Numerical Method for Solving Nonlinear Hyperbolic Equation with Unknown Inverse Coefficient under Periodic Constraints

M1-1 2nd KOCAELI SCIENCE CONGRESS (KOSC-2025) Kocaeli University, Faculty of Arts and Sciences November 19-21, 2025, İzmit, Kocaeli, Türkiye Numerical Method for Solving Nonlinear Hyperbolic Equation with Unknown Inverse Coefficient under Periodic Constraints 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.17984960 Abstract This paper investigates the inverse problem of determining unknown time-dependent coefficients in a one-dimensional nonlinear hyperbolic equation with periodic boundary conditions. The Finite Difference Method (FDM) together with the Gauss–Seidel iteration is used for numerical computation. Two implicit finite difference schemes, the fully implicit and Crank–Nicolson methods, are applied. A numerical example illustrates the effectiveness of the proposed method. Results show that while both schemes align well with experimental data, the implicit scheme provides higher accuracy with lower absolute and relative errors. Keywords: inverse problem, nonlinear hyperbolic equation, finite difference method 1. Introduction Inverse problems play a fundamental role in modern science and engineering, particularly in situations where certain system parameters cannot be measured or computed directly. Their practical significance places them among the most prominent and actively investigated areas of contemporary applied mathematics. Through inverse problem techniques, one can determine essential characteristics of a medium, including wave intensity and propagation velocity, elastic properties, conductivity, electric and magnetic permeability coefficients, as well as the nature and location of inhomogeneities in otherwise inaccessible regions [1–3]. Inverse problems associated with hyperbolic equations are of particular importance in seismology, where they have been extensively studied [4, 5]. The incorporation of non-local boundary conditions, however, significantly complicates both the theoretical analysis and numerical treatment of such problems. A notable example of nonlocal boundary conditions is the use of periodic boundary conditions, which arise in a variety of applications [6–8]. Periodic conditions appear naturally in problems involving cyclic or repetitive phenomena, such as those encountered in lunar theory [9]. M1-2 2nd Kocaeli Science Congress, November 19-21, 2025 In recent years, considerable research effort has focused on inverse coefficient problems for hyperbolic equations, employing a wide range of analytical and numerical methods. Among these approaches, the finite difference method has been frequently adopted and applied in numerous studies [10–13]. In the present study, we investigate an inverse problem of unknown time-dependent coefficient in a one-dimensional nonlinear hyperbolic equation with periodic boundary conditions. For the numerical solution, two different finite-difference schemes—an implicit scheme and the Crank–Nicolson scheme—are applied. The existence, uniqueness, and stability of the analytical solution for this problem have been previously established in [14]. 2. Statement of the Problem Consider a nonlinear hyperbolic equation defined on the domain   : (0, ), (0, ) : D z t T     ( ) ( , , ), tt zz t f z t       (1) with the initial conditions ( ,0) ( ), ( , ) ( ), t z z z t z       (2) and boundary conditions (0, ) ( , ), (0, ) ( , ), z z t t t t         (3) as well as the overdetermination condition 0 ( ) ( , ) . t z z t dz      (4) Where the functions , ,    and ( , , ) f z t  are given functions on   0,  and   , D    , respectively; while the function ( , ) z t  and the coefficient ( ) t  are unknown. Using the Fourier method to solve equations (1)–(4), we obtain 0 0 0 0 10 0 0 0 1 2 ( , ) ( ) ( ) ( , , ) 2 1 cos2 sin 2 ( ) ( , , )cos2 sin 2 ( ) cos2 2 1 cos2 sin 2 ( ) ( , , )sin 2 sin 2 ( ) 2 t t ck ck k t sk sk z t t t f d d kt kt f k k t d d kz k kt kt f k k t d d k                                                                           1 sin 2 . k 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): M1-3 2nd Kocaeli Science Congress, November 19-21, 2025   10 0 0 ( ) 2 cos2 sin 2 ( ) ( , , )sin 2 sin 2 ( ) 2 ( ) . ( , , ) t sk sk k t k kt kt f k k t d d k t zf z t dz                                  (6) Definition 2.1 The problem of finding the pair { ( ), ( , )} 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 0 0 0 01 ( ) ( ), ( ), ( ), 1,..., : max ( ) max ( ) max ( ) ck sk ck sk t T t T t T k t t t t k N t t t                             of continuous on [0, T] functions satisfying the norm   0 0 0 0 1 ( ) max ( ) max ( ) max ( ) . ck sk t T t T t T k t t t t                 Theorem 2.1 If the following assumptions satisfy: A1. 2 ( ) [0, ], ( ) [0, ]. t C T t C T     A2. 1 ( ) [0, ], ( ) [0, ]. z C z C       A3.1 Let the function ( , , ) f z t  be continuous to all arguments in   , D    and satisfies the following condition ( ) ( ) ( ) ( ) ( , , ) ( , , ) ( , ) , 0,1,2, 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   ( ), ( , ) t z t   of the problem (1)-(4) is constantly dependent on the input data , , .    3. Numerical Procedure for the Nonlinear Problem We develop an iterative algorithm to linearize the problem,           2 2 1 2 2 , , , n n n t f z t t z            (7) M1-4 2nd Kocaeli Science Congress, November 19-21, 2025                 ,0 , 0, , ,0 , 0, , n n t z z z z z z           (8)                   ( ) 0, , , 0, , 0, , 0, 0, . n n n n z z t t t T t t t T            (9) By setting       , , n z t z t    and       1 , , , n f z t f z t  , we can express the problem (7)–(9) as a linear problem.       2 2 2 2 , , , , t f z t z t D t z            (10)             ,0 , 0, , ,0 , 0, , t z z z z z z           (11)             0, , , 0, , 0, , 0, 0, . z z t t t T t t t T            (12) Initially, we employ the linearization method, after that we apply two different implicit finite difference schemes, which are implicit and Crank-Nicolson, for solving (10–12). Firstly, the implicit scheme is as follows:     2 1 2 2 2 1 1 1 1 2 2 1 1 2 2 . j j j j j j j j i i i i i i f t z                          (13) Secondly, the Crank-Nicolson scheme is as follows:   2 2 2 1 1 1 2 1 1 1 1 1 1 1 2 2 2 2 2 1 1 2 . 2 j j j j j j j j j j j i i i i i i i i i f t z z                                            (14) For time discretization second order backward difference scheme is used, and for space discretization, second order central difference scheme is applied for implicit and Crank-Nicolson scheme. 0 , i i    1 , j j Nz    2 1 , 2 j j jNz Nz       where the computational domain     0, 0, T   is discretized as flows.   1 , i z i z    1,2,...., , i Nz  j t j t   and 1,2,...., . j Nt  Where z Nz    and t T Nt   are the space in z direction and time steps, respectively. Also, Nz and Nt are two positive integers.   , , j i i j z t      , i i z      1 1 , . j i j f f z t     At time 0, t  adjustments should be made in accordance with the initial condition M1-5 2nd Kocaeli Science Congress, November 19-21, 2025 and compatibility requirements. In our numerical computation, 2 j  exhibits present time, 1 j  signifies previous of present time, and j denotes previous of previous present time. In order to determine the inverse coefficient   t  , integrating Eq. (1) with respect to x from 0 to π and using Eqs. (3) and (4), we obtain         0 " , . , z t t t zf z t dz         (15) The finite difference approximation of Eq. (15) is     1 1 1 1 1 2 1 1 2 , j j j j j Nz Nz j j t z fin                            where       1 1 0 , , . j j j j t fin zf z t dz           For 0 j    1 1 1 0 0 1 2 1 1 2 . Nz Nz t z fin                            For 1 j    2 2 2 1 0 1 2 2 1 2 . Nz Nz t z fin                            In order to calculate   1 , j fin   trapezoidal rule for integration is used. The Nz used for numerical solutions is different from the Nin used for trapezoidal rule for integration. We use Gauss Seidel iteration to compute our implicit (13) and Crank-Nicolson (14) finite difference formulation. Inverse coefficient and source term calculate with using previous present time values. The Gauss–Seidel iteration formula is given as follows:       1 2 1 2 1 2 1 1 , i Nz j s j s j s i ik ik i k k k k i ii ii ii rhs a a a a a                  where s is iteration number. In numerical computations, since time step is very small, we can take   2 0 1 . jj i i      Gauss Seidel iteration for 0, s      1 2 1 2 1 1 1 1 . i Nz j j j i ik ik i k k k k i ii ii ii rhs a a a a a                M1-6 2nd Kocaeli Science Congress, November 19-21, 2025 Right hand side terms for implicit finite difference scheme defines as; 1 1 2 1 2 . j j j j i i i rhs t f             In the same manner, the right-hand side terms ( ) i rhs for the Crank–Nicolson finite difference scheme are defined as:   2 1 1 1 1 1 2 1 1 1 2 2 2 2 j j j j j j j i i i i i i t rhs t f z                          and coefficient matrices ( ) ii a define as with respect to periodic boundary conditions and inner computational nodes. Of course, different coefficient matrices occur for implicit scheme and CrankNicolson scheme. Firstly, in every time step, maximum iteration number defines to stop Gauss Seidel iteration. However, to conclude Gauss Seidel iteration in one time step, tolerance value also defines, therefore in some time step size iteration process ends before the maximum iteration numbers. The numerical methods which are described above are implemented in the in-house implicit finite difference code. 4. Numerical Example If we consider inverse problem (1)-(4) with     4 4 , , sin2 , ( ) sin2 , , 2 t t f z t e z z z t e              0, , 0, . z t T    Then the problem becomes 4 ( ) sin2 , t tt zz t e z         0, ,0 z t T         ,0 sin2 , ,0 0, t z z z         0, , , t t         0, , , z z t t       4 0 , . 2 t z z t dz e     Verifying the analytical solution of this problem is straightforward           4 2 4 , , 12 16 4 , sin2 . t t z t t t e z      In the numerical solutions, 200 of number of meshes (Nz) are used, therefore our space discretization in z direction is 0.0157. To evaluate the integral, Nz meshes are once again employed. Time step size is 0.005s. Maximum iteration per time step is taken as 100 for Gauss Seidel iteration process. However, tolerance value of Gauss Seidel iteration is kept as 10-6. Within this tolerance values, iteration number is less than maximum iteration per time step for every time step. Figure 1 represents the inverse coefficient distribution with time. Numerical and exact results of inverse coefficient enhance with time. M1-7 2nd Kocaeli Science Congress, November 19-21, 2025 Figure 1: Exact and numerical solutions of ( ) t  from 0s to 1s. In order to compare exact and numeric solutions of inverse coefficient, true error and relative true error can be considered. Table 1 shows the exact solution, true error and relative true error of ( ) t  at specific times for implicit and Crank-Nicolson schemes. Generally, true error and relative true error enhances with time for both schemes. The implicit scheme and the Crank–Nicolson scheme have produced equivalent true error and relative true error. Maximum relative true error is observed at 1.0s, 4.1840% and 4.1740% occurred for implicit and Crank-Nicolson schemes, respectively. Relative true error is less than 5.0% for both schemes, therefore, both numerical results are close to exact results for inverse coefficient. Table 1: The exact solution, true error and relative true error of ( ) t  t[s] Exact True error for implicit Relative true error for implicit [%] True error for Crank-Nicolson Relative true error for Crank-Nicolson [%] 0.1 4.1200 0.0799 1.9392 0.1181 2.8663 0.2 4.4810 0.0429 0.9565 0.0477 1.0638 0.3 5.0917 0.0694 1.3627 0.0798 1.5668 0.4 5.9855 0.1037 1.7329 0.1234 2.0610 0.5 7.2500 0.1508 2.0805 0.1791 2.4706 0.6 9.0665 0.2211 2.4388 0.2560 2.8235 0.7 11.7624 0.3321 2.8231 0.3697 3.1435 0.8 15.8743 0.5139 3.2372 0.5482 3.4533 0.9 22.2231 0.8191 3.6860 0.8410 3.7844 1.0 32.0000 1.3389 4.1840 1.3357 4.1740 Figure 2 shows exact and numerical (implicit and Cranck-Nicolson scheme) solutions of the ( , ) z t  for (a) t=0.25s, (b) t=0.5s, (c), 0.75s and (d) t=1.0s. Hyperbolic profiles can be observed for exact and both numerical solutions at given times. With increasing time, depth of hyperbolic profiles increases. At a given time, symmetric hyperbolic profiles are produced. Solutions of exact and implicit scheme and Crank-Nicolson scheme are so close each other. M1-8 2nd Kocaeli Science Congress, November 19-21, 2025 (a) (b) (c) (d) Figure 2: Exact and numerical solutions of the ( , ) z t  for (a) t=0.25s, (b) t=0.5s, (c), 0.75s and (d) t=1.0s Figure 3 presents the hyperbolic profiles in all times for solutions of (a) exact and (b) implicit scheme and (c) CrankNicolson scheme. The same observations can be made for Figure 2. (a) Exact (b) Implicit (c) Crank-Nicolson Figure 3: Exact and numerical solutions of the ( , ) z t  for all times In the same manner, in order to correctly compare the exact solution with both numerical computations (implicit scheme and Crank–Nicolson scheme) of ( , ) z t  , the true error and the relative true error can again be observed. Figure 4 always represents the true errors and relative true errors of M1-9 2nd Kocaeli Science Congress, November 19-21, 2025 both numerical computations. True errors increase with time, and the maximum true errors are observed at the peak point of hyperbolic profiles for both numerical computations. True error Relative true error Implicit Crank-Nicolson Figure 4: True error and relative true error of ( , ) z t  for all times The maximum true errors of implicit schema and Crank-Nicolson scheme are nearly -0.04 and -0.06 according to exact solution. When examining the relative true error, it is observed that the maximum relative true errors occur at the boundary points. For the implicit scheme, this value is approximately 0.5%, whereas for the Crank–Nicolson scheme, it is around 1%. In both schemes, the relative true error remains below 1%; however, the implicit scheme is roughly twice as accurate as the Crank– Nicolson scheme. 5. Conclusions Analytical and numerical investigation of one-dimensional nonlinear hyperbolic equation with periodic conditions is carried out. This investigation contains an inverse problem of unknown unsteady coefficient. For analytical solution, the generalized Fourier method is utilized to calculate Fourier coefficients. For numerical solutions, two different implicit finite difference schemes, namely, implicit and CrankNicolson, with Gauss Seidel iteration process are applied. The major conclusions are listed below: 1. Inverse coefficient enhances with time for exact and numerical computations. 2. The implicit scheme and the Crank–Nicolson scheme have predicted the inverse coefficient to the same level of accuracy. 3. One-dimensional symmetric hyperbolic profiles are produced analytically and numerically. With time increases, the depth of the hyperbolic profiles also enhances. 4. The maximum relative true error of Crank-Nicolson scheme (1%) is doubled to implicit scheme (0.5%); therefore, estimation of implicit scheme is closer than Crank-Nicolson scheme