Numerical investigation for the nonlinear model of hepatitis-B virus with the existence of optimal solution

Naveed Shahid; Muhammad Aziz-ur Rehman; Nauman Ahmed; Dumitru Baleanu; Muhammad Sajid Iqbal; Muhammad Rafiq; Naveed Shahid; Muhammad Aziz-ur Rehman; Nauman Ahmed; Dumitru Baleanu; Muhammad Sajid Iqbal; Muhammad Rafiq

doi:10.3934/math.2021480

AIMS Mathematics

2021, Volume 6, Issue 8: 8294-8314. doi: 10.3934/math.2021480

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Numerical investigation for the nonlinear model of hepatitis-B virus with the existence of optimal solution

1.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
2.
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
3.
Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
4.
Institute of Space Sciences, Magurele-Bucharest, Romania
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
6.
Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, Pakistan

Received: 12 December 2020 Accepted: 06 May 2021 Published: 28 May 2021
MSC : 65N06, 92B99, 37M15

In the recent article, a reaction-advection-diffusion model of the hepatitis-B virus (HBV) is studied. Existence and uniqueness of the optimal solution for the proposed model in function spaces is analyzed. The advection and diffusion terms make the model more generic than the simple model. So, the numerical investigation plays a vital role to understand the behavior of the solutions. To find the existence and uniqueness of the optimal solutions, a closed and convex subset (closed ball) of the Banach space is considered. The explicit estimates regarding the solution of the system for the admissible auxiliary data is computed. On the other hand, for the numerical approximation of the solution, an elegant numerical technique is devised to find the approximate solutions. After constructing the discrete model, some fundamental properties must necessarily be possessed by the proposed numerical scheme. For instance, consistency, stability, and positivity of the solutions. These properties are carefully studied in the current article. To prove the positivity of the proposed scheme, M-matrix theory is used. All the above mentioned properties are verified by sketching the graph via simulations. Furthermore, these plots are helpful to understand the true behavior of the solutions. For this purpose, a fruitful discussion is included about the simulations to justify our results.
- reaction,
- advection,
- diffusion,
- optimal solution,
- explicit estimates,
- auxiliary data,
- structure preserving
Citation: Naveed Shahid, Muhammad Aziz-ur Rehman, Nauman Ahmed, Dumitru Baleanu, Muhammad Sajid Iqbal, Muhammad Rafiq. Numerical investigation for the nonlinear model of hepatitis-B virus with the existence of optimal solution[J]. AIMS Mathematics, 2021, 6(8): 8294-8314. doi: 10.3934/math.2021480

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Abstract

In the recent article, a reaction-advection-diffusion model of the hepatitis-B virus (HBV) is studied. Existence and uniqueness of the optimal solution for the proposed model in function spaces is analyzed. The advection and diffusion terms make the model more generic than the simple model. So, the numerical investigation plays a vital role to understand the behavior of the solutions. To find the existence and uniqueness of the optimal solutions, a closed and convex subset (closed ball) of the Banach space is considered. The explicit estimates regarding the solution of the system for the admissible auxiliary data is computed. On the other hand, for the numerical approximation of the solution, an elegant numerical technique is devised to find the approximate solutions. After constructing the discrete model, some fundamental properties must necessarily be possessed by the proposed numerical scheme. For instance, consistency, stability, and positivity of the solutions. These properties are carefully studied in the current article. To prove the positivity of the proposed scheme, M-matrix theory is used. All the above mentioned properties are verified by sketching the graph via simulations. Furthermore, these plots are helpful to understand the true behavior of the solutions. For this purpose, a fruitful discussion is included about the simulations to justify our results.

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