Finite difference schemes for time-dependent convection <i>q</i>-diffusion problem

Yasir Nawaz; Muhammad Shoaib Arif; Kamaleldin Abodayeh; Mairaj Bibi; Yasir Nawaz; Muhammad Shoaib Arif; Kamaleldin Abodayeh; Mairaj Bibi

doi:10.3934/math.2022897

AIMS Mathematics

2022, Volume 7, Issue 9: 16407-16421. doi: 10.3934/math.2022897

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Research article

Finite difference schemes for time-dependent convection q-diffusion problem

1.
Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan
2.
Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh, Saudi Arabia
3.
Stochastic Analysis and Optimization Research Group, Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan
4.
Department of Mathematics, Comsats University Islamabad, Park Road, Islamabad, 45550, Pakistan

Received: 13 April 2022 Revised: 31 May 2022 Accepted: 07 June 2022 Published: 06 July 2022
MSC : 65Mxx, 76Rxx

The energy balance ordinary differential equations (ODEs) model of climate change is extended to the partial differential equations (PDEs) model with convections and q-diffusions. Instead of integer order second-order partial derivatives, partial q-derivatives are considered. The local stability analysis of the ODEs model is established using the Routh-Hurwitz criterion. A numerical scheme is constructed, which is explicit and second-order in time. For spatial derivatives, second-order central difference formulas are employed. The stability condition of the numerical scheme for the system of convection q-diffusion equations is found. Both types of ODEs and PDEs models are solved with the constructed scheme. A comparison of the constructed scheme with the existing first-order scheme is also made. The graphical results show that global mean surface and ocean temperatures escalate by varying the heat source parameter. Additionally, these newly established techniques demonstrate predictability.
- energy balance model,
- convection and q-diffusion,
- heat source,
- second-order scheme,
- stability
Citation: Yasir Nawaz, Muhammad Shoaib Arif, Kamaleldin Abodayeh, Mairaj Bibi. Finite difference schemes for time-dependent convection q-diffusion problem[J]. AIMS Mathematics, 2022, 7(9): 16407-16421. doi: 10.3934/math.2022897

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Abstract

The energy balance ordinary differential equations (ODEs) model of climate change is extended to the partial differential equations (PDEs) model with convections and q-diffusions. Instead of integer order second-order partial derivatives, partial q-derivatives are considered. The local stability analysis of the ODEs model is established using the Routh-Hurwitz criterion. A numerical scheme is constructed, which is explicit and second-order in time. For spatial derivatives, second-order central difference formulas are employed. The stability condition of the numerical scheme for the system of convection q-diffusion equations is found. Both types of ODEs and PDEs models are solved with the constructed scheme. A comparison of the constructed scheme with the existing first-order scheme is also made. The graphical results show that global mean surface and ocean temperatures escalate by varying the heat source parameter. Additionally, these newly established techniques demonstrate predictability.

References

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