{"id":"https://openalex.org/W4309785584","doi":"https://doi.org/10.3390/sym14112462","title":"A New Numerical Approach for Variable-Order Time-Fractional Modified Subdiffusion Equation via Riemann\u2013Liouville Fractional Derivative","display_name":"A New Numerical Approach for Variable-Order Time-Fractional Modified Subdiffusion Equation via Riemann\u2013Liouville Fractional Derivative","publication_year":2022,"publication_date":"2022-11-21","ids":{"openalex":"https://openalex.org/W4309785584","doi":"https://doi.org/10.3390/sym14112462"},"language":"en","primary_location":{"id":"doi:10.3390/sym14112462","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14112462","pdf_url":"https://www.mdpi.com/2073-8994/14/11/2462/pdf?version=1669017459","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/11/2462/pdf?version=1669017459","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023666683","display_name":"Dowlath Fathima","orcid":null},"institutions":[{"id":"https://openalex.org/I120238654","display_name":"Saudi Electronic University","ror":"https://ror.org/05ndh7v49","country_code":"SA","type":"education","lineage":["https://openalex.org/I120238654"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Dowlath Fathima","raw_affiliation_strings":["Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Jeddah 23442, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-9516-1485","affiliations":[{"raw_affiliation_string":"Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Jeddah 23442, Saudi Arabia","institution_ids":["https://openalex.org/I120238654"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034707465","display_name":"Muhammad Naeem","orcid":"https://orcid.org/0000-0001-5112-3711"},"institutions":[{"id":"https://openalex.org/I199693650","display_name":"Umm al-Qura University","ror":"https://ror.org/01xjqrm90","country_code":"SA","type":"education","lineage":["https://openalex.org/I199693650"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Muhammad Naeem","raw_affiliation_strings":["Department of Mathematics of Applied Sciences, Umm-Al-Qura University, Makkah 21955, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics of Applied Sciences, Umm-Al-Qura University, Makkah 21955, Saudi Arabia","institution_ids":["https://openalex.org/I199693650"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024152345","display_name":"Umair Ali","orcid":"https://orcid.org/0000-0001-5840-2289"},"institutions":[{"id":"https://openalex.org/I197827452","display_name":"Institute of Space Technology","ror":"https://ror.org/01tmmzv45","country_code":"PK","type":"education","lineage":["https://openalex.org/I197827452"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Umair Ali","raw_affiliation_strings":["Department of Applied Mathematics and Statistics, Institute of Space Technology, P.O. Box 2750, Islamabad 44000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Statistics, Institute of Space Technology, P.O. Box 2750, Islamabad 44000, Pakistan","institution_ids":["https://openalex.org/I197827452"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048530123","display_name":"Abdul Hamid Ganie","orcid":null},"institutions":[{"id":"https://openalex.org/I120238654","display_name":"Saudi Electronic University","ror":"https://ror.org/05ndh7v49","country_code":"SA","type":"education","lineage":["https://openalex.org/I120238654"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Abdul Hamid Ganie","raw_affiliation_strings":["Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Abha 61421, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-1787-5036","affiliations":[{"raw_affiliation_string":"Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Abha 61421, Saudi Arabia","institution_ids":["https://openalex.org/I120238654"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019785333","display_name":"Farah Aini Abdullah","orcid":"https://orcid.org/0000-0002-5215-9617"},"institutions":[{"id":"https://openalex.org/I139322472","display_name":"Universiti Sains Malaysia","ror":"https://ror.org/02rgb2k63","country_code":"MY","type":"education","lineage":["https://openalex.org/I139322472"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Farah Aini Abdullah","raw_affiliation_strings":["School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang 11800, Malaysia"],"raw_orcid":"https://orcid.org/0000-0002-5215-9617","affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang 11800, Malaysia","institution_ids":["https://openalex.org/I139322472"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5019785333","https://openalex.org/A5024152345"],"corresponding_institution_ids":["https://openalex.org/I139322472","https://openalex.org/I197827452"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.4127,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.56221533,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"14","issue":"11","first_page":"2462","last_page":"2462"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.7500637173652649},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6151928305625916},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.5460647344589233},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5436015725135803},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5218683481216431},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.518467128276825},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.5129711031913757},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.48951613903045654},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.45799925923347473},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.42629337310791016},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4083574116230011},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15155139565467834}],"concepts":[{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.7500637173652649},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6151928305625916},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.5460647344589233},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5436015725135803},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5218683481216431},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.518467128276825},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.5129711031913757},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.48951613903045654},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.45799925923347473},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.42629337310791016},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4083574116230011},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15155139565467834},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14112462","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14112462","pdf_url":"https://www.mdpi.com/2073-8994/14/11/2462/pdf?version=1669017459","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:ee96733ff05e4cfea8041cc7787f0ec8","is_oa":true,"landing_page_url":"https://doaj.org/article/ee96733ff05e4cfea8041cc7787f0ec8","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 14, Iss 11, p 2462 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/11/2462/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14112462","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14112462","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14112462","pdf_url":"https://www.mdpi.com/2073-8994/14/11/2462/pdf?version=1669017459","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3779441219","display_name":null,"funder_award_id":"22UQU4310396DSR35","funder_id":"https://openalex.org/F4320324047","funder_display_name":"Umm Al-Qura University"}],"funders":[{"id":"https://openalex.org/F4320324047","display_name":"Umm Al-Qura University","ror":"https://ror.org/01xjqrm90"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4309785584.pdf","grobid_xml":"https://content.openalex.org/works/W4309785584.grobid-xml"},"referenced_works_count":44,"referenced_works":["https://openalex.org/W1571391776","https://openalex.org/W1966553738","https://openalex.org/W1969399740","https://openalex.org/W1986915637","https://openalex.org/W1987452198","https://openalex.org/W1989124327","https://openalex.org/W2000497619","https://openalex.org/W2042535847","https://openalex.org/W2057769307","https://openalex.org/W2074508281","https://openalex.org/W2077868122","https://openalex.org/W2137696066","https://openalex.org/W2186723650","https://openalex.org/W2516369676","https://openalex.org/W2563888514","https://openalex.org/W2575489637","https://openalex.org/W2730542645","https://openalex.org/W2756996718","https://openalex.org/W2794714757","https://openalex.org/W2889078787","https://openalex.org/W2910795882","https://openalex.org/W2918865039","https://openalex.org/W2923505959","https://openalex.org/W2945653375","https://openalex.org/W2973967052","https://openalex.org/W2994550214","https://openalex.org/W3011238522","https://openalex.org/W3014569360","https://openalex.org/W3023609308","https://openalex.org/W3027340946","https://openalex.org/W3038075946","https://openalex.org/W3042974430","https://openalex.org/W3082133093","https://openalex.org/W3088793405","https://openalex.org/W3092569095","https://openalex.org/W3122059167","https://openalex.org/W4312620187","https://openalex.org/W4403479997","https://openalex.org/W6647128842","https://openalex.org/W6647747937","https://openalex.org/W6686667077","https://openalex.org/W6743997808","https://openalex.org/W6758273054","https://openalex.org/W6872710421"],"related_works":["https://openalex.org/W2082446734","https://openalex.org/W4313560640","https://openalex.org/W2253351617","https://openalex.org/W4246784705","https://openalex.org/W3121927021","https://openalex.org/W2089892488","https://openalex.org/W2416906960","https://openalex.org/W2050580331","https://openalex.org/W3049300385","https://openalex.org/W2232781361"],"abstract_inverted_index":{"Fractional":[0],"differential":[1],"equations":[2,48],"describe":[3,12],"nature":[4],"adequately":[5],"because":[6],"of":[7,68],"the":[8,26,59,66,69,80,88],"symmetry":[9],"properties":[10],"that":[11,35,79,87],"physical":[13],"and":[14,63,65,86,94],"biological":[15],"processes.":[16],"In":[17],"this":[18],"paper,":[19],"a":[20],"new":[21],"approximation":[22],"is":[23,41,53,72],"found":[24],"for":[25,43],"variable-order":[27],"(VO)":[28],"Riemann\u2013Liouville":[29],"fractional":[30],"derivative":[31],"(RLFD)":[32],"operator;":[33],"on":[34],"basis,":[36],"an":[37],"efficient":[38],"numerical":[39,76],"approach":[40,71],"formulated":[42],"VO":[44],"time-fractional":[45],"modified":[46],"subdiffusion":[47],"(TFMSDE).":[49],"Complete":[50],"theoretical":[51],"analysis":[52],"performed,":[54],"such":[55],"as":[56],"stability":[57],"by":[58],"Fourier":[60],"series,":[61],"consistency,":[62],"convergence,":[64],"feasibility":[67],"proposed":[70,81],"also":[73],"discussed.":[74],"A":[75],"example":[77],"illustrates":[78],"scheme":[82],"demonstrates":[83],"high":[84],"accuracy,":[85],"obtained":[89],"results":[90],"are":[91],"more":[92],"feasible":[93],"accurate.":[95]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2022-11-29T00:00:00"}
