On numerical solutions of telegraph, viscous, and modified Burgers equations via Bernoulli collocation method

Document Type : Article

Authors

1 -Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, 35511, Egypt - Laboratoire Interdisciplinaire de l Universite Francaise d Egypte (UFEID Lab), Universite Francaise d'Egypte, Cairo 11837, Egypt

2 Faculty of Modern Technologies Engineering, Amol University of Special Modern Technologies, Amol, 4615664616, Iran

3 - Department of Mathematics, Firat University, 23119, Elazig, Turkey - Department of Medical Research, China Medical University, Taichung, Taiwan

Abstract

The presented work aims to develop a novel technique for solving a general form of both linear and nonlinear partial differential equations (PDEs). This technique is based on applying a collocation method with the aid of Bernoulli polynomials and the use of such an algorithm to solve different types of PDEs. The method applies the regular finite difference scheme to convert the model equation into a system of a linear or nonlinear algebraic equation and then this system is solved using a novel iterative technique. Then, by solving this system an unknown coefficient is acquired and an approximate solution for the problems is achieved. Some test results of famous equations including the telegraph, viscous Burger, and modified Burger equations are presented to demonstrate the effectiveness of the proposed algorithm along with a comparison with other related techniques. The method proves to provide accurate results in terms of absolute error and through graphical representation of the solution.

Keywords

Main Subjects

References

References:
1. Gao, F. and Chi, C. "Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation", Appl. Math. Comput., 187, pp. 1272-1276 (2007).
2. Rabczuk, T., Huilong, R., and Xiaoying, Z. "A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem", Comput. Mater. Contin., 59, pp. 31-55 (2019).
3. Raissi, M., Paris, P., and George, E.K. "Physicsinformed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations", J. Comput. Phys., 378, pp. 686-707 (2019).
4. Zang, Y., Gang, B., Xiaojing, Y., et al. "Weak adversarial networks for high-dimensional partial differential equations", J. Comput. Phys., 411, 109409 (2020).
5. Alesemi, M., Iqbal, N., and Botmart, T. "Novel analysis of the fractional-order system of non-linear partial differential equations with the exponentialdecay kernel", Mathematics, 10(4), p. 615 (2022).
6. Rawani, M.K., Lajja, V., Verma, A.K., et al. "On a weakly L-stable time integration formula coupled with nonstandard finite difference scheme with application to nonlinear parabolic partial differential equations", Math. Methods Appl. Sci., 45(3), pp. 1276-1298 (2022).
7. Bateman, H. "Some recent researches on the motion of  fluids", Mon. Weather Rev., 43(4), pp. 163-170 (1915). 
8. Burgers, J.M. "A mathematical model illustrating the theory of turbulence", Adv. Appl. Mech., 1, pp. 171-199 (1948). 
9. Cole, J.D. "On a quasi-linear parabolic equation occurring in aerodynamics", Quart. Appl. Math., 9(3), pp. 225-236 (1951).
10. Tari, H., Ganji, D.D., and Babazadeh, H. "The application of He's variational iteration method to nonlinear equations arising  in heat transfer", Phys. Lett. A., 363(7), pp. 213-217 (2007).
11. Bratsos, A.G. "A fourth-order numerical scheme for solving the modified Burgers equation", Comput. Math. Appl., 60(5), pp. 1393-1400 (2010).
12. Korkmaz, A. and Dag, I. "Shock wave simulations using sinc differential quadrature method", Eng. Comput., 28(6), pp. 654-674 (2011).
13. Korkmaz, A., Aksoy, M., and Dag, I. "Quartic B-spline differential quadrature method", Int. J. Nonlinear Sci., 11(4), pp. 403-411 (2011).
14. Korkmaz, A. and Dag, I. "Polynomial based differential quadrature method for numerical solution of nonlinear Burgers' equation", J. Franklin Inst., 348(10), pp. 2863-2875 (2011).
15. Wang, K., Wang, G.D., and Zhu, H.W. "A new perspective on the study of the fractal coupled Boussinesq- Burger equation in shallow water", Fractals, 29(5), 2150122 (2021).
16. Usman, M., Hamid, M., and Liu, M. "Novel operational matrices-based finite difference/spectral algorithm for a class of time-fractional Burger equation in multidimensions", Chaos Solitons Fractals, 144, 110701 (2021).
17. Hashmi, M.S., Misbah, W., Yao, S.W., et al. "Cubic spline based differential quadrature method: A numerical approach for fractional Burger equation", Results Phys., 26, 104415 (2021).
18. Ahmad, S., Ullah, A., Akgul, A., et al. "A novel homotopy perturbation method with applications to nonlinear fractional order KdV and Burger equation with exponential-decay kernel", J. Funct. Spaces, 2021, 8770488 (2021).
19. Zhang, R.P., Yu, X.J., and Zhao, G.Z. "Modified Burgers' equation by the local discontinuous Galerkin method", Chin. Phys. B., 22(3), 030210 (2013).
20. Bratsos, A.G. and Abdul, K. "An exponential time differencing method of lines for the Burgers and the modified Burgers equations", Numer. Methods Partial Differ. Equ., 34(6), pp. 2024-2039 (2018).
21. Seydaoglu, M. "An accurate approximation algorithm for Burgers' equation in the presence of small viscosity", J. Comput. Appl. Math., 344, pp. 473-481 (2018).
22. Mohamed, N. "Fully implicit scheme for solving Burgers' equation based on finite difference method", Egypt. J. Eng. Sci. Technol., 26(26), pp. 38-44 (2018).
23. Arora, G. and Varun, J. "A computational approach using modified trigonometric cubic B-spline for numerical solution of Burgers' equation in one and two dimensions", Alex. Eng. J., 57(2), pp. 1087-1098 (2018).
24. Aswin, V.S., Ashish, A., and Mohammad, M.R. "A differential quadrature based numerical method for highly accurate solutions of Burgers' equation", Numer. Methods Partial Differ. Equ., 33(6), pp. 2023- 2042 (2017).
25. Du, M.J., Wang, Y.L., Temuer, C.L., et al. "A modified reproducing kernel method for solving Burgers' equation arising from diffusive waves in fluid dynamics", Appl. Math. Comput., 315, pp. 500-506 (2017).
26. Zhang, X., Jiang, Y., Hu, Y., et al. "High-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers' equations", Math. Comput. Simulation, 196, pp. 151-165 (2022).
27. El-Gamel, M., Adel, W., and El-Azab, M.S. "Collocation method based on Bernoulli polynomial and shifted Chebychev for solving the Bratu equation", J. Appl. Computat. Math., 7, p. 3 (2018).
28. El-Gamel, M. "Two very accurate and efficient methods for solving time-dependent problems", Appl. Math., 9(11), p. 1270 (2018).
29. Adel, W. and Sabir, Z. "Solving a new design of nonlinear second-order Lane-Emden pantograph delay differential model via Bernoulli collocation method", Eur. Phys. J. Plus, 135(6), p. 427 (2020).
30. Bazm, S. and Hosseini, A. "Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra-Fredholm integral equations of Hammerstein type", Comput. Appl. Math., 39(2), p. 49 (2020).
31. Zeghdane, R. "Numerical solution of stochastic integral equations by using Bernoulli operational matrix", Math. Comput. Simulation, 165, pp. 238-254 (2019).
32. Adel, Waleed. "A numerical technique for solving a class of fourth-order singular singularly perturbed and emden-fowler problems arising in astrophysics", Int. J. Appl. Comput. Math., 8, pp. 1-18 (2022).
33. Adel, Waleed, Kubra Erdem Bicer, and Mehmet Sezer. "A novel numerical approach for simulating the nonlinear MHD jeffery-hamel flow problem", Int. J. Appl. Comput. Math. 7(3), pp. 1-15 (2021).
34. Pandey, K., Verma, L., and Verma, A.K. "On a finite difference scheme for Burgers' equation", Appl. Math. Comput., 215(6), pp. 2206-2214 (2009).
35. Mittal, R.C. and Rohila, R. "A study of onedimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method", J. Math. Chem., 55(2), pp. 673-695 (2017).
Scientia Iranica
Volume 31, Issue 1
Transactions on Mechanical Engineering (B)
January and February 2024
Pages 43-54

Files

Share

How to cite

Statistics