A positive and bounded convergent scheme for general space-fractional diffusion-reaction systems with inertial times

ABSTRACT

We consider a multidimensional system of hyperbolic equations with fractional diffusion, constant damping and nonlinear reactions. The system considers fractional Riesz derivatives, and generalizes many models from science. In particular, the system describes the dynamics of populations with temporal delays, whence the need to approximate nonnegative and bounded solutions is an important numerical task. Motivated by these facts, we propose a scheme to approximate the solutions. We prove the existence of the solutions under suitable regularity assumptions on the reaction functions. We prove that the scheme is capable of preserving positivity and boundedness. The technique has consistency of the second order in space and time. Using a discrete form of the energy method, we establish the stability and the convergence. As a corollary, we prove the uniqueness of the solutions. Some computer simulations in the two- and the three-dimensional scenarios are provided at the end of this work for illustration purposes.

Keywords:

• Hyperbolic system of partial differential equations
• Riesz-type fractional diffusion
• positivity- and boundedness-preserving scheme
• stability and convergence analyses
• discrete fractional energy method

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M22
• 65Q10

Acknowledgements

The authors would like to thank the anonymous reviewers for their invaluable comments. All of their criticisms and suggestions were taken into account, resulting in a substantial improvement in the overall quality of this manuscript. Conceptualization, J.E.M.-D.; methodology, J.E.M.-D.; validation, J.E.M.-D.; formal analysis, J.A.-P. and J.E.M.-D.; investigation, J.E.M.-D.; resources, J.A.-P. and J.E.M.-D.; data curation, J.E.M.-D.; writing–original draft preparation, J.A.-P. and J.E.M.-D.; writing–review and editing, J.E.M.-D.; software, J.E.M.-D.; visualization, J.E.M.-D.; supervision, J.E.M.-D.; project administration, J.E.M.-D.; funding acquisition, J.E.M.-D.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author would like to acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACYT) to perform doctoral studies at the Universidad Autónoma de Aguascalientes. Meanwhile, the corresponding author wishes to acknowledge the financial support from CONACYT through grant A1-S-45928, in support of the research project ‘Conservative methods for fractional hyperbolic systems: analysis and applications’.