Numerical Investigation of an Inverse Coefficient Problem in a Nonlinear Hyperbolic Equation

M2-1 2nd KOCAELI SCIENCE CONGRESS (KOSC-2025) Kocaeli University, Faculty of Arts and Sciences November 19-21, 2025, İzmit, Kocaeli, Türkiye Numerical Investigation of an Inverse Coefficient Problem in a Nonlinear Hyperbolic Equation Akbala Yernazar1, İrem Bağlan2, Erman Aslan3 1Department of Mathematics, Ahmet Yassawi University, Turkistan, Kazakhstan 2Department of Mathematics, Kocaeli University, Kocaeli, TR-41380, Turkey 3Department of Mechanical Engineering, Kocaeli University, Kocaeli, TR-41380, Turkey Corresponding author: [email protected] ORCID IDs: First Author: 0000-0002-0068-4954 Second Author: 0000-0003-4900-6027 Third Author: 0000-0001-8595-6092 DOI : 10.5281/zenodo.18011497 Abstract In this study, an inverse coefficient problem for a one-dimensional nonlinear hyperbolic equation with periodic boundary conditions is investigated. The numerical solution is obtained through an implicit finite difference scheme. A comparison of the analytical and numerical results demonstrates that the estimated inverse coefficient and the corresponding  values exhibit excellent agreement, indicating the accuracy and reliability of the proposed approach. Keywords: inverse problem, nonlinear hyperbolic equation, finite difference method 1. Introduction The advent of new scientific and engineering knowledge has increased the significance of inverse problems. The scope of inverse problems is vast, encompassing fields such as physics, chemistry, biology, astrophysics, geophysics, medicine, and economics. Inverse problems have numerous and diverse practical applications in areas like microwave heating, climate modeling, heat conduction, biological processes, seismology, and seawater desalination. In the field of medicine, for instance, techniques such as computed tomography are based on the principles of inverse problems. In geophysics, the investigation of subterranean structures and the exploration of underground resources can be conducted more cost-effectively using inverse problems, as an alternative to deep drilling. Furthermore, the use of inverse problems is pervasive in domains such as radio astronomy, spectroscopy and nuclear physics. In most cases, such problems are modelled by partial differential equations. Predictions can be made regarding the behavior of the system if the input data, including initial boundary conditions, solution geometry and coefficients, are known with certainty. In some cases, however, these inputs may be absent or unknown. In such instances, the output from experimental measurements can be used to reconstruct the missing data. This process gives rise to illposed inverse problems, where minor alterations in the input can result in significant changes to the M2-2 2nd Kocaeli Science Congress, November 19-21, 2025 solution. The theory of ill-posed inverse problems was first proposed by Tikhonov [1] and later developed into the theory of regularization, which aims to solve ill-posed inverse problems, by Tikhonov [2] and Morozov [3]. In recent years, inverse problems for hyperbolic equations with unknown coefficients have attracted significant interest from numerous researchers [4-11]. Here, the emphasis is on determining the unknown coefficient such wave speed or damping from empirical observations typically obtained from measurements of the solution at different times and locations. Beylina [11] investigated inverse problems with integral overdetermination conditions in which they demonstrated the existence and uniqueness of solutions using priori estimates and Galerkin procedures. A study by Kozhanov [12] solved nonlinear inverse problems for second-order hyperbolic equations, establishing existence and uniqueness theorems for regular solutions with weak derivatives in the Sobolev sense. Similar to other studies, the Fourier method has been employed to investigate inverse boundary value problems for hyperbolic equations [13-16]. The works of Akhundov & Habibova [17] and Kozhanov [12] explored the determination of unknown coefficients of hyperbolic equations and proved theorems on uniqueness, stability, and existence of solutions. These studies employed integral overdetermination conditions to solve the time-dependent unknown coefficients. These approaches underscore the importance of mathematical techniques in simplifying complex inverse problems. In mathematical modeling, however, problems often specify relationships between the values of the desired function both on the boundary and within the domain, rather than relying solely on classical boundary conditions. Such conditions are referred to as non-local conditions [18]. In this study, we will focus on periodic boundary conditions, which are relevant to the problem at hand. Huntul et al. [14] investigated the reconstruction of a time-dependent potential in a hyperbolic problem with periodic boundary conditions. The inverse problem for parabolic, hyperbolic, and Euler-Bernoulli equations under periodic boundary conditions has been investigated by [19-22]. These works have contributed to the understanding and solution of inverse problems for hyperbolic equations with periodic boundary conditions and further limitations. Numerical methods such as finite difference method [23, 24], finite element method [25, 26] and finite volume method [27, 28], are widely employed for investigating inverse problems for hyperbolic equations. The work by Huntul et al. [14] employed the Crank-Nicolson finite difference method and Tikhonov regularization to investigate the reconstruction of a time-dependent potential, establishing both accuracy and stability. Lin and Ewing [29] proposed a direct numerical method for solving a coefficient inverse problem. The study used finite differences in one dimensional for efficient computation. Le et al. [30] proposed a quasi-reversibility method for solving an inverse source problem. The researchers estimated solution using Fourier series and proved convergence with decrease in noise levels. The Carleman estimates were used to establish a local stability result for an inverse source problem in a study by Jiang et al. [31], which subsequently implemented a Tikhonov regularization-based numerical algorithm. The overall literature demonstrates the growing development of stable and efficient numerical techniques for solving inverse problems in hyperbolic equations. In the present study, we investigate an inverse problem involving unknown time-dependent coefficient in a one-dimensional nonlinear hyperbolic equation under periodic boundary conditions. For the numerical solution, implicit finite-difference schemes is applied. The existence, uniqueness, and stability of the analytical solution for this problem have been previously established in [32]. 2. Statement of the Problem Consider a nonlinear hyperbolic equation defined on the domain   : (0, ), (0, ) : D z t T     ( ) ( , , ), tt zz а t f z t        (1) with the initial conditions M2-3 2nd Kocaeli Science Congress, November 19-21, 2025 ( ,0) ( ), ( ,0) ( ), t z z z z       (2) and boundary conditions (0, ) ( , ), (0, ) ( , ), z z t t t t         (3) as well as the overdetermination condition 0 ( ) ( , ) . E t z t dz    (4) Where the functions , , E   and ( , , ) f z t  are given functions on   0,  and   , D    , respectively; while the function ( , ) z t  and the coefficient ( ) a t are unknown. Using the Fourier method to solve equations (1)–(4), we obtain       0 0 0 0 10 0 1 2 ( , ) ( ) ( ) ( , ) ( , , ) 2 1 1 cos2 sin 2 ( ) ( , ) ( , , ) cos2 sin 2 ( ) cos2 2 1 1 cos2 sin 2 ( ) ( , ) ( , , ) sin 2 s 2 t t ck ck k sk sk z t t t а f d d kt kt а f k k t d d kz k k kt kt а f k k k                                                                        10 0 in 2 ( ) sin 2 . t k k t d d kz               (5) By employing the overdetermination condition (4) together with the solution (5), the inverse coefficient in equations (1)–(4) is determined as shown in (6):   0 ( ) , , ( ) . ( ) E t f z t dz а t E t      (6) Definition 2.1 The problem of finding the pair   ( ), ( , ) а t z t  in (1)–(4) will be called an inverse problem. Definition 2.2 B is called Banach space when the following set     0 00 0 0 1 ( ) ( ) ( ), ( ), ( ), 1,..., max max ( ) max ( ) 2 sk ck ck sk t T t T t T k t t t t t k N t t                             of continuous on [0, T] functions satisfying the norm   0 0 0 0 1 ( ) ( ) max max ( ) max ( ) . 2ck sk t T t T t T k t t t t                 Theorem 2.1 If the following assumptions satisfy: A1. 2 ( ) [0, ], ( ) [0, ]. E t C T а t C T   M2-4 2nd Kocaeli Science Congress, November 19-21, 2025 A2. ( ) [0, ], ( ) [0, ]. z C z C       A3.1 ( , , ) f z t  is continuous across all arguments in   , D    and satisfies the condition: ( ) ( ) ( ) ( ) ( , , ) ( , , ) ( , ) , 0,2, k k k k f z t f z t b z t k z z             where   2 ( , ) , ( , ) 0. b z t L D b z t   A3.2 ( , , ) [0, ], [0, ], ( , , ) . f z t C t T f z t M       A3.3 0 ( , , ) 0, [0, ]. f z t dz t T       Then the problem (1)–(4) has a unique solution. Theorem 2.2 If the assumptions of the theorem 1 be satisfied, then the pair of the solution   ( ), ( , ) a t z t  of the problem (1)–(4) is constantly dependent on the input data , , . E   3. Numerical Procedure for the Nonlinear Problem We develop an iterative algorithm to linearize the problem,             2 2 1 2 2 , , , m m m m а t f z t t z             (7)         ( ,0) ( ), 0, , ( ,0) ( ), 0, , m m t z z z z z z           (8)             (0, ) ( , ), 0, , (0, ) ( , ), 0, . m m m m z z t t t T t t t T           (9) By setting     ( , ) ( , ) m m z t v z t   and       1 , , , m f z t f z t  , we can express the problem (7)-(9) as a linear problem.     2 2 2 2 , . v v а t v f z t t z         (10) After this linearization process, an implicit finite difference scheme is used for the numerical solution of the above equation. In the time discretization used in Eq. (10), a first-order accurate backward finite difference scheme is employed. For spatial discretization, a second-order accurate central finite difference scheme is utilized;     2 1 2 2 2 1 1 1 1 1 2 2 1 1 2 2 . j j j j j j j j j i i i i i i i i v v v v v v а v f t z                    (11) The initial condition is defined as; 0 . i i v   (12) M2-5 2nd Kocaeli Science Congress, November 19-21, 2025 The Dirichlet boundary condition is a first-type boundary condition that assigns a fixed value, while the Neumann boundary condition is a second-type boundary condition that assigns a fixed gradient. The periodic boundary condition is a combination of the two. While defining the periodic boundary condition, the parts of the coefficient matrix corresponding to the boundary values were modified according to Eqs. (13) and (14). 1 , j j Nz v v  (13) 2 1 . 2 j j jNz Nz v v v   (14) The computational domain extends from   0,  in the z-direction and   0, T in time. It’s divided into discrete intervals, where spatial points are defined as   1 i z i z    for 1,2,...., i Nz  and temporal points are given by j t j t   for 1,2,...., . j Nt  Here, z  is the spatial step size, determined by z Nz    , and t  is the time step, calculated as t T Nt   . The integers Nz and Nt specify the number of intervals in space and time, respectively. The values v ,  and f are discretized as   , j i i j v v z t ,   i i z   and   2 2 , j i j f f z t    , respectively. The initial condition is specified at 0. t  In the numerical calculation, 2 j  indicates the current time, 1 j  refers the previous time, and j corresponds two steps before the current time. To compute the inverse coefficient   a t , Eq. (1) is integrated over the interval   0,  with respect to z , utilizing the relationships provided in Eqs. (3) and (4). This process yields:         0 0 " , . zz E t v dx f z t dz а t E t        (15) The   " E t term in (15) is discretized as follows:   2 1 2 2 '' . j j j E E E E t t      (16) The two integral expressions 0 zz v dz   and   0 , f z t dz   in Eq. (15) are evaluated using the trapezoidal rule. The spatial discretization points used in the numerical solution are also utilized in the trapezoidal rule for integration. For the numerical solution of Eq. (11), no iterative methods are employed; instead, a direct method is used instead. The right-hand side matrix constitutes from previous values, and it is used in direct method. The right-hand side matrix is   1 2 1 1 2 ( ) . j j j j i i i i i rhs v v t a t v f         4. Numerical Example Considering inverse problem,       22 2 , , 2 5 6 5 cos 2 t f z t e t t z          M2 - 6 with the over determination condition is given as;   E t z t dz e In that case, the problem transforms as tt zz          The initial condition is defined as     The periodic boundary condition is determined as (0, ) ( , ), (0, ) ( , ),0 . t t t t t T       The analytical solution of this problem is expressed as follows   a t z t t z e 4.1 Grid Independence Study, Time Step Size Determination Before starting the main solution, a grid independence study and a time step determination study are conducted to establish the necessary grid size and time step. For the grid independence study, eight different grid densities are used, each progressively 2560. The grid independence study is performed for all pre sequentially as follows: 0.01s, 0.005s, 0.0025s, 0.00125s, 0.000625s, and 0.0003125s. Figure 1 shows the grid independence studies conducted for the maximum value of υ at 1 second for each time step. To facilitate a better comparison, the graphs are presented on the same scale, and the results from the analytical solution are also included. As the time step decrea values is observed. For all time steps, the differences between solutions decrease as the number of grid points increases. A grid number of 1280 is determined to be the grid steps. Furthermore, the solution at a time step of 0.0025s and a grid number of 1280 aligns very closely with the analytical solution. ∆t=0.01s 2nd Kocaeli Science Congress, November 19 with the over determination condition is given as;      2 0 1 ( , ) 2 sin 2 . 2 t E t z t dz e         In that case, the problem transforms as     22 2 ( ) 2 5 6 5 cos 2 . t tt zz a t e t t z                    ,0 1 cos2 , 0, .z z z z        condition is determined as (0, ) ( , ), (0, ) ( , ),0 . z z t t t t t T           The analytical solution of this problem is expressed as follows           2 2 , , 9 , 1 cos 2 . t a t z t t z e    4.1 Grid Independence Study, Time Step Size Determination Before starting the main solution, a grid independence study and a time step determination study are conducted to establish the necessary grid size and time step. For the grid independence study, eight different grid densities are used, each progressively doubled: 20, 40, 80, 160, 320, 640, 1280, and 2560. The grid independence study is performed for all pre - determined time steps, which are halved sequentially as follows: 0.01s, 0.005s, 0.0025s, 0.00125s, 0.000625s, and 0.0003125s. independence studies conducted for the maximum value of υ at 1 second for each time step. To facilitate a better comparison, the graphs are presented on the same scale, and the results from the analytical solution are also included. As the time step decrea ses, an increase in the values is observed. For all time steps, the differences between solutions decrease as the number of grid points increases. A grid number of 1280 is determined to be the grid - independent number for all time solution at a time step of 0.0025s and a grid number of 1280 aligns very closely with the analytical solution. ∆t=0.005s ∆ t=0.0025s 2nd Kocaeli Science Congress, November 19 - 21, 2025 Before starting the main solution, a grid independence study and a time step determination study are conducted to establish the necessary grid size and time step. For the grid independence study, eight doubled: 20, 40, 80, 160, 320, 640, 1280, and determined time steps, which are halved independence studies conducted for the maximum value of υ at 1 second for each time step. To facilitate a better comparison, the graphs are presented on the same scale, and the ses, an increase in the υ values is observed. For all time steps, the differences between solutions decrease as the number of grid independent number for all time solution at a time step of 0.0025s and a grid number of 1280 aligns very t=0.0025s M2 - 7 ∆t=0.00125s Figure 1: Although Figure 1 states that the numerical solution with a grid number of 1280 and a time step of 0.0025s is very close to the analytical solution, Figure 2 shows the variation of υ values with different time steps for a gridindependent mesh number of 12 independence study (Figure 1), the grid independence study, the υ values obtained from numerical solutions do not stabilize as the time step decreases further. Figure 1: Variation of υ values with the time steps for a grid independent mesh number of 1280 4.2 Numerical Predictions Figure 3 presents: (a) the timedependent variation of the inverse coefficient calculated both analytically and numerically, (b) the time solutions, and (c) the timedependent variation of the relative true error between these two solutions. In Figure 3(a), the analytically and numerically calculated inverse co observed that the two inverse coefficients are very close to each other, However, this closeness can be more clearly identified by examining the true error in Figure 3(b). According to Figure 3(b), the true error increa ses as time progresses; however, these increases are very small and remain within reasonable limits. To facilitate a better comparison of the difference between the two inverse coefficients, the relative true errors are shown as percentages in Figure 3(c). be observed that the relative error starts at approximately around 0.25% as time increases. This demonstrates that the analytically calculated inverse coefficient is very close to the num erically calculated inverse coefficient. 2nd Kocaeli Science Congress, November 19 ∆t=0.000625s ∆ t=0.0003125s Figure 1: Grid independence study Although Figure 1 states that the numerical solution with a grid number of 1280 and a time step of 0.0025s is very close to the analytical solution, Figure 2 shows the variation of υ values with different independent mesh number of 12 80. Similar to the observations in the grid independence study (Figure 1), the υ values increase as the time step decreases. However, unlike the values obtained from numerical solutions do not stabilize as the time Variation of υ values with the time steps for a grid independent mesh number of 1280 dependent variation of the inverse coefficient calculated both numerically, (b) the time - dependent variation of the true error between these two dependent variation of the relative true error between these two solutions. In Figure 3(a), the analytically and numerically calculated inverse co efficients increase over time. It is observed that the two inverse coefficients are very close to each other, However, this closeness can be more clearly identified by examining the true error in Figure 3(b). According to Figure 3(b), the true ses as time progresses; however, these increases are very small and remain within reasonable limits. To facilitate a better comparison of the difference between the two inverse coefficients, the relative true errors are shown as percentages in Figure 3(c). From Figure 3(c), it can be observed that the relative error starts at approximately - 3% in the initial stages and decreases to around 0.25% as time increases. This demonstrates that the analytically calculated inverse coefficient erically calculated inverse coefficient. 2nd Kocaeli Science Congress, November 19 - 21, 2025 t=0.0003125s Although Figure 1 states that the numerical solution with a grid number of 1280 and a time step of 0.0025s is very close to the analytical solution, Figure 2 shows the variation of υ values with different 80. Similar to the observations in the grid values increase as the time step decreases. However, unlike the values obtained from numerical solutions do not stabilize as the time Variation of υ values with the time steps for a grid independent mesh number of 1280 dependent variation of the inverse coefficient calculated both dependent variation of the true error between these two dependent variation of the relative true error between these two solutions. efficients increase over time. It is observed that the two inverse coefficients are very close to each other, However, this closeness can be more clearly identified by examining the true error in Figure 3(b). According to Figure 3(b), the true ses as time progresses; however, these increases are very small and remain within reasonable limits. To facilitate a better comparison of the difference between the two inverse From Figure 3(c), it can 3% in the initial stages and decreases to around 0.25% as time increases. This demonstrates that the analytically calculated inverse coefficient M2 - 8 (a) Figure 3: The timedependent variation of (a) the inverse coefficient, (b) true error and (c) relative true Figure 4 presents the timedependent variation of the numerical solution. Both solutions are found to be very close to each other. The concave shape at all time instances. Due to the periodic boundary condition, both ends of the domain have the same value. In the initial moments, the concavity is less pronounced, but it deepens as time progresses. While the minimum value of the edges of concavity increase as time advances. (a) Figure 4: The timedependent variation of the υ value for (a) exact solution, (b) for numerical solution 2nd Kocaeli Science Congress, November 19 (b) (c) dependent variation of (a) the inverse coefficient, (b) true error and (c) relative true error dependent variation of the υ value for (a) the exact solution and (b) the numerical solution. Both solutions are found to be very close to each other. The υ value exhibits a concave shape at all time instances. Due to the periodic boundary condition, both ends of the domain same value. In the initial moments, the concavity is less pronounced, but it deepens as time progresses. While the minimum value of concavity remains relatively constant over time, the values at the edges of concavity increase as time advances. (b) dependent variation of the υ value for (a) exact solution, (b) for numerical solution 2nd Kocaeli Science Congress, November 19 - 21, 2025 dependent variation of (a) the inverse coefficient, (b) true error and (c) relative true value for (a) the exact solution and (b) the value exhibits a concave shape at all time instances. Due to the periodic boundary condition, both ends of the domain same value. In the initial moments, the concavity is less pronounced, but it deepens as time remains relatively constant over time, the values at dependent variation of the υ value for (a) exact solution, (b) for numerical solution M2 - 9 In Figure 5, it can be observed that there is very little difference between the analytical and numerical solutions. However, to better compare the difference between the two calculations, Figure 5 illustrates the timedependent variations of (a) the true error increases over time, forming a shape resembling the letter “m”. It is negative in regions with periodic boundary conditions and positive in areas immediately adjacent to these boundary conditio In Figure 5(b), the absolute relative true error increases with time and is concentrated in the middle of the domain. In the one second solution, the maximum absolute relative error is around 1%, which indicates that the analytical and numerical soluti (a) Figure 5: The timedependent variation of (a) the true error and (b) the absolute relative true error 5. Conclusions In this study, numerical solutions of a one hyperbolic equation with periodic boundary conditions were carried out. An implicit finite difference scheme was used for the numerical solutions. The temporal discretiza order accuracy, while the spatial discretization has second independence study conducted for each time step, an appropriate grid size and time step were determined. The inverse coefficient and step, and accuracy levels were found to align very well with the analytical results. It has been demonstrated that the nonlinear hyperbolic equation representing the wave equation, along with periodic boundary condit ions, can be solved numerically. References [1] Tikhonov, A. N. (1943). "On the Académiedes Sciences de l’URSS, 39(5), 195 [2] Tikhonov, A. N. (1963). "On the Compres Rendus de l’ Académiedes [3] Morozov, V. A. (1996). "Regularization of incorrectly regularization parameters." USSR Computational 252. https://doi.org/10.1016/00415553(66)90046 2nd Kocaeli Science Congress, November 19 In Figure 5, it can be observed that there is very little difference between the analytical and numerical solutions. However, to better compare the difference between the two calculations, Figure 5 illustrates dependent variations of (a) the true error and (b) the absolute relative true error. The true error increases over time, forming a shape resembling the letter “m”. It is negative in regions with periodic boundary conditions and positive in areas immediately adjacent to these boundary conditio absolute relative true error increases with time and is concentrated in the middle of the domain. In the one second solution, the maximum absolute relative error is around 1%, which indicates that the analytical and numerical soluti ons are very close to each other. (b) dependent variation of (a) the true error and (b) the absolute relative true error In this study, numerical solutions of a one - dimensional inverse coefficient nonlinear hyperbolic equation with periodic boundary conditions were carried out. An implicit finite the numerical solutions. The temporal discretiza order accuracy, while the spatial discretization has second - order accuracy. Based on the grid independence study conducted for each time step, an appropriate grid size and time step were determined. The inverse coefficient and υ values obtai ned with the selected grid step, and accuracy levels were found to align very well with the analytical results. It has been demonstrated that the nonlinear hyperbolic equation representing the wave equation, along ions, can be solved numerically. [1] Tikhonov, A. N. (1943). "On the stability of inverse problems." Compres Sciences de l’URSS, 39(5), 195 –198. [2] Tikhonov, A. N. (1963). "On the solution of ill-posed problems and the method of regularization." Académiedes Sciences de l’URSS, 151(3), 501–504. [3] Morozov, V. A. (1996). "Regularization of incorrectly posed problems and parameters." USSR Computational Mathematics and Mathematical Physics, 6(1), 242 5553(66)90046 -2 2nd Kocaeli Science Congress, November 19 - 21, 2025 In Figure 5, it can be observed that there is very little difference between the analytical and numerical solutions. However, to better compare the difference between the two calculations, Figure 5 illustrates error and (b) the absolute relative true error. The true error increases over time, forming a shape resembling the letter “m”. It is negative in regions with periodic boundary conditions and positive in areas immediately adjacent to these boundary conditio ns. absolute relative true error increases with time and is concentrated in the middle of the domain. In the one second solution, the maximum absolute relative error is around 1%, which dependent variation of (a) the true error and (b) the absolute relative true error dimensional inverse coefficient nonlinear hyperbolic equation with periodic boundary conditions were carried out. An implicit finite the numerical solutions. The temporal discretiza tion has firstorder accuracy. Based on the grid independence study conducted for each time step, an appropriate grid size and time step were ned with the selected grid number, time step, and accuracy levels were found to align very well with the analytical results. It has been demonstrated that the nonlinear hyperbolic equation representing the wave equation, along Rendus de l’ method of regularization." the choice of and Mathematical Physics, 6(1), 242 –