A Numerical Study of Initial and Boundary Conditions Problem for Wave Equation Using the Operational Matrices

Myasar Obaid Enadi; Majeed A.  AL-Jawary

doi:10.30526/38.4.4097

Authors

Myasar Obaid Enadi Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq. https://orcid.org/0009-0005-2426-4558
Majeed A. AL-Jawary Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq. https://orcid.org/0000-0003-3967-0012

DOI:

https://doi.org/10.30526/38.4.4097

Keywords:

Wave problem; Operational matrices, Orthogonal Polynomials, Approximate solutions

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.

Author Biographies

Myasar Obaid Enadi, Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq.

Department of Mathematics
Majeed A. AL-Jawary, Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq.

Department of Mathematics

References

1. Wazwaz AM. Linear and nonlinear integral equations. Berlin: Springer; 2011.

2. 2. He JH. Variational iteration method for delay differential equations. Commun Nonlinear Sci Numer Simul. 1997;2(4):235–236. https://doi.org/10.1016/S1007-5704(97)90008-3

3. 3. He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput. 2000;114(2–3):115–123. https://doi.org/10.1016/S0096-3003(99)00104-6

4. 4. He JH. Variational iteration method-Some recent results and new interpretations. J Comput Appl Math. 2007;207(1):3–17. https://doi.org/10.1016/j.cam.2006.07.009

5. 5. Wazwaz AM. The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations. Comput Math with Appl. 2007;54(7–8):933–939. https://doi.org/10.1016/j.camwa.2006.12.038

6. 6. Wazwaz AM. The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations. Comput Math with Appl. 2007;54(7–8):926–932. https://doi.org/10.1016/j.camwa.2006.12.038

7. 7. Wazwaz AM. Partial differential equations and solitary waves theory. Springer Science & Business Media; 2010.

8. 8. Mansour MB. Accurate computation of traveling wave solutions of some nonlinear diffusion equations. Wave Motion. 2007;44(3): 222-230.‏ doi:10.1016/j.wavemoti.2006.10.001

9. 9. Biazar J, Ghazvini H. An analytical approximation to the solution of a wave equation by a variational iteration method. Appl Math Lett. 2008;21(8):780–5. https://doi.org/10.1016/j.aml.2007.08.004

10. 10. Britt S, Tsynkov S, Turkel E. Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials. J Comput Phys. 2018;354:26–42. https://doi.org/10.1016/j.jcp.2017.10.049

11. 11. Cheniguel A. Numerical Method for Solving Wave Equation with Non Local Boundary Conditions. Proc Int MultiConference Eng Comput Sci. 2013;2:13–5.

12. 12. Ullah, H., Islam, S., Dennis, L.C.C., Abdelhameed, T.N., Khan, I. and Fiza M. Approximate solution of two-dimensional nonlinear wave equation by optimal homotopy asymptotic method. Math Probl Eng. 2015;2015(1):380104. https://doi.org/10.1155/2015/380104

13. 13. Abolarin OE. New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems. Am J Comput Math. 2013;03(02):110–113. https://doi.org/10.4236/ajcm.2013.32018

14. 14. Kumar S, Kumar A, Momani S, Aldhaifallah M, Nisar KS. Numerical solutions of nonlinear fractional model arising in the appearance of the stripe patterns in two-dimensional systems. Adv Differ Equations. 2019;2019(1):1-19. https://doi.org/10.1186/s13662-019-2334-7

15. 15. Kumar S, Kumar A, Abbas S, Al Qurashi M, Baleanu D. A modified analytical approach with existence and uniqueness for fractional Cauchy reaction–diffusion equations. Adv Differ Equations. 2020;2020(1):28. https://doi.org/10.1186/s13662-019-2488-3

16. 16. Corrington MS. Solution of Differential and Integral Equations with Walsh Functions. IEEE Trans Circuit Theory. 1973;20(5):470–476.

17. 17. Sparis PD, Mouroutsos SG. The operational matrix of differentiation for orthogonal polynomial series. Int J Control. 1986;44(1):1–15. https://doi.org/10.1080/00207178608933579

18. 18. Kosunalp HY, G¨ulsu M. Operational Matrix by Hermite Polynomials for Solving Nonlinear Riccati Differential Equations. Int J Math Comput Sci. 2020;16(2):525–536.

19. 19. Bagherpoorfard M, Ghassabzade FA. Hermite Matrix Polynomial Collocation Method for Linear Complex Differential Equations and Some Comparisons. J Appl Math Phys. 2013;01(05):58–64. http://dx.doi.org/10.4236/jamp.2013.15009

20. 20. Isah A, Phang C. Operational matrix based on Genocchi polynomials for solution of delay differential equations. Ain Shams Eng J. 2018;9(4):2123–2128. https://doi.org/10.1016/j.asej.2016.09.015

21. 21. Abdelkawy MA, Taha TM. An operational matrix of fractional derivatives of laguerre polynomials. Walailak J Sci Technol. 2014;11(12):1041–1055. http://dx.doi.org/10.14456/WJST.2014.59

22. 22. Yousefi SA, Behroozifar M. Operational matrices of Bernstein polynomials and their applications. Int J Syst Sci. 2010;41(6):709–716. https://doi.org/10.1080/00207720903154783

23. 23. Yousefi SA, Behroozifar M, Dehghan M. The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass. Journal of computational and applied mathematics. 2011;235(17):5272-83. doi:10.1016/j.cam.2011.05.038

24. 24. Ibraheem GH, Al-Jawary MA. The operational matrix of Legendre polynomials for solving nonlinear thin film flow problems. Alexandria Eng J. 2020;59(5):4027–4033. https://doi.org/10.1016/j.aej.2020.07.008

25. 25. Sharma B, Kumar S, Paswan MK, and Mahato D. Chebyshev Operational Matrix Method for Lane-Emden Problem. Nonlinear Eng. 2019;8(1):1–9. https://doi.org/10.1515/nleng-2017-0157

26. 26. El-Gamel M E, Adel W, El-Azab MS. Collocation Method Based on Bernoulli Polynomial and Shifted Chebychev for Solving the Bratu Equation. J Appl Comput Math. 2018;07(03).

27. 27. Singh S, Patel VK, Singh VK, Tohidi E. Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions. Comput Math with Appl. 2018;75(7):2280–2294. https://doi.org/10.1016/j.camwa.2017.12.003

28. 28. Bencheikh A, Chiter L. Boubaker Operational Matrix Method for Fractional Emden-Fowler Problem. J Sib Fed Univ - Math Phys. 2023;16(3):357–369. https://www.mathnet.ru/eng/jsfu1085

29. 29. Turkyilmazoglu M. Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type. Appl Math Model. 2013;37(14–15):7539–7548. https://doi.org/10.1016/j.apm.2013.02.014

30. 30. Turkyilmazoglu M. An effective approach for numerical solutions of high-order Fredholm integro-differential equations. Appl Math Comput. 2014;227:384–398. https://doi.org/10.1016/j.amc.2013.10.079

31. 31. Turkyilmazoglu M. Effective computation of solutions for nonlinear heat transfer problems in Fins. J Heat Transfer. 2014;136(9):1–7. https://doi.org/10.1115/1.4027772

32. 32. Turkyilmazoglu M. Solution of Initial and Boundary Value Problems by an Effective Accurate Method. Int J Comput Methods. 2017;14(6). https://doi.org/10.1142/S0219876217500694

33. 33. Turkyilmazoglu M. Parabolic Partial Differential Equations with Nonlocal Initial and Boundary Values. Int J Comput Methods. 2015;12(5):1–10. https://doi.org/10.1142/S0219876215500243

34. 34. Salih OM, AL-Jawary MA. Computational Methods for Solving Nonlinear Ordinary Differential Equations Arising in Engineering and Applied Sciences. Iraqi J Sci. 2023;64(8):4070–4091. DOI: 10.24996/ijs.2023.64.8.30

35. 35. Enadi MO, Al-Jawary MA. The operational matrices for Elliptic Partial Differential Equations with mixed boundary conditions. J Phys Conf Ser. 2024;2857(1):1–18. DOI 10.1088/1742-6596/2857/1/012027

36. 36. Enadi MO, AL-Jawary MA, Turkyilmazoglu M. The operational matrices methods for solving the Poisson equation with nonlocal boundary conditions. J Appl Math Comput. 2025;71(2):2385–2432. https://doi.org/10.1007/s12190-024-02313-y

37. 37. Davaeifar S, Rashidinia J. Boubaker polynomials collocation approach for solving systems of nonlinear Volterra–Fredholm integral equations. J Taibah Univ Sci. 2017;11(6):1182–1199. https://doi.org/10.1016/j.jtusci.2017.05.002

38. 38. Bhrawy AH, Tohidi E, Soleymani F. A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl Math Comput. 2012;219(2):482–497.‏ https://doi.org/10.1016/j.amc.2012.06.020

39. 39. Toutounian F, Tohidi E. A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis. Appl Math Comput. 2013;223:298–310. https://doi.org/10.1016/j.amc.2013.07.094

40. 40. Jung, C.Y., Liu, Z., Rafiq, A., Ali, F. and Kang SM. Solution of second order linear and nonlinear ordinary differential equations using Legendre operational matrix of differentiation. Int J Pure Appl Math. 2014;93(2):285–295. doi: http://dx.doi.org/10.12732/ijpam.v93i2.12

41. Doha EH, Bhrawy AH, Ezz-Eldien SS. A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput Math with Appl. 2011;62(5):2364–2373. ‏ https://doi.org/10.1016/j.camwa.2011.07.024.