On the comparative performance of fourth order Runge-Kutta and the Galerkin-Petrov time discretization methods for solving nonlinear ordinary differential equations with application to some mathematical models in epidemiology

Attaullah; Mansour F. Yassen; Sultan Alyobi; Fuad S. Al-Duais; Wajaree Weera; Attaullah; Mansour F. Yassen; Sultan Alyobi; Fuad S. Al-Duais; Wajaree Weera

doi:10.3934/math.2023185

AIMS Mathematics

2023, Volume 8, Issue 2: 3699-3729. doi: 10.3934/math.2023185

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On the comparative performance of fourth order Runge-Kutta and the Galerkin-Petrov time discretization methods for solving nonlinear ordinary differential equations with application to some mathematical models in epidemiology

1.
Department of Mathematics & Statistics, Bacha Khan University, Charsadda 24461, Pakistan
2.
Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Aflaj 11912, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517 Damietta, Egypt
4.
King Abdulaziz University, College of Science & Arts, Department of Mathematics, Rabigh, Saudi Arabia
5.
Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam bin Abdulaziz University, Al-Kharj, Al-Aflaj 11942, Saudi Arabia
6.
Administration Department, Administrative Science College, Thamar University, Thamar, Yemen
7.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

Received: 01 July 2022 Revised: 05 November 2022 Accepted: 10 November 2022 Published: 25 November 2022
MSC : 34A12, 34K28

Anti-viral medication is comparably incredibly beneficial for individuals who are infected with numerous viruses. Mathematical modeling is crucial for comprehending the various relationships involving viruses, immune responses and health in general. This study concerns the implementation of a continuous Galerkin-Petrov time discretization scheme with mathematical models that consist of nonlinear ordinary differential equations for the hepatitis B virus, the Chen system and HIV infection. For the Galerkin scheme, we have two unknowns on each time interval which have to be computed by solving a $ 2 \times 2 $ block system. The proposed method is accurate to order 3 in the whole time interval and shows even super convergence of order 4 in the discrete time points. The study presents the accurate solutions achieved by means of the aforementioned schemes, presented numerically and graphically. Further, we implemented the classical fourth-order Runge-Kutta scheme accurately and performed various numerical tests for assessing the efficiency and computational cost (in terms of time) of the suggested schemes. The performances of the fourth order Runge-Kutta and the Galerkin-Petrov time discretization approaches for solving nonlinear ordinary differential equations were compared, with applications towards certain mathematical models in epidemiology. Several simulations were carried out with varying time step sizes, and the efficiency of the Galerkin and Runge Kutta schemes was evaluated at various time points. A detailed analysis of the outcomes obtained by the Galerkin scheme and the Runge-Kutta technique indicates that the results presented are in excellent agreement with each other despite having distinct computational costs in terms of time. It is observed that the Galerkin scheme is noticeably slower and requires more time in comparison to the Runge Kutta scheme. The numerical computations demonstrate that the Galerkin scheme provides highly precise solutions at relatively large time step sizes as compared to the Runge-Kutta scheme.
- HBV model,
- the cGP(2)-method,
- RK4-method,
- numerical comparison,
- efficiency of methods
Citation: Attaullah, Mansour F. Yassen, Sultan Alyobi, Fuad S. Al-Duais, Wajaree Weera. On the comparative performance of fourth order Runge-Kutta and the Galerkin-Petrov time discretization methods for solving nonlinear ordinary differential equations with application to some mathematical models in epidemiology[J]. AIMS Mathematics, 2023, 8(2): 3699-3729. doi: 10.3934/math.2023185

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Abstract

Anti-viral medication is comparably incredibly beneficial for individuals who are infected with numerous viruses. Mathematical modeling is crucial for comprehending the various relationships involving viruses, immune responses and health in general. This study concerns the implementation of a continuous Galerkin-Petrov time discretization scheme with mathematical models that consist of nonlinear ordinary differential equations for the hepatitis B virus, the Chen system and HIV infection. For the Galerkin scheme, we have two unknowns on each time interval which have to be computed by solving a $ 2 \times 2 $ block system. The proposed method is accurate to order 3 in the whole time interval and shows even super convergence of order 4 in the discrete time points. The study presents the accurate solutions achieved by means of the aforementioned schemes, presented numerically and graphically. Further, we implemented the classical fourth-order Runge-Kutta scheme accurately and performed various numerical tests for assessing the efficiency and computational cost (in terms of time) of the suggested schemes. The performances of the fourth order Runge-Kutta and the Galerkin-Petrov time discretization approaches for solving nonlinear ordinary differential equations were compared, with applications towards certain mathematical models in epidemiology. Several simulations were carried out with varying time step sizes, and the efficiency of the Galerkin and Runge Kutta schemes was evaluated at various time points. A detailed analysis of the outcomes obtained by the Galerkin scheme and the Runge-Kutta technique indicates that the results presented are in excellent agreement with each other despite having distinct computational costs in terms of time. It is observed that the Galerkin scheme is noticeably slower and requires more time in comparison to the Runge Kutta scheme. The numerical computations demonstrate that the Galerkin scheme provides highly precise solutions at relatively large time step sizes as compared to the Runge-Kutta scheme.

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