{"id":"https://openalex.org/W2809985998","doi":"https://doi.org/10.3390/computation6030040","title":"Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein\u2013Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz\u2013Galerkin Method","display_name":"Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein\u2013Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz\u2013Galerkin Method","publication_year":2018,"publication_date":"2018-06-27","ids":{"openalex":"https://openalex.org/W2809985998","doi":"https://doi.org/10.3390/computation6030040","mag":"2809985998"},"language":"en","primary_location":{"id":"doi:10.3390/computation6030040","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computation6030040","pdf_url":"https://www.mdpi.com/2079-3197/6/3/40/pdf?version=1530095895","source":{"id":"https://openalex.org/S2738402919","display_name":"Computation","issn_l":"2079-3197","issn":["2079-3197"],"is_oa":true,"is_in_doaj":true,"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":"Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2079-3197/6/3/40/pdf?version=1530095895","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023251238","display_name":"Afshan Kanwal","orcid":"https://orcid.org/0000-0001-7062-7624"},"institutions":[{"id":"https://openalex.org/I930072361","display_name":"Tun Hussein Onn University of Malaysia","ror":"https://ror.org/01c5wha71","country_code":"MY","type":"education","lineage":["https://openalex.org/I930072361"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Afshan Kanwal","raw_affiliation_strings":["Department of Mathematics and Statistics, Universiti Tun Hussein Onn Malaysia, Pagoh Campus, 84600 Pagoh, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Universiti Tun Hussein Onn Malaysia, Pagoh Campus, 84600 Pagoh, Malaysia","institution_ids":["https://openalex.org/I930072361"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014529962","display_name":"Chang Phang","orcid":"https://orcid.org/0000-0002-0291-3327"},"institutions":[{"id":"https://openalex.org/I930072361","display_name":"Tun Hussein Onn University of Malaysia","ror":"https://ror.org/01c5wha71","country_code":"MY","type":"education","lineage":["https://openalex.org/I930072361"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Chang Phang","raw_affiliation_strings":["Department of Mathematics and Statistics, Universiti Tun Hussein Onn Malaysia, Pagoh Campus, 84600 Pagoh, Malaysia"],"raw_orcid":"https://orcid.org/0000-0002-0291-3327","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Universiti Tun Hussein Onn Malaysia, Pagoh Campus, 84600 Pagoh, Malaysia","institution_ids":["https://openalex.org/I930072361"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083535368","display_name":"Umer Iqbal","orcid":"https://orcid.org/0000-0002-2888-9919"},"institutions":[{"id":"https://openalex.org/I195024194","display_name":"Riphah International University","ror":"https://ror.org/02kdm5630","country_code":"PK","type":"education","lineage":["https://openalex.org/I195024194"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Umer Iqbal","raw_affiliation_strings":["Riphah College of Computing, Riphah International University, Faisalabad Campus, 38000 Faisalabad, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Riphah College of Computing, Riphah International University, Faisalabad Campus, 38000 Faisalabad, Pakistan","institution_ids":["https://openalex.org/I195024194"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5014529962"],"corresponding_institution_ids":["https://openalex.org/I930072361"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":2.1587,"has_fulltext":true,"cited_by_count":23,"citation_normalized_percentile":{"value":0.87332986,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"6","issue":"3","first_page":"40","last_page":"40"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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":0.9998999834060669,"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7823177576065063},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.6831009984016418},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.5989377498626709},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.554796040058136},{"id":"https://openalex.org/keywords/ritz-method","display_name":"Ritz method","score":0.4788539409637451},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.4755901098251343},{"id":"https://openalex.org/keywords/jacobi-polynomials","display_name":"Jacobi polynomials","score":0.4672168493270874},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.44856661558151245},{"id":"https://openalex.org/keywords/diffusion-equation","display_name":"Diffusion equation","score":0.4298684895038605},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.4184033274650574},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.34654414653778076},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07112637162208557}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7823177576065063},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.6831009984016418},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.5989377498626709},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.554796040058136},{"id":"https://openalex.org/C21575042","wikidata":"https://www.wikidata.org/wiki/Q905132","display_name":"Ritz method","level":3,"score":0.4788539409637451},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.4755901098251343},{"id":"https://openalex.org/C48490523","wikidata":"https://www.wikidata.org/wiki/Q371631","display_name":"Jacobi polynomials","level":3,"score":0.4672168493270874},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.44856661558151245},{"id":"https://openalex.org/C571446","wikidata":"https://www.wikidata.org/wiki/Q6510183","display_name":"Diffusion equation","level":3,"score":0.4298684895038605},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4184033274650574},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.34654414653778076},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07112637162208557},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C136264566","wikidata":"https://www.wikidata.org/wiki/Q159810","display_name":"Economy","level":1,"score":0.0},{"id":"https://openalex.org/C2780378061","wikidata":"https://www.wikidata.org/wiki/Q25351891","display_name":"Service (business)","level":2,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/computation6030040","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computation6030040","pdf_url":"https://www.mdpi.com/2079-3197/6/3/40/pdf?version=1530095895","source":{"id":"https://openalex.org/S2738402919","display_name":"Computation","issn_l":"2079-3197","issn":["2079-3197"],"is_oa":true,"is_in_doaj":true,"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":"Computation","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:29db0285b9dd4c5e84fca6d69046de73","is_oa":true,"landing_page_url":"https://doaj.org/article/29db0285b9dd4c5e84fca6d69046de73","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":"Computation, Vol 6, Iss 3, p 40 (2018)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2079-3197/6/3/40/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/computation6030040","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":"Computation","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/computation6030040","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computation6030040","pdf_url":"https://www.mdpi.com/2079-3197/6/3/40/pdf?version=1530095895","source":{"id":"https://openalex.org/S2738402919","display_name":"Computation","issn_l":"2079-3197","issn":["2079-3197"],"is_oa":true,"is_in_doaj":true,"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":"Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320310112","display_name":"Universiti Tun Hussein Onn Malaysia","ror":"https://ror.org/01c5wha71"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2809985998.pdf","grobid_xml":"https://content.openalex.org/works/W2809985998.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W1542702287","https://openalex.org/W1968501254","https://openalex.org/W1975514399","https://openalex.org/W1983159263","https://openalex.org/W1998312161","https://openalex.org/W2043742654","https://openalex.org/W2120597557","https://openalex.org/W2226597399","https://openalex.org/W2271858403","https://openalex.org/W2279965344","https://openalex.org/W2291960678","https://openalex.org/W2516962718","https://openalex.org/W2562724410","https://openalex.org/W2584261301","https://openalex.org/W2588125567","https://openalex.org/W2592174115","https://openalex.org/W2602989791","https://openalex.org/W2610386373","https://openalex.org/W2625501393","https://openalex.org/W2773580519","https://openalex.org/W2964280453","https://openalex.org/W4247953052","https://openalex.org/W6689036548","https://openalex.org/W6739377547"],"related_works":["https://openalex.org/W2489863694","https://openalex.org/W2353026275","https://openalex.org/W2099676275","https://openalex.org/W4297612202","https://openalex.org/W2949498383","https://openalex.org/W1651872008","https://openalex.org/W2493096353","https://openalex.org/W1652500118","https://openalex.org/W2793709684","https://openalex.org/W4294834596"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"two-dimensional":[3,55],"Genocchi":[4,56],"polynomials":[5,57],"and":[6,20,34,65,96],"the":[7,14,21,32,48,52,60,68,75,79,83,91,94,104],"Ritz\u2013Galerkin":[8,61],"method":[9,85],"were":[10,71],"developed":[11],"to":[12,73,88],"investigate":[13],"Fractional":[15,22],"Diffusion":[16],"Wave":[17],"Equation":[18,24],"(FDWE)":[19],"Klein\u2013Gordon":[23],"(FKGE).":[25],"A":[26,39],"satisfier":[27],"function":[28],"that":[29],"satisfies":[30],"all":[31],"initial":[33],"boundary":[35],"conditions":[36],"was":[37,45,86],"used.":[38],"linear":[40,76],"system":[41,77],"of":[42,54,78,93,103],"algebraic":[43,80],"equations":[44],"obtained":[46],"for":[47],"considered":[49],"equation":[50],"with":[51,59],"help":[53],"along":[58],"method.":[62],"The":[63,101],"FDWE":[64],"FKGE,":[66],"including":[67],"nonlinear":[69],"case,":[70],"reduced":[72],"solve":[74],"equation.":[81],"Hence,":[82],"proposed":[84,105],"able":[87],"greatly":[89],"reduce":[90],"complexity":[92],"problems":[95],"provide":[97],"an":[98],"accurate":[99],"solution.":[100],"effectiveness":[102],"technique":[106],"is":[107],"demonstrated":[108],"through":[109],"several":[110],"examples.":[111]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2018-07-10T00:00:00"}