction accuracy of the potential term, we use root mean squares error rmse, 

which is given by the following expression (51): 

 

𝑟𝑚𝑠𝑒(𝑝)

= √
1

𝑁 
∑(𝑝𝑗 − 𝑝𝑒𝑥𝑎𝑐𝑡(𝑡𝑗)

)
2

𝑁 

𝑗=1

 

,                                                                                                              (64) 

 

 

3.2 Numerical results for IP- I 

Assume the inverse problem I with T = 1 and input data 𝑎 = 𝑏 = 0.01: 

𝑢(𝑥, 0) =
cos(2𝜋𝑥)

𝑒1
,       𝑥

∈ [0,1],                                                                                                                            (65) 

 

𝑝(𝑡) = cos(2 𝜋 𝑡) ,        𝑡

∈ [0, 𝑇],                                                                                                                           (66) 

 



IHJPAS. 2025, 38 (1) 
 

480 
 

𝑓(𝑥, 𝑡) = 𝑒−𝑡(−0.367879 − 0.367879 cos(2𝜋 𝑡)) cos(2𝜋𝑥),     (𝑥, 𝑡)

∈ 𝑄𝑇                                                                                                                                (67) 

 

with the analytic solution 

𝑢(𝑥, 𝑡) = 𝑒−1−𝑡 cos(2𝜋𝑥),     (𝑥, 𝑡)

∈ 𝑄𝑇 ,                                                                                                                               (68) 

and overdetermination condition  

ℎ1(𝑡) = −𝑒
−1−𝑡,        𝑡

∈ [0, 𝑇]                                                                                                                           (69) 

that can be checked by direct substitution. 

 

Figure 5. shows the numerical solution of the time-dependent potential term from 

overdetermination Equation (6) in comparison with the exact solution (𝑝(𝑡) = cos(2 𝜋𝑥)), 

obtained by solving the IP-I with the above input data using the FDM, described in Section 2, 

with 𝑀 = 𝑁 ∈ {10, 20,30, 40}. This figure shows that as mesh size increases, the retrieved 

coefficients converge to the exact solution, revealing that mesh independence occurs. 

Figure 6. shows that the convergent objective function Equation (59) reaches a very low 

threshold stationary value of 𝑂(10−8) and is plotted with  𝑀 = 𝑁 ∈ {10, 20,30, 40}. From these 

figures, it can be observed a speed convergence is achieved in no more than 15 iterations only to 

reach a meagre value when 𝑀 = 𝑁 = 40, for example. 

Next, we choose 𝑁 = 𝑀 = 40 for the rest of the numerical investigation, with cases (𝑞 =

0%, 0.5%, 1%) included in the measurement data Equation (6). Figure 7. explains the plotting 

of the exact solution and numerical results for 𝑝(𝑡) with no regularization (𝛽 =  0) and no noise 

(𝑞 = 0%) and noise 𝑞 ∈ {0. 5%, 1%}. It is clear that as the noise percentage increases from 0% 

to 1%, the identified coefficient has oscillatory behaviour, which is expected since the problem 

under investigation is ill-posed. Therefore, a sort of stabilization should be applied in order to 

restore stability and reduce the oscillatory behaviour. 

 

 
Figure 5. Numerical and exact solution for potential term 𝑝(𝑡) when 𝑀 = 𝑁  ∈  {10, 20, 30,40}. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-1

-0.5

0

0.5

1

1.5

p(
t)

exact

M=N=10

M=N=20

M=N=30

M=N=40



IHJPAS. 2025, 38 (1) 
 

481 
 

 
Figure 6. The unregularized objective function Equation (59), with (𝑞 = 0) and  𝑀 = 𝑁  ∈  {10, 20, 30,40}. 

 

 The associated numerical results for 𝑝(𝑡) after applying Tikhonov’s regularization method with 

𝛽 ∈ {10−4, 10−5, 10−6} for case 𝑞 = {0. 5%} and 𝛽 ∈ {10−4, 10−5} for case 𝑞 = {1%} are 

presented in Figures 8. and 9., respectively. From this figure, one can deduce that as 𝛽 = 10−4 

recover adequate identification with reasonable accuracy with 𝑟𝑚𝑠𝑒(𝑝) = {0.2987, 0.3328} for 

𝑞 ∈ {0.5, 1}% noise, respectively. The 3D graph for exact, numerical and absolute error between 

the exact solution and numerical solution for temperatures (𝑈𝑋, 𝑡) plotted in Figure 10. with (a) 

𝑞 = 0.5% and (b) 𝑞 =  1%  with 𝛽 = 10−4 and also accurate identification is obtained in terms 

of free oscillation. Next, in Table 1. we compute the 𝑟𝑚𝑠𝑒 values Equation (64) for 𝛽 ∈

{10−𝑖, 𝑖 = 4, 5, 6} and 𝑞 ∈ {0.5, 1}%. Figures 8, 9, 10. and Table 1.  show good correspondence 

and convergence between the numerical solutions of 𝑝(𝑡) and 𝑢(𝑥, 𝑡) with their corresponding 

exact solutions when 𝑞 decreases from 1 % 𝑡𝑜 0.5% and then to 0%. 

 
Figure 7. Numerical reconstructions and exact solution for 𝑝(𝑡), with noise level 𝑞 =  {0, 0.5%, 1%}, without 

regularization applied for IP- I. 

 

0 5 10 15

Number of Iterations 

10
-8

10
-7

10
-6

10
-5

10
-4

10
-3

10
-2

10
-1

10
0

O
b

je
c
ti
v
e
 f

u
n

c
ti
o

n
M=N=10

M=N=20

M=N=30

M=N=40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-6

-4

-2

0

2

4

6

p
(t

)

exact

q=0%

q=0.5%

q=1%



IHJPAS. 2025, 38 (1) 
 

482 
 

 
Figure 8. Numerical reconstructions and exact solution for 𝑝(𝑡), with regularization parameter 𝛽 =

{10−4, 10−5, 10−6} and 𝑞 =  0.5% noise. 

 
Figure 9. Numerical reconstructions and exact solution for 𝑝(𝑡), with regularization parameter 𝛽 =  {10−4, 10−5} 

and 𝑞 =  1% noise, for IP- I. 

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

p
(t

)

exact

1
=10 -4

2
=10 -5

2
=10 -6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

p(
t)

exact

1
=10 -4

2
=10 -5



IHJPAS. 2025, 38 (1) 
 

483 
 

 
(a) 

 

 
                                                              (b) 

Figure 10. Numerical and exact temperature 𝑢(𝑥, 𝑡) with (a) 𝑞 =  0.5%  and 𝛽 =  10−4, (b) 𝑞 =  1% noise 

and 𝛽 =  10−4, for IP- I. 

 

 

 

 

 



IHJPAS. 2025, 38 (1) 
 

484 
 

Table 1. Numerical information for IP- I with various noise levels. 

𝒒 =  𝟎. 𝟓% 𝛽 =  10−4 𝛽 =  10−5 𝛽 =  10−6 

No. of iterations 48 40 37 

Objective function Equation (59) at the final 

iteration 

0.0015 2.8078E-04 5.4229E-05 

𝒓𝒎𝒔𝒆(𝒑) 0.2987 0.3678 0.7326 

𝐪 =  𝟏% 𝛽 =  10−4 𝛽 =  10−5 𝛽 =  10−6 

No. of iterations 42 45 48 

Objective function Equation (59) at the final 

iteration 

0.0020 6.1036E-04 1.6380E-04 

𝒓𝒎𝒔𝒆(𝒑) 0.3328 0.6661 1.4340 

 

3.3 .Numerical results for IP- II 

 Consider the IP- II Equations (1)-(3),(5) and (7) with input data in Example 2.3.2. To solve this 

problem, we employ the same process presented in section 3. 

Here, all the conditions of inverse problem II are satisfied, and hence, the unique solvability of 

the solution is guaranteed. Initially, we start with an initial guess when t = 0 (i. e 𝑝(0) = 0) and 

retrieve the function 𝑝(𝑡) and 𝑢(𝑥, 𝑡) for noise-free case (q = 0) (see Figure 11.), then for q ∈

 {1%, 3%} noisy data. From this figure, it is clear to observe the excellent matching when the 

mesh size is chosen as 𝑀 = 𝑁 = 40. the objective function Equation (60) is plotted as a function 

of the number of iterations in Figure 12. for noise-free cases and for noise included. The fast 

convergence can be seen to reach a very low value of O (10−9) in just 11 iterations. 

 

 
Figure 11. Numerical and exact solution for potential term 𝑝(𝑡) when 𝑀 = 𝑁 = 40, for Example, in IP-II. 

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-1

-0.5

0

0.5

1

1.5

p
(t

)

exact

numerical



IHJPAS. 2025, 38 (1) 
 

485 
 

 
Figure 12. The unregularized objective function Equation (60), with q = {0, 1%, 3%} noise data included in 

measurements Equation (7), for Example, in IP-II. 

 
Figure 13. Numerical reconstructions and exact solution for 𝑝(𝑡), with noise level 𝑞 =  {0, 1%, 3%}, without 

regularization applied for IP- II. 

 

For the cases plotted in Figure 13., the results obtained were inaccurate and unstable when the 

regularization parameter was set 𝛽 = 0 and q ∈  {1%, 3%}—the Tikhonov regularization 

method employed to obtain stable reconstructions for 𝑝(𝑡). Regularization parameters 𝛽 =

 { 10−5, 10−4, 10−3} were chosen by trial and error strategy, which is based on starting from a 

small value for 𝛽 and gradually increasing it until the oscillatory behaviour starts to disappear as 

applied in )40(, for noise data 𝑞 =  1%. Figure 15. shows the objective function Equation (60) 

decreases steadily in just below 50 iterations. Tikhonov's approach with the selected parameters 

gives a reasonable and stable approximate solution of the potential term 𝑝(𝑡) (see Figure 14.). 

When 𝑞 =  3%,  we deduce that the regularization parameters 𝛽 =  {10−4 and 10−3} give the 

stable and accurate approximate solution for 𝑝(𝑡)(see Figures 16. and 17). The 3D graphs of the 

exact and numerical solutions for 𝑢(𝑥, 𝑡), and the absolute error between are plotted in Figure 

0 5 10 15 20 25 30 35

Number of Iterations 

10
-9

10
-8

10
-7

10
-6

10
-5

10
-4

10
-3

10
-2

10
-1

10
0

O
b

je
ct

iv
e

 f
u

n
ct

io
n

q=0%

q=1%

q=3%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-8

-6

-4

-2

0

2

4

6

8

10

12

p
(t

)

exact

q=0%

q=1%

q=3%



IHJPAS. 2025, 38 (1) 
 

486 
 

18. with (i) 𝑞 = 1% and 𝛽 = 10−4, (ii) 𝑞 = 3% and 𝛽 =  10−3. Other information about the 

number of iterations, the value of the objective function Equation (60) at the final iteration and 

the rmse of 𝑝(𝑡) are given in Table 2. From Figures 14, 16. and 18. and Table 2., it can be seen 

that there is an adequate agreement between the numerical results of 𝑝(𝑡) and 𝑢(𝑥, 𝑡) for their 

analytical solutions. 

 

 
Figure 14. Numerical reconstructions and exact solution for 𝑝(𝑡), with regularization parameter 𝛽 =

{ 10−5, 10−4, 10−3} and 𝑞 =  1% noise. 

 

 
Figure 15. The regularized objective function Equation (60), with regularization parameter 𝛽 =

{ 10−5, 10−4, 10−3} and 𝑞 =  1% noise. 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

p
(t

)

exact

1
=10 -3

2
=10 -4

2
=10 -5

0 5 10 15 20 25 30 35 40 45

Number of Iterations 

10
-3

10
-2

10
-1

R
e

g
u
la

ri
z
e
d

 o
b

je
c
ti
v
e

 f
u

n
c
ti
o
n

1
=10 -3

2
=10 -4

2
=10 -5



IHJPAS. 2025, 38 (1) 
 

487 
 

 

Figure 16. Numerical reconstructions and exact solution for 𝑝(𝑡), with regularization parameter 𝛽 =

 {10−4, 10−3} and q =  3% noise. 

 

 
Figure 17. The regularized objective function Equation (60), with regularization parameter 𝛽 =  {10−4, 10−3} 

and 𝑞 =  3% noise. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

p(
t)

exact

1
=10 -3

2
=10 -4

0 5 10 15 20 25 30 35 40 45 50

Number of Iterations 

10
-2

10
-1

R
e

g
u
la

ri
ze

d
 o

b
je

ct
iv

e
 f
u

n
ct

io
n

1
=10 -3

2
=10 -4



IHJPAS. 2025, 38 (1) 
 

488 
 

 
(a) 

 
(b) 

Figure 18. Exact and numerical temperature 𝑢(𝑥, 𝑡) with (a) 𝑞 =  1%  and 𝛽 =  10−4, (b) 𝑞 =  3% noise and 𝛽 =

 10−3. 

 
Table 2. Numerical information for inverse problem II with noisy data and regularization. 

𝑞 =  1% 𝛽 =  10−3 𝛽 =  10−4 𝛽 =  10−5 

No. of iterations 36 45 44 

Objective function (60) at final iteration 0.0141 0.0028 5.9268E-04 

𝑟𝑚𝑠𝑒(𝑝) 0.3054 0.1684 0.6502 

q =  3% 𝛽 =  10−3 𝛽 =  10−4 𝛽 =  10−5 

No. of iterations 43 49 52 

Objective function (60) at final iteration 0.0237 0.0089 0.0036 

𝑟𝑚𝑠𝑒(𝑝) 0.2919 0.4942 1.9405 

 

4. Conclusions  

         In this work, an investigation was conducted by considering the pseudo-parabolic equations 

of the third-order with initial and various boundary conditions and overdetermination data to 



IHJPAS. 2025, 38 (1) 
 

489 
 

recover the time–dependent potential terms. The direct problems were solved by the FDM. Von 

Neumann technique was employed to study the stability of the proposed numerical direct 

algorithm. The inverse problems were reformulated as a nonlinear optimization problem and 

solved numerically by lsqnonlin iterative routine from MATLAB. To stabilize the ill-posed 

problem under investigation, Tikhonov's regularization method was applied. The numerical test 

examples for each problem confirmed the applicability of the proposed algorithm to obtain an 

accurate and stable solution. As a future work, it can apply this process to solve the other different 

inverse problems of higher dimensions. 

 

Acknowledgements  

     We would like to express our gratitude other referees for their valuable comments and 

suggestions that led to a truly significant improvement of the paper. 

 

Conflict of Interest 

              The authors declare that there are no competing interests regarding the publication of this paper. 

 

Funding 

              This work is not supported by any the Foundation. 

 

Ethical Clearance 

              Ethics of scientific research were carried out in accordance with international conditions. 

 

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