Get quick answers to your questions about the article from our AI researcher chatbot
{'id': 'https://openalex.org/W3083794939', 'doi': 'https://doi.org/10.1155/2020/9057387', 'title': 'Numerical Solution of Sine-Gordon Equation with the Local Kriging Meshless Method', 'display_name': 'Numerical Solution of Sine-Gordon Equation with the Local Kriging Meshless Method', 'publication_year': 2020, 'publication_date': '2020-09-08', 'ids': {'openalex': 'https://openalex.org/W3083794939', 'doi': 'https://doi.org/10.1155/2020/9057387', 'mag': '3083794939'}, 'language': 'en', 'primary_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1155/2020/9057387', 'pdf_url': 'https://downloads.hindawi.com/journals/mpe/2020/9057387.pdf', 'source': {'id': 'https://openalex.org/S178238111', 'display_name': 'Mathematical Problems in Engineering', 'issn_l': '1024-123X', 'issn': ['1024-123X', '1026-7077', '1563-5147'], 'is_oa': True, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310319869', 'host_organization_name': 'Hindawi Publishing Corporation', 'host_organization_lineage': ['https://openalex.org/P4310319869'], 'host_organization_lineage_names': ['Hindawi Publishing Corporation'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'type': 'article', 'type_crossref': 'journal-article', 'indexed_in': ['crossref', 'doaj'], 'open_access': {'is_oa': True, 'oa_status': 'gold', 'oa_url': 'https://downloads.hindawi.com/journals/mpe/2020/9057387.pdf', 'any_repository_has_fulltext': False}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5026378402', 'display_name': 'Pengfei Guo', 'orcid': 'https://orcid.org/0000-0001-5712-3122'}, 'institutions': [{'id': 'https://openalex.org/I152033890', 'display_name': 'Hainan Normal University', 'ror': 'https://ror.org/031dhcv14', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I152033890']}], 'countries': ['CN'], 'is_corresponding': False, 'raw_author_name': 'Pengfei Guo', 'raw_affiliation_strings': ['School of Mathematics and Statistics, Key Laboratory of Data Science and Intelligence Education, Key Laboratory of Computer and Application of Hainan Province, Hainan Normal University, Haikou 571158, China'], 'affiliations': [{'raw_affiliation_string': 'School of Mathematics and Statistics, Key Laboratory of Data Science and Intelligence Education, Key Laboratory of Computer and Application of Hainan Province, Hainan Normal University, Haikou 571158, China', 'institution_ids': ['https://openalex.org/I152033890']}]}, {'author_position': 'middle', 'author': {'id': 'https://openalex.org/A5047126629', 'display_name': 'Ariunkhishig Boldbaatar', 'orcid': None}, 'institutions': [{'id': 'https://openalex.org/I152033890', 'display_name': 'Hainan Normal University', 'ror': 'https://ror.org/031dhcv14', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I152033890']}], 'countries': ['CN'], 'is_corresponding': False, 'raw_author_name': 'Ariunkhishig Boldbaatar', 'raw_affiliation_strings': ['School of Mathematics and Statistics, Key Laboratory of Data Science and Intelligence Education, Key Laboratory of Computer and Application of Hainan Province, Hainan Normal University, Haikou 571158, China'], 'affiliations': [{'raw_affiliation_string': 'School of Mathematics and Statistics, Key Laboratory of Data Science and Intelligence Education, Key Laboratory of Computer and Application of Hainan Province, Hainan Normal University, Haikou 571158, China', 'institution_ids': ['https://openalex.org/I152033890']}]}, {'author_position': 'middle', 'author': {'id': 'https://openalex.org/A5080462017', 'display_name': 'Dutao Yi', 'orcid': 'https://orcid.org/0000-0002-0541-4283'}, 'institutions': [{'id': 'https://openalex.org/I16609230', 'display_name': 'Hunan University', 'ror': 'https://ror.org/05htk5m33', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I16609230']}], 'countries': ['CN'], 'is_corresponding': True, 'raw_author_name': 'Dutao Yi', 'raw_affiliation_strings': ['Key Laboratory for Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China'], 'affiliations': [{'raw_affiliation_string': 'Key Laboratory for Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China', 'institution_ids': ['https://openalex.org/I16609230']}]}, {'author_position': 'last', 'author': {'id': 'https://openalex.org/A5062641516', 'display_name': 'Pengxiang Dai', 'orcid': None}, 'institutions': [{'id': 'https://openalex.org/I16609230', 'display_name': 'Hunan University', 'ror': 'https://ror.org/05htk5m33', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I16609230']}], 'countries': ['CN'], 'is_corresponding': False, 'raw_author_name': 'Pengxiang Dai', 'raw_affiliation_strings': ['Key Laboratory for Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China'], 'affiliations': [{'raw_affiliation_string': 'Key Laboratory for Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China', 'institution_ids': ['https://openalex.org/I16609230']}]}], 'institution_assertions': [], 'countries_distinct_count': 1, 'institutions_distinct_count': 2, 'corresponding_author_ids': ['https://openalex.org/A5080462017'], 'corresponding_institution_ids': ['https://openalex.org/I16609230'], 'apc_list': {'value': 2400, 'currency': 'USD', 'value_usd': 2400, 'provenance': 'doaj'}, 'apc_paid': {'value': 2400, 'currency': 'USD', 'value_usd': 2400, 'provenance': 'doaj'}, 'fwci': 0.701, 'has_fulltext': True, 'fulltext_origin': 'pdf', 'cited_by_count': 8, 'citation_normalized_percentile': {'value': 0.621448, 'is_in_top_1_percent': False, 'is_in_top_10_percent': False}, 'cited_by_percentile_year': {'min': 84, 'max': 85}, 'biblio': {'volume': '2020', 'issue': None, 'first_page': '1', 'last_page': '11'}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T10514', 'display_name': 'Numerical methods in engineering', 'score': 0.998, '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'}}, 'topics': [{'id': 'https://openalex.org/T10514', 'display_name': 'Numerical methods in engineering', 'score': 0.998, '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'}}, {'id': 'https://openalex.org/T10288', 'display_name': 'Fractional Differential Equations Solutions', 'score': 0.9735, '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/T12293', 'display_name': 'Dam Engineering and Safety', 'score': 0.9595, 'subfield': {'id': 'https://openalex.org/subfields/2205', 'display_name': 'Civil and Structural Engineering'}, '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/regularized-meshless-method', 'display_name': 'Regularized meshless method', 'score': 0.74798894}, {'id': 'https://openalex.org/keywords/interpolation', 'display_name': 'Interpolation', 'score': 0.589954}], 'concepts': [{'id': 'https://openalex.org/C177562122', 'wikidata': 'https://www.wikidata.org/wiki/Q7309629', 'display_name': 'Regularized meshless method', 'level': 5, 'score': 0.74798894}, {'id': 'https://openalex.org/C39482219', 'wikidata': 'https://www.wikidata.org/wiki/Q192826', 'display_name': 'Kronecker delta', 'level': 2, 'score': 0.7167002}, {'id': 'https://openalex.org/C73000952', 'wikidata': 'https://www.wikidata.org/wiki/Q17007827', 'display_name': 'Discretization', 'level': 2, 'score': 0.71636534}, {'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.71518856}, {'id': 'https://openalex.org/C137800194', 'wikidata': 'https://www.wikidata.org/wiki/Q11713455', 'display_name': 'Interpolation (computer graphics)', 'level': 3, 'score': 0.589954}, {'id': 'https://openalex.org/C134306372', 'wikidata': 'https://www.wikidata.org/wiki/Q7754', 'display_name': 'Mathematical analysis', 'level': 1, 'score': 0.51019573}, {'id': 'https://openalex.org/C158622935', 'wikidata': 'https://www.wikidata.org/wiki/Q660848', 'display_name': 'Nonlinear system', 'level': 2, 'score': 0.49145162}, {'id': 'https://openalex.org/C81692654', 'wikidata': 'https://www.wikidata.org/wiki/Q225926', 'display_name': 'Kriging', 'level': 2, 'score': 0.48103338}, {'id': 'https://openalex.org/C186899397', 'wikidata': 'https://www.wikidata.org/wiki/Q1491980', 'display_name': 'Galerkin method', 'level': 3, 'score': 0.4807591}, {'id': 'https://openalex.org/C182310444', 'wikidata': 'https://www.wikidata.org/wiki/Q1332643', 'display_name': 'Boundary value problem', 'level': 2, 'score': 0.45974192}, {'id': 'https://openalex.org/C82047721', 'wikidata': 'https://www.wikidata.org/wiki/Q7524247', 'display_name': 'Singular boundary method', 'level': 4, 'score': 0.43032667}, {'id': 'https://openalex.org/C28826006', 'wikidata': 'https://www.wikidata.org/wiki/Q33521', 'display_name': 'Applied mathematics', 'level': 1, 'score': 0.3876772}, {'id': 'https://openalex.org/C41008148', 'wikidata': 'https://www.wikidata.org/wiki/Q21198', 'display_name': 'Computer science', 'level': 0, 'score': 0.12027663}, {'id': 'https://openalex.org/C135628077', 'wikidata': 'https://www.wikidata.org/wiki/Q220184', 'display_name': 'Finite element method', 'level': 2, 'score': 0.07145789}, {'id': 'https://openalex.org/C63632240', 'wikidata': 'https://www.wikidata.org/wiki/Q1935004', 'display_name': 'Boundary element method', 'level': 3, 'score': 0.06351167}, {'id': 'https://openalex.org/C121332964', 'wikidata': 'https://www.wikidata.org/wiki/Q413', 'display_name': 'Physics', 'level': 0, 'score': 0.056915373}, {'id': 'https://openalex.org/C502989409', 'wikidata': 'https://www.wikidata.org/wiki/Q11425', 'display_name': 'Animation', 'level': 2, 'score': 0.0}, {'id': 'https://openalex.org/C105795698', 'wikidata': 'https://www.wikidata.org/wiki/Q12483', 'display_name': 'Statistics', 'level': 1, 'score': 0.0}, {'id': 'https://openalex.org/C121684516', 'wikidata': 'https://www.wikidata.org/wiki/Q7600677', 'display_name': 'Computer graphics (images)', 'level': 1, 'score': 0.0}, {'id': 'https://openalex.org/C62520636', 'wikidata': 'https://www.wikidata.org/wiki/Q944', 'display_name': 'Quantum mechanics', 'level': 1, 'score': 0.0}, {'id': 'https://openalex.org/C97355855', 'wikidata': 'https://www.wikidata.org/wiki/Q11473', 'display_name': 'Thermodynamics', 'level': 1, 'score': 0.0}], 'mesh': [], 'locations_count': 2, 'locations': [{'is_oa': True, 'landing_page_url': 'https://doi.org/10.1155/2020/9057387', 'pdf_url': 'https://downloads.hindawi.com/journals/mpe/2020/9057387.pdf', 'source': {'id': 'https://openalex.org/S178238111', 'display_name': 'Mathematical Problems in Engineering', 'issn_l': '1024-123X', 'issn': ['1024-123X', '1026-7077', '1563-5147'], 'is_oa': True, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310319869', 'host_organization_name': 'Hindawi Publishing Corporation', 'host_organization_lineage': ['https://openalex.org/P4310319869'], 'host_organization_lineage_names': ['Hindawi Publishing Corporation'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, {'is_oa': False, 'landing_page_url': 'https://doaj.org/article/c840a98fac66422ab94943087ee15fbf', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4306401280', 'display_name': 'DOAJ (DOAJ: Directory of Open Access Journals)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': None, 'host_organization_name': None, 'host_organization_lineage': [], 'host_organization_lineage_names': [], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': None, 'is_accepted': False, 'is_published': False}], 'best_oa_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1155/2020/9057387', 'pdf_url': 'https://downloads.hindawi.com/journals/mpe/2020/9057387.pdf', 'source': {'id': 'https://openalex.org/S178238111', 'display_name': 'Mathematical Problems in Engineering', 'issn_l': '1024-123X', 'issn': ['1024-123X', '1026-7077', '1563-5147'], 'is_oa': True, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310319869', 'host_organization_name': 'Hindawi Publishing Corporation', 'host_organization_lineage': ['https://openalex.org/P4310319869'], 'host_organization_lineage_names': ['Hindawi Publishing Corporation'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'sustainable_development_goals': [], 'grants': [{'funder': 'https://openalex.org/F4320322866', 'funder_display_name': 'Natural Science Foundation of Hainan Province', 'award_id': '119MS039'}], 'datasets': [], 'versions': [], 'referenced_works_count': 46, 'referenced_works': ['https://openalex.org/W1968358120', 'https://openalex.org/W1973124775', 'https://openalex.org/W1976571953', 'https://openalex.org/W1976848078', 'https://openalex.org/W1977362569', 'https://openalex.org/W1980666715', 'https://openalex.org/W1980884737', 'https://openalex.org/W1987503845', 'https://openalex.org/W1989495978', 'https://openalex.org/W1996615110', 'https://openalex.org/W2000867446', 'https://openalex.org/W2006499752', 'https://openalex.org/W2008314186', 'https://openalex.org/W2017288915', 'https://openalex.org/W2017308992', 'https://openalex.org/W2025246194', 'https://openalex.org/W2035238356', 'https://openalex.org/W2036272282', 'https://openalex.org/W2040959667', 'https://openalex.org/W2064215350', 'https://openalex.org/W2068988900', 'https://openalex.org/W2088206150', 'https://openalex.org/W2088771027', 'https://openalex.org/W2090701613', 'https://openalex.org/W2122811504', 'https://openalex.org/W2125518390', 'https://openalex.org/W2132243470', 'https://openalex.org/W2142136198', 'https://openalex.org/W2604406592', 'https://openalex.org/W2786459026', 'https://openalex.org/W2800366705', 'https://openalex.org/W2901513775', 'https://openalex.org/W2969940537', 'https://openalex.org/W2971795017', 'https://openalex.org/W2981976950', 'https://openalex.org/W2991173198', 'https://openalex.org/W3000933102', 'https://openalex.org/W3008886629', 'https://openalex.org/W3015568041', 'https://openalex.org/W3020499586', 'https://openalex.org/W3022359638', 'https://openalex.org/W3035074051', 'https://openalex.org/W3037428033', 'https://openalex.org/W3098196229', 'https://openalex.org/W4233794908', 'https://openalex.org/W4254602140'], 'related_works': ['https://openalex.org/W2984212906', 'https://openalex.org/W2501927888', 'https://openalex.org/W2388359891', 'https://openalex.org/W2367496310', 'https://openalex.org/W2350605469', 'https://openalex.org/W2170611593', 'https://openalex.org/W2094296621', 'https://openalex.org/W2012062214', 'https://openalex.org/W1975248355', 'https://openalex.org/W1008049870'], 'abstract_inverted_index': {'This': [0], 'paper': [1], 'develops': [2], 'a': [3, 61], 'local': [4, 50], 'Kriging': [5, 23], 'meshless': [6, 17, 98], 'solution': [7], 'to': [8, 26, 39, 80, 91, 100], 'the': [9, 33, 49, 54, 75, 86, 89, 93, 96, 102], 'nonlinear': [10, 64], '2': [11, 103], '+': [12, 104], '1-dimensional': [13, 105], 'sine-Gordon': [14, 106], 'equation.': [15], 'The': [16, 69], 'shape': [18], 'function': [19, 30], 'is': [20, 67], 'constructed': [21], 'by': [22], 'interpolation': [24], 'method': [25, 57, 99], 'have': [27], 'Kronecker': [28], 'delta': [29], 'property': [31], 'for': [32, 58], 'two-dimensional': [34], 'field': [35], 'function,': [36], 'which': [37], 'leads': [38], 'convenient': [40], 'implementation': [41], 'of': [42, 63, 95], 'imposing': [43], 'essential': [44], 'boundary': [45], 'conditions.': [46], 'Based': [47], 'on': [48], 'Petrov–Galerkin': [51], 'formulation': [52], 'and': [53, 74], 'center': [55], 'difference': [56], 'time': [59], 'discretization,': [60], 'system': [62], 'discrete': [65], 'equations': [66], 'obtained.': [68], 'numerical': [70, 76], 'examples': [71], 'are': [72, 78], 'presented': [73], 'solutions': [77], 'found': [79], 'be': [81], 'in': [82, 88], 'good': [83], 'agreement': [84], 'with': [85], 'results': [87], 'literature': [90], 'validate': [92], 'ability': [94], 'present': [97], 'handle': [101], 'equation': [107], 'related': [108], 'problems.': [109]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W3083794939', 'counts_by_year': [{'year': 2024, 'cited_by_count': 3}, {'year': 2023, 'cited_by_count': 2}, {'year': 2021, 'cited_by_count': 3}], 'updated_date': '2025-01-10T09:53:51.242563', 'created_date': '2020-09-14'}