376 © 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 A Numerical Study of Initial and Boundary Conditions Problem for Wave Equation Using the Operational Matrices Myasar Obaid Enadi 1* and Majeed A. AL-Jawary 2 1,2 Department of Mathematics, College of Education for Pure Sciences (Ibn AL-Haitham), University of Baghdad, Baghdad, Iraq. *Corresponding Author. Received: 6 December 2024 Accepted: 12 May 2025 Published: 20 October 2025 doi.org/10.30526/38.4.4097 Abstract In this paper, orthogonal polynomials and their operational matrices will be utilized to address the initial and boundary value problems of the one-dimensional wave problem, where the domain of the space variable is bounded, which covers a variety of scientific and engineering operations. Six types of orthogonal polynomials, Include instead of such as the Genocchi, Bernoulli, Legendre, Boubaker, Chebyshev and Standard polynomials. The linear problem with its initial and boundary conditions are transformed to a linear algebraic equations, which can then be solved by utilizing to get an approximate solution for this problem. Some test problems related to the one-dimensional wave equation with different conditions are discussed and solved to show how reliable and efficient the proposed methods. The error norm and the mean square error , were computed; these are presented through analytical tables and graphics showing the rapid convergence for these methods. Keywords: Wave problem; Operational matrices,Orthogonal Polynomials,Approximate solutions. 1. Introduction The mathematical foundation for many problems in mathematics, physics, engineering, and chemistry can be explained mathematically using partial differential equations (PDEs). In physics, for example, partial differential equations clear describe wave propagation and heat transfer. Furthermore, partial differential equations were utilized to describe the majority of physical processes in domains such as electricity, quantum mechanics, wave propagation in shallow water, fluid dynamics, plasma physics, and others (1). There is a need to find effective and reliable approximate or analytical techniques that can handle PDEs because of these massive applications, there is a need to develop effective and trustworthy approximate or analytical procedures for dealing with PDEs. Many mathematicians and engineers have solved a wide range of functional equations using the VIM by He (2–4). Many tapes of scientific, physical, and engineering problems can be classified as initial boundary value problems (IBVP). With a few exceptions, we are unable to find accurate analytical answers to the majority of these difficulties. There have been several attempts to https://orcid.org/0009-0005-2426-4558 mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-3967-0012 mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0005-2426-4558 mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-3967-0012 mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0005-2426-4558 mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-3967-0012 mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0005-2426-4558 mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-3967-0012 mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0005-2426-4558 mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-3967-0012 mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0005-2426-4558 mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-3967-0012 mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq IHJPAS. 2025,38(4) 377 develop approximate and analytical methods for solving the non-linear diffusion and wave problems, see (5–8). Wave equations can only be solved analytically in highly particular situations, hence many realistic cases cannot be solved using commonly used analytical methods. In addition to discretization methods such as finite difference, finite volume, and finite element approaches, various more ways have been proposed to solve the wave problem. For example, the analytical approach to solving the wave equation (9), the numerical solution of the wave issue (10,11), and the optimal homotopy strategy for solving the nonlinear wave equation (12). Furthermore, there are numerous techniques that offer an approximate resolve for the differing kinds of the differential equations, see, (13–15). Corrington suggested in 1973 that a system of algebraic linear equations could be created from linear differential and integral equations by using a least squares approximation and repeatedly integrating Walsh functions (16). In addition, Sparis and Mouroutsos in 1986 using the orthogonal polynomial series operational matrices to solve differential equations (17), many researchers have used orthogonal polynomials to find approximate solutions for many applications. See; (18–28). The authors were highly interested in since they were practical methods for resolving an extensive number of approximation theory and numerical analysis problems. On the other hand, the orthogonal polynomials and operational matrices stand out above other types due to their effective reduction of the required solution, which is achieved by applying the operational matrices technique to transform the linear differential equations into linear algebraic systems of equations, where any computer program can be used to solve them. Farther more, Turkyilmazoglu in 2013 proposed an approximate analytical technique for resolving differential equations based on standard polynomials, which is used on various types of problems (29–33). In 2023, Othman, et al. introduced several orthogonal polynomials, for instance, Hermite, Laguerre, Chebyshev, and others polynomials with inner product, to develop the computational method (CM) (34). Myasar, et al. (35,36), further introduced the Genocchi, Bernoulli, and Boubaker polynomials, which contribute to the effective computational method (ECM). The outline for this paper is as follows: Section two shows the second-order linear wave equation's mathematical formulation. Section three: Preliminary of the orthogonal polynomials. Section four: Main results and applications of orthogonal polynomials to resolve some examples for the wave problem. Section five: gives the conclusions. 2. The Mathematical Model of the Wave Equation Let us consider the one-dimensional wave problem (7): ( ) (1) with initial conditions: ( ) ( ) ( ) ( ) (2) and boundary conditions: ( ) ̂( ) (3) ( ) ̂( ) . (4) where ( ) is a particular function, Ω is the bounded domain, is a constant, ̂( ) and ̂( ) are known functions, , -( ) ( ), and ( ) the external normal vector to the boundary , when Equation 1 will be homogeneous wave equation, and if Equation 1 will be non-homogeneous wave problem. IHJPAS. 2025,38(4) 378 3. Preliminary of the orthogonal polynomials and their operational matrices The orthogonal polynomials and their operational matrices play an important part applied and pure mathematics as well as numerical calculation. Six types of these orthogonal polynomials will be used: standard polynomials ( ), Legendre, Chebyshev, Boubaker, Genocchi, and Bernoulli polynomials to resolve the wave problem with initial and boundary conditions. 3.1. The operational matrices for standard polynomials Assume that the wave problem (1), with the initial and boundary conditions (2-4), has a unique solution, this can be implemented using a suitable linear transformation. Taking the basic functions * ( ) ( ) ( ) +, and * ( ) ( ) ( ) +, Let the solution of Equation 1, may be presented by using the double series extension in terms of a base functions ( ) ∑ ∑ ( ) ( ) , (5) The Equation 5 may be written using the dot product, as follows: ( ) ∑ ∑ ( ) ( ) ( ) ( ) , (6) where the coefficients will be evaluated later, by use the definition: ( ) , -, ( ) , - , and [ ] , Furthermore, the -order partial derivatives for and with respect and are obtained by: ( ) ( ) , ( ) ( ) ( ), where is the derivative matrix that is dimensional ( ) ( ) is given by: [ ] , Thus, the derivatives of ( ) will be obtained using Equation 6 have the following forms: ( ) ( ) ( ) ( ) ( ) ( ) ( ) (7) 3.2. The operational matrices for Boubaker polynomials The -degree of Boubaker polynomials ( ) are defined as follows (28,37): ( ) ∑ ( ) ( ) ( ⁄ ) also Boubaker polynomials can be computed by used the iterative procedure as follows: ( ) ( ) ( ), . Furthermore, suppose that the unknown function ( ), may be approximated by applying the double series based on the Boubaker polynomial: ( ) ∑ ∑ ( ) ( ) ( ) ( ) , (8) where ( ) , ( ) ( )-, ( ) , ( ) ( )- , and [ ] , where are the Boubaker polynomials coefficients will be evaluated latter. IHJPAS. 2025,38(4) 379 The derivatives of ( ) may be transformed into matrices by utilizing the following formulation: ( ) ( ) ( ) ( ) ( ) ( ) ( ) (9) The derivatives matrix of Boubaker polynomials denoted by , and if is even or is odd, Defined as follows, respectively: [ ] , [ ] . The following relations can be used to calculate the components { } 0 1 : ( ) ( ) ∏ ( ) 0 1. 3.3. The operational matrices for Bernoulli polynomials Bernoulli polynomials ( ) of degree is given by (27,38,39): ( ) ∑ . / , where is the Bernoulli number, such that ∑ ( ) . /∑ , for , , The first Bernoulli numbers define as follows: . Additionally, the unknown function ( ) can be approximated by using the following dot product based on the Bernoulli polynomials: ( ) ∑ ∑ ( ) ( ) ( ) ( ) , (10) where ( ) , ( ) ( )-, ( ) , ( ) ( )- , and [ ] , where is the unknown coefficients of Bernoulli polynomials determined latter. Moreover, the derivatives of ( ) can be written in matrices form by the following formula: ( ) ( ) ( ) ( ) ( ) ( ) ( ) (11) where is the derivative matrix of Bernoulli polynomials, is given as below: IHJPAS. 2025,38(4) 380 [ ] , 3.4. The operational matrices for Legendre polynomials Legendre polynomials ( ) of degree is given as follows (24,40): ( ) , ( ) , …. , ( ) ( ) ( ) ( ) , . Furthermore, the Legendre polynomials will be get by using the formula as follows: ( ) ∑ ( ) ( ) ( ) ( ) ( ) By using, the linear combination as follow can be approximated the unknown function ( ). ( ) ∑ ∑ ( ) ( ) ( ) ( ) , (12) where ( ) , ( ) ( )-, ( ) , ( ) ( )- , and [ ] , Moreover, is the coefficients of Legendre polynomials evaluated latter. The derivative of ( ) with a respect to and may be written in matrices as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) (13) The derivatives matrix of the Legendre polynomials denoted by , and we obtained by (34): 2 ( ) If is even then , and if is odd then . 3.5. The operational matrices for Genocchi polynomials The Genocchi polynomials ( ) of degree is given by (20): ( ) ∑ . / , By using, the linear combination as follow can be approximated the function ( ). ( ) ∑ ∑ ( ) ( ) ( ) ( ) , (14) where ( ) , ( ) ( )- , ( ) ( ( ) ( )) and [ ] Moreover, is the unknown coefficient of Genocchi polynomials evaluated latter. The derivatives of ( ) can be written in matrices form as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) (15) The derivative matrix of Genocchi polynomials is denoted by and given by: [ ] , IHJPAS. 2025,38(4) 381 3.6. The operational matrices for Chebyshev polynomials Chebyshev polynomials ( ) of degree are defined as follows (25,41): ( ) ∑ ( ) ( ) ( ) ( ) ( ) , In addition, the function ( ) may be approximated by using the linear combination as following: ( ) ∑ ∑ ( ) ( ) ( ) ( ) , (16) where ( ) , ( ) ( )- , ( ) , ( ) ( )-, and [ ] , Furthermore, represents the unknown coefficients of Chebyshev polynomials calculated latter. The derivatives of ( ) may be rewritten in matrices using the following formula: ( ) ( ) ( ) ( ) ( ) ( ) ( ) (17) The derivative matrix of the Chebyshev polynomials is denoted by , and obtained by: { If is odd then and if is even then , and , and for . 3.7. Algorithm 1. Input (integer) n. Input (double series) tools. Input (array) = , (initial approximation, with dimension, are chosen so that the boundary conditions are satisfied). 2. ̂ ( ) ̂ is a system of linear algebraic equations that has been solved and is obtained. Go to (2). 2.1 If | | tol then , break (the program is finished). 2.2 Else then . 3. Go to (2). 4. Main Results and Applications of Orthogonal Polynomials in Applied Science In this section, will be applied the proposed methods to resolve some test problems of the non-homogeneous wave equation to get the approximate solution. We use two distinct error criteria to assess the accuracy of suggested methods, which were calculated by using . The norm is defined by: ( ∑ . ( ) ( )/ ∑ . ( )/ ) ⁄ , and the norm which is defined by: | ( ) ( )|. where and represent the numerical and analytical solutions, respectively. Example 1. Consider the following inhomogeneous wave problem: (18) with initial conditions: IHJPAS. 2025,38(4) 382 ( ) ( ) (19) and Dirichlet boundary conditions: ( ) ( ) . (20) and the analytical solution is: ( ) . By applying the proposed method to resolve this problem with initial and boundary condition gives in Equations 19, and 20, respectively. More specifically, by converting the unknown function ( ) with partial derivatives in to linear equations system. Furthermore, assume that . First: Applying the standard polynomials. By inserting the Equations 6, and 7 into Equations 18, 19 and 20 the -order partial derivatives and conditions are converted into matrices, such that: ( ) ( ) ( ) ( ) ( ) , - [ ] [ ] , - , - [ ] [ ] , - , , (21) For the Equation 6, becomes as following: ( ) , - [ ] , - , ( ) (22) From Equation 7, the first derivatives of ( ) with a respective its: ( ) ( ) ( ) ( ) , - [ ] [ ] , - , , (23) By substituting the IC from Equation 19, when in to Equations 22, and 23, will be obtained: ( ) ( ) ( ) , (24) ( ) ( ) ( ) ( ) , (25) Also, by inserting the BC from Equation 20 when in to Equation 22, will be get: ( ) ( ) ( ) , (26) ( ) ( ) ( ), ( ) ( ) ( ) ( ) , (27) By resolve the linear algebraic system consisting of the Equations 21, 24, 25, 26, and 27, to find the values of , , , , , , , and , using the , we get: [ ] [ ]. By replacing these values in Equations 6 or 22, therefore, the approximate solution using IHJPAS. 2025,38(4) 383 standard polynomials ( ) as follows: ( ) ( ) ( ) Finally, by using the other proposed methods such as: Boubaker, Bernoulli, Chebyshev, Genocchi, and Legendre, the approximate solution for test example 1 as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Table 1. The values of and for example 1, when . IHJPAS. 2025,38(4) 384 Norm . Norm . Figure 1. The plots logarithmic of and , for example 1, when . In Table 1 and Figure 1, it can be seen the values of the norm and . They depend on the value of , which represents the degree of the polynomial. The higher the value of gives the lower error. The best results were achieved using the standard polynomials when, , the values of and equals, , , respectively, followed by Bernoulli polynomials. Figure 2. Plots the norm for example 1, . Exact solution Approximate solution Figure 3. Comparison between analytical and numerical solution for example 1 by the standard poly. IHJPAS. 2025,38(4) 385 Example 2. Consider the inhomogeneous wave problem as follows: ( ) ( ) (28) with initial conditions: ( ) ( ) (29) and Dirichlet boundary conditions: ( ) ( ) ( ). (30) and the analytical solution is: ( ) ( ). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) such that represented the operational matrices of all proposed methods. Let us use the to solve the given test problem and determine the and : Table 2. The values of and for example 2, . The error norm and mean square error values are readily seen in Table 2, and Figure 4, which can be obtained by solving example 2 using Mathematica. Some methods gave close IHJPAS. 2025,38(4) 386 results, but the Bernoulli polynomials gave better results when , such that equals , respectively, but when the Bernoulli and standard polynomials give the same results. Norm . Norm . Figure 4. The plots logarithmic of and , for example 2, when . Figure 5. Plots the norm for example 2, . Exact solution Approximate solution Figure 6. Comparison between analytical and numerical solution for example 2 by the Bernoulli poly. Example 3. Consider the homogeneous wave problem as follows (7): (31) IHJPAS. 2025,38(4) 387 with initial conditions: ( ) ( ) ( ) (32) and Neumann boundary conditions: ( ) ( ) ( ) ( ). (33) and the analytical solution is: ( ) ( ) ( ). By implementing the proposed methods will be get the following linear equations system: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). Table 3. The values of and for example 3, when . From Table 3. and Figure 7., the error norm and mean square error values are visible, as these values decrease as the value of increases, which represents the degree of the polynomial. The best methods for approximation are the Bernoulli polynomials, such that when , the values of and equals , , IHJPAS. 2025,38(4) 388 respectively, but when , it can be seen the values of and gives , . Norm . Norm . Figure 7. The plots logarithmic of and , for example 3, when . Figure 8. Plots the norm for example 3, . Exact solution Approximate solution Figure 9. Comparison between analytical and numerical solution for example 3 by the Bernoulli poly. Example 4: Consider the following inhomogeneous wave problem (7): (34) with initial conditions: IHJPAS. 2025,38(4) 389 ( ) ( ) ( ) (35) and Neumann boundary conditions: ( ) ( ) ( ) ( ). (36) and the analytical solution is: ( ) ( ) ( ). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). Table 4. The values error norm and for example 4, when . From Table 4, and Figure 10, it can be seen the values of the and . The best approximation method was the Bernoulli polynomial, where the value of the error norm was equal to and the value of the mean square error was equal to when , also when , the values of , and are , , respectively. IHJPAS. 2025,38(4) 390 Norm . Norm . Figure 10. The plots logarithmic of and , for example 4, when . Figure 11. Plots the norm for example 4, . Exact solution Approximate solution Figure 12. Comparison between analytical and numerical solution for example 4 by the Bernoulli poly. Example 5. Consider the following homogeneous wave problem: , (37) with initial conditions: ( ) ( ) (38) IHJPAS. 2025,38(4) 391 and mixed boundary conditions: ( ) ( ) . (39) and the analytical solution is: ( ) . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . Table 5. The values and for example 5, when . The mean square error and the error norm values can be easily observed in Table 5, and Figure 13, which we obtained by solving example 5 using Mathematica. Some methods gave close results; except for the Bernoulli polynomial, which gave better results in general. The values of , and when can be observed to be equals to , and equals , , respectively, when . IHJPAS. 2025,38(4) 392 Norm . Norm . Figure 13. The plots logarithmic of and , for example 5, when . Figure 14. Plots the norm for example 5, when . Exact solution Approximate solution Figure 15. Comparison between analytical and numerical solution for example 5 by the Bernoulli poly. Example 6. Consider the following homogeneous wave problem: (40) with initial conditions: ( ) ( ) ( ) (41) and mixed boundary conditions: ( ) ( ) ( ) ( ) ( ) (42) IHJPAS. 2025,38(4) 393 and the analytical solution is: ( ) ( ) ( ). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). Table 6. The values of and for example 6, when . From Table 6, and Figure 16, the error norm and mean square error values are presented. The best approximation method was the Bernoulli polynomial, where the value of the error norm was equal to and the value of the mean square error was equal to when . IHJPAS. 2025,38(4) 394 Norm . Norm . Figure 16. The plots logarithmic of and , for example 6, when . Figure 17. Plots the norm for example 6, when . Exact solution Approximate solution Figure 18. Comparison between analytical and numerical solution for example 6 by the Bernoulli poly. 5. Conclusion In this work, the orthogonal polynomials and operational matrices based on standard polynomials were proposed and applied to resolve non-homogeneous wave problems. The approximate solutions were achieved and showed to be reliable and effective, even for low- order polynomials. Additionally, the mean square error and the norm were calculated to assess the accuracy, dependability, and validity of the approaches. The results IHJPAS. 2025,38(4) 395 show that the recommended solutions have high accuracy and lower error rates. 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