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Modeling of fractional order DPG model insight global warming and pollution effect on desertification for control mechanism

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Abstract

This study presents a novel fractional-order mathematical model that investigates the dynamic interplay between dust pollutants, plant biomass, and global warming, referred to as the DPG system. This research introduces a fractional-order formulation that incorporates ecological memory and long-range interactions, providing a more realistic representation of desertification dynamics than classical integer-order models. It also establishes a comprehensive analytical numerical framework designed to capture system feedback, assess instability patterns, and evaluate the effectiveness of chaos control mechanisms. The model utilizes Caputo derivatives to capture the memory effects inherent in ecological and atmospheric processes. Key parameters such as dust emission, plant decay, and global warming feedback mechanisms are integrated into a nonlinear system of differential equations. Analytical evaluations ensure the existence, uniqueness, and generalized Hyers-Ulam-Rassias stability of the proposed system. Sensitivity analysis identifies the parameters that have the most significant influence on desertification risk. Furthermore, a Newton polynomial-based numerical scheme is constructed to efficiently simulate system behavior under varying fractional orders. Chaos control strategies are implemented to stabilize the system near critical equilibrium points. Numerical simulations reveal that lower fractional orders dampen dust accumulation, slow plant biomass regeneration, and delay global warming trends, highlighting the efficacy of fractional modeling in capturing real-world environmental inertia and feedback. This research provides a robust analytical and computational foundation for understanding ecosystem resilience in the face of both anthropogenic and climatic stressors.

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Funding

Open access funding provided by Symbiosis International (Deemed University).

Author information

Authors and Affiliations

  1. Mathematics Research Center, Department of Mathematics, Near East University, 99138, Nicosia, Turkey

    Muhammad Farman & Saba Jamil

  2. Jadara University Research Center,, Jadara University, Irbid, Jordan

    Muhammad Farman

  3. Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan

    Khadija Jamil & Saba Jamil

  4. Symbiosis Institute of Technology, Hyderabad Campus, Symbiosis International (Deemed University), Pune, India

    Ausif Padder

  5. Operational Research Center in Healthcare, Near East University, 99138 Mersin 10, Nicosia/TRNC, Turkey

    Hijaz Ahmad

  6. Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, 02841, Seoul, South Korea

    Hijaz Ahmad

  7. Department of Mathematics, Sonari College, 785690, Sonari, Assam, India

    Kaushik Dehingia

  8. Mathematics and Computer Engineering, Biruni University, Istanbul, Turkey

    Mustafa Bayram

  9. International Center for Interdisciplinary Research in Sciences, The University of Lahore, , Lahore,, Pakistan

    Khadija Jamil

  10. Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan

    Muhammad Farman

Authors

  1. Muhammad Farman
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  3. Saba Jamil
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  4. Ausif Padder
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Contributions

Muhammad Farman: Formal analysis, methodology, writing, reviewing, editing, software, and resources. Khadija Jamil: Data curation, investigation, software, writing, and original draft. Saba Jamil: Formal analysis, software, editing, and resources. Ausif Padder: Conceptualization, formal analysis, software, methodology, writing, reviewing, and editing. Hijaz Ahmadf: Data curation, investigation, software, writing, and original draft. Kaushik Dehingia: Formal analysis, software, editing, and resources. Mustafa Bayram: Formal analysis, software, editing, and resources.

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Ausif Padder.

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Appendix

Appendix

For simplicity, we make the following observations about the given system

$$begin{aligned} begin{aligned}&{}^CD_t^omega Dleft( t right) = {mathscr {Y} _1}left( {t,D} right) ,\ &{}^CD_t^omega {P}left( t right) = {mathscr {Y} _2}left( {t,P} right) ,\ &{}^CD_t^omega {G}left( t right) = {mathscr {Y} _3}left( {t,{D}} right) . end{aligned} end{aligned}$$

So,

$$begin{aligned} Dleft( t right) – Dleft( 0 right) = frac{{1 – omega }}{{Gamma left( omega right) }}int limits _0^t {{mathscr {Y} _1}left( {sigma ,Dleft( sigma right) } right) {{left( {t – sigma } right) }^{omega – 1}}dsigma .} end{aligned}$$

At the point (t_{u +1} = (u + 1)Delta t), the system can be expressed as

$$begin{aligned} Dleft( {{t_{u + 1}}} right) – Dleft( 0 right) = frac{{1 – omega }}{{Gamma left( omega right) }}int limits _0^{{t_{u + 1}}} {{mathscr {Y} _1}left( {sigma ,Dleft( sigma right) } right) {{left( {{t_{u + 1}} – sigma } right) }^{omega – 1}}dsigma .} end{aligned}$$

As a result,

$$begin{aligned} Dleft( {{t_{u + 1}}} right) = Dleft( 0 right) + frac{{1 – omega }}{{Gamma left( omega right) }}sum limits _{v = 2}^u {int limits _{{t_o}}^{{t_{o + 1}}} {{mathscr {Y} _1}left( {sigma ,Dleft( sigma right) } right) {{left( {{t_{u + 1}} – sigma } right) }^{omega – 1}}dsigma .} } end{aligned}$$
(25)

Equation (25) can be employed as a substitute for the Newton polynomial to generate the numerical solution of the system.

$$begin{aligned} begin{aligned} D ^{u + 1}&= {{D}_0} + frac{{1 – omega }}{{Gamma left( omega right) }}sum limits _{o = 2}^u {int limits _{{t_o}}^{{t_{o + 1}}} {left{ {{mathscr {Y} _1}left( {{t_{o – 2}},{D^{o – 2}}} right) + frac{{{mathscr {Y} _1}left( {{t_{o – 1}},{D^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D^{o – 2}}} right) }}{{Delta t}}} right. } } left( {omega – {t_{o – 2}}} right) \ &left. {frac{{{mathscr {Y} _1}left( {{t_o},{D^o}} right) – 2{mathscr {Y} _1}left( {{t_{o – 1}},{D^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D^{o – 2}}} right) }}{{2{{left( {Delta t} right) }^2}}}left( {sigma – {t_{o – 2}}} right) left( {sigma – {t_{o – 1}}} right) } right} times {left( {{t_{o + 1}} – omega } right) ^{omega – 1}}dsigma .end{aligned} end{aligned}$$
(26)

Accordingly, the preceding equation can be reformulated as follows

$$begin{aligned} begin{aligned}{D ^{u + 1}}&= {{D}_0} + frac{{1 – omega }}{{Gamma left( omega right) }}sum limits _{o = 2}^u {left{ {int limits _{{t_o}}^{{t_{o + 1}}} {{mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) {{left( {{t_{u + 1}} – sigma } right) }^{omega – 1}}dsigma } } right. } \ &+ int limits _{{t_o}}^{{t_{o + 1}}} {frac{{{mathscr {Y} _1}left( {{t_{o – 1}},{D ^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) }}{{Delta t}}} left( {sigma – {t_{o – 2}}} right) {left( {{t_{u + 1}} – sigma } right) ^{omega – 1}}dsigma \ &+ int limits _{{t_o}}^{{t_{o + 1}}} {left. {frac{{{mathscr {Y} _1}left( {{t_o},{D ^o}} right) – 2{mathscr {Y} _1}left( {{t_{o – 1}},{D ^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) }}{{2{{left( {Delta t} right) }^2}}}left( {sigma – {t_{o – 2}}} right) left( {sigma – {t_{o – 1}}} right) {{left( {{t_{u + 1}} – omega } right) }^{omega – 1}}dsigma } right} } .end{aligned} end{aligned}$$
(27)

Consequently,

$$begin{aligned} begin{aligned}{D ^{u + 1}}&= {{D}_0} + frac{{1 – omega }}{{Gamma left( omega right) }}sum limits _{o = 2}^u {{mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) int limits _{{t_o}}^{{t_{o + 1}}} {{{left( {{t_{u + 1}} – sigma } right) }^{omega – 1}}dsigma } } \ &+ frac{1}{{Gamma left( omega right) }}sum limits _{o = 2}^u {frac{{{mathscr {Y} _1}left( {{t_{o – 1}},{D ^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) }}{{Delta t}}} int limits _{{t_o}}^{{t_{o + 1}}} {left( {sigma – {t_{o – 2}}} right) {{left( {{t_{u + 1}} – sigma } right) }^{omega – 1}}dsigma } \ &+ frac{1}{{Gamma left( omega right) }}sum limits _{o = 2}^u {frac{{{mathscr {Y} _1}left( {{t_o},{D ^o}} right) – 2{mathscr {Y} _1}left( {{t_{o – 1}},{D ^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) }}{{2{{left( {Delta t} right) }^2}}}} int limits _{{t_o}}^{{t_{o + 1}}} {left( {sigma – {t_{o – 2}}} right) left( {sigma – {t_{o – 1}}} right) {{left( {{t_{u + 1}} – omega } right) }^{omega – 1}}dsigma .} end{aligned} end{aligned}$$
(28)

Equation (28) can be employed to evaluate the aforementioned integrals

$$begin{aligned} int limits _{{t_o}}^{{t_{o + 1}}} {{{left( {{t_{o + 1}} – sigma } right) }^{omega – 1}}dsigma = frac{{{{left( {Delta t} right) }^omega }}}{omega }left[ {{{left( {u – o + 1} right) }^omega } – {{left( {u – o} right) }^omega }} right] ,} end{aligned}$$
$$begin{aligned} int limits _{{t_o}}^{{t_{o + 1}}} {left( {sigma – {t_{o – 2}}} right) {{left( {{t_{u + 1}} – sigma } right) }^{omega – 1}}dsigma } = frac{{{{left( {Delta t} right) }^{omega + 1}}}}{{omega left( {omega + 1} right) }}left[ {{{left( {u – o + 1} right) }^omega }left( {u – o + 3 + 2omega } right) – {{left( {u – o} right) }^omega }left( {u – o + 3 + 3omega } right) } right] , end{aligned}$$
$$begin{aligned} begin{array}{l}int limits _{{t_o}}^{{t_{o + 1}}} {left( {sigma – {t_{o – 2}}} right) left( {sigma – {t_{o – 1}}} right) {{left( {{t_{u + 1}} – omega } right) }^{omega – 1}}dsigma = frac{{{{left( {Delta t} right) }^{omega + 2}}}}{{omega left( {omega + 1} right) left( {omega + 2} right) }}} \ times left[ {begin{array}{l}{{{left( {u – o + 1} right) }^omega }left[ {begin{array}{l}{2{{left( {u – o} right) }^2} + left( {3omega + 10} right) left( {u – o} right) }\ { + 2{omega ^2} + 9omega + 12}end{array}} right] }\ { – {{left( {u – o} right) }^omega }left[ {begin{array}{l}{2{{left( {u – o} right) }^2} + left( {5omega + 10} right) left( {u – o} right) }\ { + 6{omega ^2} + 18omega + 12}end{array}} right] }end{array}} right] .end{array} end{aligned}$$

Substituting these values into equation (28) yields the following computational procedure

$$begin{aligned} begin{aligned}{D ^{u + 1}}&= {{D}_0} + frac{{{{left( {Delta t} right) }^omega }}}{{Gamma left( {omega + 1} right) }}sum limits _{o = 2}^u {{mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) times mathfrak {Q}} \ &+ frac{{{{left( {Delta t} right) }^omega }}}{{Gamma left( {omega + 2} right) }}sum limits _{o = 2}^u {left[ {{mathscr {Y} _1}left( {{t_{o – 1}},{D ^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) } right] times {mathfrak {Q}_1}} \ &+ frac{{{{left( {Delta t} right) }^{omega + 2}}}}{{omega left( {omega + 1} right) left( {omega + 2} right) }}sum limits _{o = 2}^u {left[ {{mathscr {Y} _1}left( {{t_o},{D ^o}} right) – 2{mathscr {Y} _1}left( {{t_{o – 1}},{D ^{o – 1}}} right) – {mathscr {Y} _1}left( {{t_{o – 2}},{D ^{o – 2}}} right) } right] times {mathfrak {Q}_2}.} end{aligned} end{aligned}$$
(29)

Similarly, corresponding expressions can be derived for the remaining system equations in (24).

$$begin{aligned} begin{aligned}{P ^{u + 1}}&= {{P}_0} + frac{{{{left( {Delta t} right) }^omega }}}{{Gamma left( {omega + 1} right) }}sum limits _{o = 2}^u {{mathscr {Y} _2}left( {{t_{o – 2}},{P ^{o – 2}}} right) times mathfrak {Q}} \ &+ frac{{{{left( {Delta t} right) }^omega }}}{{Gamma left( {omega + 2} right) }}sum limits _{o = 2}^u {left[ {{mathscr {Y} _2}left( {{t_{o – 1}},{P ^{o – 1}}} right) – {mathscr {Y} _2}left( {{t_{o – 2}},{P ^{o – 2}}} right) } right] times {mathfrak {Q}_1}} \ &+ frac{{{{left( {Delta t} right) }^{omega + 2}}}}{{omega left( {omega + 1} right) left( {omega + 2} right) }}sum limits _{o = 2}^u {left[ {{mathscr {Y} _2}left( {{t_o},{P ^o}} right) – 2{mathscr {Y} _2}left( {{t_{o – 1}},{P ^{o – 1}}} right) – {mathscr {Y} _2}left( {{t_{o – 2}},{P ^{o – 2}}} right) } right] times {mathfrak {Q}_2},} end{aligned} end{aligned}$$
$$begin{aligned} begin{aligned}{G ^{u + 1}}&= {{G}_0} + frac{{{{left( {Delta t} right) }^omega }}}{{Gamma left( {omega + 1} right) }}sum limits _{o = 2}^u {{mathscr {Y} _3}left( {{t_{o – 2}},{G ^{o – 2}}} right) times mathfrak {Q}} \ &+ frac{{{{left( {Delta t} right) }^omega }}}{{Gamma left( {omega + 2} right) }}sum limits _{o = 2}^u {left[ {{mathscr {Y} _3}left( {{t_{o – 1}},{G ^{o – 1}}} right) – {mathscr {Y} _2}left( {{t_{o – 2}},{G ^{o – 2}}} right) } right] times {mathfrak {Q}_1}} \ &+ frac{{{{left( {Delta t} right) }^{omega + 2}}}}{{omega left( {omega + 1} right) left( {omega + 2} right) }}sum limits _{o = 2}^u {left[ {{mathscr {Y} _3}left( {{t_o},{D ^o}} right) – 2{mathscr {Y} _3}left( {{t_{o – 1}},{G ^{o – 1}}} right) – {mathscr {Y} _3}left( {{t_{o – 2}},{G ^{o – 2}}} right) } right] times {mathfrak {Q}_2},} end{aligned} end{aligned}$$

Where

$$begin{aligned} begin{aligned}&mathfrak {Q} = left[ {{{left( {u – o + 1} right) }^omega } – {{left( {u – o} right) }^omega }} right] ,\ &{mathfrak {Q}_1} = left[ {{{left( {u – o + 1} right) }^omega }left( {u – o + 3 + 2omega } right) – {{left( {u – o} right) }^omega }left( {u – o + 3 + 3omega } right) } right] ,\ &{mathfrak {Q}_2} = left[ begin{array}{l}{left( {u – o + 1} right) ^omega }left[ begin{array}{l}2{left( {u – o} right) ^2} + left( {3omega + 10} right) left( {u – o} right) \ + 2{omega ^2} + 9omega + 12end{array} right] \ – {left( {u – o} right) ^omega }left[ begin{array}{l}2{left( {u – o} right) ^2} + left( {5omega + 10} right) left( {u – o} right) \ + 6{omega ^2} + 18omega + 12end{array} right] end{array} right] . end{aligned} end{aligned}$$

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Farman, M., Jamil, K., Jamil, S. et al. Modeling of fractional order DPG model insight global warming and pollution effect on desertification for control mechanism.
Sci Rep (2026). https://doi.org/10.1038/s41598-026-47606-3

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Keywords

  • Caputo derivative
  • Hyers-Ulam-Rassias stability
  • Phase Surface Simulations
  • Chaos Control
  • Modeling

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