Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-07T10:53:36.090613Z
Paper Citation Record · LEDGER
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation
As of 8 August 2026, this Paper Citation Record lists 18 of 18 outbound references and 0 inbound Pith citation observations for arXiv:2506.04420.
A citation records a reference. It does not transfer a finding from one paper to another.
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One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-07T06:34:17.273281+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links
A source-named dated measurement, never combined with another source.
Source: cited_works
Reference resolution
18 of 18 outbound references displayed
- verified exact0
- verified fuzzy18
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External citation measurements
No source-named external measurement is stored.
Outbound references
Observation 38de4bf8-6e94-46f4-991d-b07eea212d92 · outbound
This paper cites Uniqueness and nonuniqueness criteria for ordinary differential equations, volume 6.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Uniqueness and nonuniqueness criteria for ordinary differential equations, volume 6
Reference 1
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:38.813424Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation b41e6a44-2886-4c1c-a3ac-1614078bf195 · outbound
This paper cites Gradient flows: in metric spaces and in the space of probability measures.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Gradient flows: in metric spaces and in the space of probability measures
Reference 2
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:38.657198Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 2d5b37be-de33-4e7b-8805-19217a1ecd89 · outbound
This paper cites Dynamical an alysis of fractional-order chemostat model.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Dynamical an alysis of fractional-order chemostat model
Reference 3
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:38.519189Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation d6ad8bbf-c1aa-40fe-a171-bb0ec6d8a63f · outbound
This paper cites Im provement of performances of the chemostat used for continuous biological water treatment with periodic co ntrols.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Im provement of performances of the chemostat used for continuous biological water treatment with periodic co ntrols
Reference 4
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:38.367923Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation b1e933a6-1d25-4c0e-bf46-5a0a96d24582 · outbound
This paper cites On periodic solutions of fractional-order differ- ential systems with a fixed length of sliding memory.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation On periodic solutions of fractional-order differ- ential systems with a fixed length of sliding memory
Reference 5
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:38.226252Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 4150e490-f81a-4a4a-84fd-713ed4700935 · outbound
This paper cites Theory of ordinary differential equations.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Theory of ordinary differential equations
Reference 6
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:38.088648Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 4dcb4d5a-3d15-48cf-8839-43a498dc59d2 · outbound
This paper cites The analysis of fractional diff erential equations.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation The analysis of fractional diff erential equations
Reference 7
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:37.952696Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation d753898a-60fe-4b8f-a4b0-65293c7d8b6e · outbound
This paper cites Unsteady state process operat ion.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Unsteady state process operat ion
Reference 8
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:37.792145Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 91b3c93a-cbad-4f6d-8b29-4b5ac595892b · outbound
This paper cites New optimal periodic control policy fo r the optimal periodic performance of a chemostat using a Fourier–Gegenbauer-based predictor-corrector me thod.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation New optimal periodic control policy fo r the optimal periodic performance of a chemostat using a Fourier–Gegenbauer-based predictor-corrector me thod
Reference 9
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:37.646883Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation fd851a90-f2c1-457d-9d20-7d5f6b0490f7 · outbound
This paper cites Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems
Reference 10
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:37.465945Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation ceb93cfc-c3a6-471f-91ae-22173a6f2d73 · outbound
This paper cites Fourier–Gegenbauer pseudospectral method for solving periodic higher-order fractional optimal control problems.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Fourier–Gegenbauer pseudospectral method for solving periodic higher-order fractional optimal control problems
Reference 11
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:37.305925Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 0e8f1ad8-cd0d-46df-a980-146ebd59e40a · outbound
This paper cites Fourier-Gegenbauer pseudospectral method for solving time-dependent one-dimensional fractional partial differential equations with variable c oefficients and periodic solutions.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Fourier-Gegenbauer pseudospectral method for solving time-dependent one-dimensional fractional partial differential equations with variable c oefficients and periodic solutions
Reference 12
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:37.152200Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 19abd33b-d9ec-4262-aa4a-15ddf3ebd21c · outbound
This paper cites Stability a nalysis of fractional-order chemostat model with time delay.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Stability a nalysis of fractional-order chemostat model with time delay
Reference 13
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:37.042313Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation f2d7723f-0fba-46fc-bfc6-026902bc3cee · outbound
This paper cites Real and complex analysis.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Real and complex analysis
Reference 14
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:36.903407Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 1a9239b9-3043-4259-81fe-c06bf4aec9f6 · outbound
This paper cites Fractio nal Adams–Bashforth/Moulton methods: An application to the fractional Keller–Segel chemotaxis system.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Fractio nal Adams–Bashforth/Moulton methods: An application to the fractional Keller–Segel chemotaxis system
Reference 15
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:36.724603Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 9d6703ed-ee08-4d56-95a2-f1f5ea41a559 · outbound
This paper cites Approximation of fractional- order Chemostat model with nonstandard finite dif- ference scheme.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Approximation of fractional- order Chemostat model with nonstandard finite dif- ference scheme
Reference 16
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:36.553263Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation a971ff75-2d7a-4279-99b1-9d762fa9dd70 · outbound
This paper cites A numerical method for discret e fractional-order chemostat model derived from nonstandard numerical scheme.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation A numerical method for discret e fractional-order chemostat model derived from nonstandard numerical scheme
Reference 17
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:36.411362Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Observation 7c8edebb-a319-4f59-9f62-a8252d6db1e2 · outbound
This paper cites Fo rward dynamics and memory effect in a fractional order chemostat minimal model with non-monotonic growth.
Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation Fo rward dynamics and memory effect in a fractional order chemostat minimal model with non-monotonic growth
Reference 18
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-07T10:53:36.257068Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.
Pith citing papers
No inbound Pith citation observations are available.