{"id":"https://openalex.org/W4415836210","doi":"https://doi.org/10.1007/s00366-025-02225-w","title":"Stability analysis of multi-spatial Riesz and multi-fractional non-linear damped wave equations involving Caputo-Fabrizio derivative","display_name":"Stability analysis of multi-spatial Riesz and multi-fractional non-linear damped wave equations involving Caputo-Fabrizio derivative","publication_year":2025,"publication_date":"2025-11-04","ids":{"openalex":"https://openalex.org/W4415836210","doi":"https://doi.org/10.1007/s00366-025-02225-w"},"language":"en","primary_location":{"id":"doi:10.1007/s00366-025-02225-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00366-025-02225-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00366-025-02225-w.pdf","source":{"id":"https://openalex.org/S89333158","display_name":"Engineering With Computers","issn_l":"0177-0667","issn":["0177-0667","1435-5663"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Engineering with Computers","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00366-025-02225-w.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060251362","display_name":"Pratibha Verma","orcid":"https://orcid.org/0000-0001-7391-3028"},"institutions":[{"id":"https://openalex.org/I46597724","display_name":"Pozna\u0144 University of Technology","ror":"https://ror.org/00p7p3302","country_code":"PL","type":"education","lineage":["https://openalex.org/I46597724"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Pratibha Verma","raw_affiliation_strings":["Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5, 60-965, Poznan, Poland"],"affiliations":[{"raw_affiliation_string":"Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5, 60-965, Poznan, Poland","institution_ids":["https://openalex.org/I46597724"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006989674","display_name":"Wojciech Sumelka","orcid":"https://orcid.org/0000-0002-8317-748X"},"institutions":[{"id":"https://openalex.org/I46597724","display_name":"Pozna\u0144 University of Technology","ror":"https://ror.org/00p7p3302","country_code":"PL","type":"education","lineage":["https://openalex.org/I46597724"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Wojciech Sumelka","raw_affiliation_strings":["Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5, 60-965, Poznan, Poland"],"affiliations":[{"raw_affiliation_string":"Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5, 60-965, Poznan, Poland","institution_ids":["https://openalex.org/I46597724"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5060251362"],"corresponding_institution_ids":["https://openalex.org/I46597724"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":5.0529,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.95809823,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"41","issue":"6","first_page":"4877","last_page":"4897"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9890000224113464,"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":0.9890000224113464,"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":0.004100000020116568,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.00139999995008111,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/bounded-function","display_name":"Bounded function","score":0.7379000186920166},{"id":"https://openalex.org/keywords/laplace-transform","display_name":"Laplace transform","score":0.6419000029563904},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.6103000044822693},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.6093999743461609},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5777999758720398},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.5557000041007996},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.5178999900817871},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.4611000120639801}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7379000186920166},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7020999789237976},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.6419000029563904},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.6103000044822693},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6093999743461609},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5777999758720398},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5662000179290771},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5557000041007996},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.5178999900817871},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.4611000120639801},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4058000147342682},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3993000090122223},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3903000056743622},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3865000009536743},{"id":"https://openalex.org/C66908091","wikidata":"https://www.wikidata.org/wiki/Q1127660","display_name":"Damped wave","level":3,"score":0.33959999680519104},{"id":"https://openalex.org/C17685861","wikidata":"https://www.wikidata.org/wiki/Q6960532","display_name":"Time derivative","level":2,"score":0.33869999647140503},{"id":"https://openalex.org/C571446","wikidata":"https://www.wikidata.org/wiki/Q6510183","display_name":"Diffusion equation","level":3,"score":0.3328000009059906},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.30730000138282776},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3003000020980835},{"id":"https://openalex.org/C143170015","wikidata":"https://www.wikidata.org/wiki/Q17007850","display_name":"Stability conditions","level":3,"score":0.29789999127388},{"id":"https://openalex.org/C175454919","wikidata":"https://www.wikidata.org/wiki/Q2342396","display_name":"Bounded operator","level":3,"score":0.26100000739097595},{"id":"https://openalex.org/C98724732","wikidata":"https://www.wikidata.org/wiki/Q1935235","display_name":"Mittag-Leffler function","level":3,"score":0.2572999894618988},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.2556999921798706}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00366-025-02225-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00366-025-02225-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00366-025-02225-w.pdf","source":{"id":"https://openalex.org/S89333158","display_name":"Engineering With Computers","issn_l":"0177-0667","issn":["0177-0667","1435-5663"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Engineering with Computers","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00366-025-02225-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00366-025-02225-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00366-025-02225-w.pdf","source":{"id":"https://openalex.org/S89333158","display_name":"Engineering With Computers","issn_l":"0177-0667","issn":["0177-0667","1435-5663"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Engineering with Computers","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4415836210.pdf"},"referenced_works_count":59,"referenced_works":["https://openalex.org/W171286733","https://openalex.org/W1493657643","https://openalex.org/W1964583879","https://openalex.org/W2005075959","https://openalex.org/W2016823491","https://openalex.org/W2036315978","https://openalex.org/W2055783248","https://openalex.org/W2057119166","https://openalex.org/W2066116605","https://openalex.org/W2078055801","https://openalex.org/W2109779582","https://openalex.org/W2327218125","https://openalex.org/W2443361655","https://openalex.org/W2503400331","https://openalex.org/W2548627090","https://openalex.org/W2562292996","https://openalex.org/W2605402017","https://openalex.org/W2768153787","https://openalex.org/W2771059297","https://openalex.org/W2902428408","https://openalex.org/W2935109770","https://openalex.org/W2963186934","https://openalex.org/W2963855174","https://openalex.org/W2969557983","https://openalex.org/W2972719794","https://openalex.org/W2980570378","https://openalex.org/W2999290147","https://openalex.org/W3022633872","https://openalex.org/W3031174519","https://openalex.org/W3036839271","https://openalex.org/W3086361146","https://openalex.org/W3088768568","https://openalex.org/W3089051038","https://openalex.org/W3102324609","https://openalex.org/W3108196499","https://openalex.org/W3123720740","https://openalex.org/W3150161930","https://openalex.org/W3157686638","https://openalex.org/W3161801719","https://openalex.org/W3174489310","https://openalex.org/W3195715075","https://openalex.org/W4311448608","https://openalex.org/W4390292116","https://openalex.org/W4390315331","https://openalex.org/W4390399037","https://openalex.org/W4390967215","https://openalex.org/W4392682402","https://openalex.org/W4396234986","https://openalex.org/W4400581955","https://openalex.org/W4400592836","https://openalex.org/W4401839209","https://openalex.org/W4402630211","https://openalex.org/W4403074183","https://openalex.org/W4405819997","https://openalex.org/W4406792158","https://openalex.org/W4408640779","https://openalex.org/W4409211691","https://openalex.org/W4410090313","https://openalex.org/W4410418219"],"related_works":[],"abstract_inverted_index":{"This":[0],"research":[1],"investigates":[2],"a":[3,11,16],"multi-fractional":[4,109],"non-linear":[5,84],"generalized":[6,85],"diffusion":[7],"wave":[8],"equation":[9],"involving":[10],"multi-spatial":[12],"Riesz":[13],"operator":[14,34,53],"and":[15,78,92,144,162,174,203,209],"damping":[17,76],"term,":[18],"analyzed":[19],"using":[20],"the":[21,26,31,45,75,80,83,103,107,116,127,142,150,157,178,184,189,194,207,212],"Caputo\u2013Fabrizio":[22],"fractional":[23,32,46],"derivative":[24,70],"in":[25,35,57,115,121,141,188],"Caputo":[27],"sense.":[28],"We":[29],"employ":[30],"Laplace":[33],"symbolic":[36],"form,":[37],"denoted":[38],"by":[39,96],"$$(-\\Delta":[40],")^{\\frac{\\alpha":[41],"}{2}}$$":[42],",":[43],"where":[44],"order":[47,134],"$$\\alpha":[48],"\\in":[49],"(0,2)$$":[50],".":[51],"The":[52,68,88,172],"is":[54,71],"widely":[55],"used":[56],"dynamic":[58],"systems.":[59,135],"In":[60],"bounded":[61,170],"domains,":[62,205],"it":[63,167],"effectively":[64],"handles":[65],"boundary":[66],"conditions.":[67],"Caputo-Fabrizio":[69],"incorporated":[72],"to":[73,148],"capture":[74],"effects":[77],"explore":[79],"behavior":[81],"of":[82,106,129,177,191,211],"diffusion-wave":[86],"equation.":[87],"outcomes":[89],"concerning":[90],"existence":[91],"uniqueness":[93],"are":[94,139,197],"obtained":[95],"employing":[97],"fixed-point":[98],"theorems.":[99],"Furthermore,":[100,193],"we":[101],"examine":[102],"Hyers-Ulam":[104],"stability":[105,195],"proposed":[108,213],"model,":[110],"establishing":[111],"that":[112,166,183],"small":[113],"perturbations":[114],"system":[117,185],"do":[118],"not":[119],"result":[120],"divergence.":[122],"Moreover,":[123],"this":[124],"study":[125],"improves":[126],"accuracy":[128],"model":[130],"approximations":[131],"for":[132],"non-integer":[133],"Multiple":[136],"case":[137],"studies":[138],"explored":[140],"one-":[143],"two-dimensional":[145],"spatial":[146],"domains":[147],"support":[149],"theoretical":[151],"findings.":[152],"Visual":[153],"representations":[154],"illustrate":[155],"how":[156],"error":[158],"evolves":[159],"over":[160,200],"space":[161,204],"time,":[163],"consistently":[164],"showing":[165],"stays":[168],"within":[169],"limits.":[171],"2D":[173],"3D":[175],"plots":[176],"simulation":[179],"results":[180],"further":[181],"demonstrate":[182],"remains":[186],"stable":[187],"presence":[190],"perturbations.":[192],"properties":[196],"thoroughly":[198],"established":[199],"various":[201],"time":[202],"underscoring":[206],"robustness":[208],"reliability":[210],"approach.":[214]},"counts_by_year":[{"year":2026,"cited_by_count":4}],"updated_date":"2026-04-15T08:11:43.952461","created_date":"2025-11-04T00:00:00"}