Numerical solution of second-order two-dimensional hyperbolic equation by bi-cubic B-spline collocation method
A method based on B-splines has been introduced for the solution of second-order nonlinear hyperbolic equation in 2-dimensions subject to appropriate initial and Dirichlet boundary conditions. We first convert the second-order equation into a system of first-order partial differential equations. The...
Saved in:
| Published in | Mathematical sciences (Karaj, Iran) Vol. 14; no. 3; pp. 201 - 213 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.09.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2008-1359 2251-7456 |
| DOI | 10.1007/s40096-020-00331-y |
Cover
| Abstract | A method based on B-splines has been introduced for the solution of second-order nonlinear hyperbolic equation in 2-dimensions subject to appropriate initial and Dirichlet boundary conditions. We first convert the second-order equation into a system of first-order partial differential equations. Then, collocation of bi-cubic B-splines is used to discretize spatial variables and their derivatives to further obtain first-order ordinary differential equations which have block tri-diagonal structure. Computation technique is discussed to handle the thus obtained block tri-diagonal matrices, which are then solved by two-step, second-order strong-stability-preserving Runge--Kutta method (SSP RK-22). The efficiency and accuracy of the proposed method are demonstrated by its application to a few test problems and by comparing the results with analytic solutions and with the results obtained by using other numerical methods available in the literature. |
|---|---|
| AbstractList | A method based on B-splines has been introduced for the solution of second-order nonlinear hyperbolic equation in 2-dimensions subject to appropriate initial and Dirichlet boundary conditions. We first convert the second-order equation into a system of first-order partial differential equations. Then, collocation of bi-cubic B-splines is used to discretize spatial variables and their derivatives to further obtain first-order ordinary differential equations which have block tri-diagonal structure. Computation technique is discussed to handle the thus obtained block tri-diagonal matrices, which are then solved by two-step, second-order strong-stability-preserving Runge--Kutta method (SSP RK-22). The efficiency and accuracy of the proposed method are demonstrated by its application to a few test problems and by comparing the results with analytic solutions and with the results obtained by using other numerical methods available in the literature. |
| Author | Arora, Rajni Singh, Swarn Singh, Suruchi |
| Author_xml | – sequence: 1 givenname: Rajni surname: Arora fullname: Arora, Rajni organization: Department of Mathematics, University of Delhi – sequence: 2 givenname: Swarn orcidid: 0000-0003-0547-6625 surname: Singh fullname: Singh, Swarn email: ssingh@svc.ac.in organization: Department of Mathematics, Sri Venkateswara College, University of Delhi – sequence: 3 givenname: Suruchi surname: Singh fullname: Singh, Suruchi organization: Department of Mathematics, Aditi Mahavidyalaya, University of Delhi |
| BookMark | eNp9kEtLxDAYRYOM4KjzB1wVXEfzaJJ2qYMvGHSj65AmqZOhbTpJi_TfG6eC4GKySfhyzsflnoNF5zsLwBVGNxghcRtzhEoOEUEQIUoxnE7AkhCGocgZX6Q3QgXElJVnYBXjDqUjRIlytgTd69ja4LRqsuibcXC-y3ydRat9Z6APxoZs-PLQuNZ2Mf0mcDv1NlS-cTqz-1EdnGrKKgf1WKXhPYx94zqbad80Xs9Aa4etN5fgtFZNtKvf-wJ8PD68r5_h5u3pZX23gZqyYoAqL3FOFVemFJpjoYVRNsfKalIIXBeCEs5Kw7XBxlKjK0ZKQpmpqSFCaU4vwPW8tw9-P9o4yJ0fQwofJcnTDlZwUSSqmCkdfIzB1lK74RB3CMo1EiP5U7CcC5apYHkoWE5JJf_UPrhWhem4RGcpJrj7tOEv1RHrGxSLkl8 |
| CitedBy_id | crossref_primary_10_1515_nleng_2022_0298 crossref_primary_10_1515_nleng_2022_0016 crossref_primary_10_31466_kfbd_929302 crossref_primary_10_1016_j_joes_2022_06_034 crossref_primary_10_1155_2022_1782229 crossref_primary_10_1155_2024_3886554 crossref_primary_10_1186_s13662_021_03596_2 |
| Cites_doi | 10.1002/num.1034 10.1016/S0893-9659(04)90019-5 10.5899/2012/cna-00129 10.1002/num.1029 10.1080/00207160801965271 10.1002/num.20341 10.1016/0045-7930(75)90022-5 10.1002/htj.21030 10.1137/S0036142901389025 10.1002/mma.2517 10.1016/j.rinp.2014.08.009 10.1002/sapm1962411212 10.1108/09615531111162837 10.1080/0020716022000002800 10.1155/2011/420608 10.1016/j.compbiomed.2013.05.019 10.1016/0377-0427(95)00201-4 10.1007/s11075-010-9377-x 10.1016/j.cam.2009.09.023 10.1016/j.cpc.2009.09.001 10.1002/num.20357 10.1016/j.matcom.2018.11.003 10.1016/j.enganabound.2009.10.010 10.1080/14786444108521297 10.1016/j.cam.2009.01.001 10.1090/S0025-5718-1975-0375785-7 10.1002/num.20350 10.1007/s40314-019-0838-0 10.1080/00036811.2014.998654 10.1007/s40096-019-00295-8 |
| ContentType | Journal Article |
| Copyright | Islamic Azad University 2020 Islamic Azad University 2020. |
| Copyright_xml | – notice: Islamic Azad University 2020 – notice: Islamic Azad University 2020. |
| DBID | AAYXX CITATION 8FE 8FG ABJCF ABUWG AFKRA AZQEC BENPR BGLVJ CCPQU DWQXO HCIFZ L6V M7S PHGZM PHGZT PIMPY PKEHL PQEST PQGLB PQQKQ PQUKI PRINS PTHSS |
| DOI | 10.1007/s40096-020-00331-y |
| DatabaseName | CrossRef ProQuest SciTech Collection ProQuest Technology Collection Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland ProQuest Central Essentials ProQuest Central Technology collection ProQuest One Community College ProQuest Central SciTech Premium Collection ProQuest Engineering Collection Engineering Database ProQuest Central Premium ProQuest One Academic (New) Publicly Available Content Database ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection |
| DatabaseTitle | CrossRef Publicly Available Content Database Engineering Database Technology Collection ProQuest One Academic Middle East (New) ProQuest Central Essentials ProQuest One Academic Eastern Edition ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Technology Collection ProQuest SciTech Collection ProQuest Central China ProQuest Central ProQuest One Applied & Life Sciences ProQuest Engineering Collection ProQuest One Academic UKI Edition ProQuest Central Korea Materials Science & Engineering Collection ProQuest Central (New) ProQuest One Academic ProQuest One Academic (New) Engineering Collection |
| DatabaseTitleList | Publicly Available Content Database |
| Database_xml | – sequence: 1 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 2251-7456 |
| EndPage | 213 |
| ExternalDocumentID | 10_1007_s40096_020_00331_y |
| GroupedDBID | -A0 0R~ 2VQ 406 40G 5VS 8FE 8FG AACDK AAHNG AAJBT AASML AATNV AAUYE ABAKF ABECU ABJCF ABJNI ABMQK ABSXP ABTEG ABTKH ABTMW ACAOD ACDTI ACGFS ACHSB ACIPV ACOKC ACPIV ACZOJ ADBBV ADINQ ADKNI ADYFF AEFQL AEMSY AESKC AFBBN AFKRA AFQWF AGMZJ AGQEE AHBYD AHSBF AHYZX AIGIU AILAN AJZVZ ALMA_UNASSIGNED_HOLDINGS AMKLP AMXSW AMYLF BCNDV BENPR BGLVJ C24 C6C CCPQU DPUIP EBLON FIGPU FNLPD GGCAI GROUPED_DOAJ H13 HCIFZ HG6 IAO IKXTQ IPNFZ ITC IWAJR J9A JZLTJ KQ8 L6V LLZTM M7S M~E NPVJJ NQJWS OK1 PIMPY PROAC PT4 PTHSS RIG ROL RSV SJYHP SMT SNE SNPRN SOHCF SOJ SRMVM SSLCW UOJIU UTJUX ZMTXR AAYXX ABBRH ABDBE ABFSG ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP ATHPR AYFIA CITATION PHGZM PHGZT PQGLB ABUWG AZQEC DWQXO PKEHL PQEST PQQKQ PQUKI PRINS |
| ID | FETCH-LOGICAL-c358t-a49143a6ad97c617c7dae41aec2871f8732659d6cd1de3dcb529235df3d27ac63 |
| IEDL.DBID | 8FG |
| ISSN | 2008-1359 |
| IngestDate | Fri Jul 25 11:11:38 EDT 2025 Thu Jul 10 07:45:27 EDT 2025 Thu Apr 24 23:10:41 EDT 2025 Fri Feb 21 02:35:41 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 3 |
| Keywords | SSP RK-22 65M06 Telegraph equation 65N06 Damped wave equation 65M60 Collocation method 65N30 Tri-diagonal solver |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c358t-a49143a6ad97c617c7dae41aec2871f8732659d6cd1de3dcb529235df3d27ac63 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0003-0547-6625 |
| OpenAccessLink | https://www.proquest.com/docview/2428758678?pq-origsite=%requestingapplication% |
| PQID | 2428758678 |
| PQPubID | 2034691 |
| PageCount | 13 |
| ParticipantIDs | proquest_journals_2428758678 crossref_citationtrail_10_1007_s40096_020_00331_y crossref_primary_10_1007_s40096_020_00331_y springer_journals_10_1007_s40096_020_00331_y |
| PublicationCentury | 2000 |
| PublicationDate | 20200900 2020-09-00 20200901 |
| PublicationDateYYYYMMDD | 2020-09-01 |
| PublicationDate_xml | – month: 9 year: 2020 text: 20200900 |
| PublicationDecade | 2020 |
| PublicationPlace | Berlin/Heidelberg |
| PublicationPlace_xml | – name: Berlin/Heidelberg – name: Heidelberg |
| PublicationTitle | Mathematical sciences (Karaj, Iran) |
| PublicationTitleAbbrev | Math Sci |
| PublicationYear | 2020 |
| Publisher | Springer Berlin Heidelberg Springer Nature B.V |
| Publisher_xml | – name: Springer Berlin Heidelberg – name: Springer Nature B.V |
| References | Parsa, Rashidi, Bég, Sadri (CR4) 2013; 43 Dehghan, Ghesmati (CR12) 2010; 34 Mittal, Bhatia (CR35) 2014; 244 Rashidinia, Ghasemi, Jalilian (CR8) 2010; 233 Mohanty (CR18) 2007; 190 Nojavana, Abbasbandya, Mohammadi (CR34) 2018; 330 Dehghan, Salehi (CR13) 2012; 35 Rashidinia, Mohammadi (CR6) 2011; 56 CR14 CR33 Bülbül, Sezer (CR24) 2004; 17 CR32 Rashidi, Ganji, Dinarvand (CR5) 2009; 25 Ahlberg, Ito (CR40) 1975; 29 Ding, Zhang (CR26) 2009; 230 Bouhiri, Lamnii, Lamnii (CR31) 2019; 164 Chawla, Al-Zanaidi (CR10) 2003; 80 Mohanty, Jain (CR20) 2001; 17 Rashidinia, Mohammadi (CR7) 2010; 181 Mohanty, Arora, Jain (CR21) 2001; 17 Dehghan, Shokri (CR11) 2009; 25 Zarebnia, Parvaz (CR30) 2013; 7 Mohanty, Singh (CR17) 2011; 2011 Smith (CR43) 1978 Bég, Rashidi, Bég, Asad (CR3) 2012; 12 Dosti, Nazemi (CR37) 2012; 7 Mohanty (CR15) 2009; 86 Jiwari, Pandit, Mittal (CR25) 2012; 218 Mohanty (CR16) 2014; 4 CR27 Boor (CR39) 1962; 41 Dehghan, Mohebbi (CR9) 2009; 25 Panton, Salle (CR29) 1975; 3 Rashidi, Hayat, Keimanesh, Yousefian (CR2) 2013; 42 CR23 Bickley (CR28) 1941; 32 Rashidi, Erfani (CR1) 2011; 21 CR41 Mohanty, Jain, George (CR22) 1996; 70 Mohanty (CR19) 2004; 152 Spiteri, Ruuth (CR42) 2002; 40 Mittal, Jain (CR38) 2012; 2012 Rashidinia, Jamalzadeh, Esfahani (CR36) 2014; 2014 M Dehghan (331_CR9) 2009; 25 RK Mohanty (331_CR16) 2014; 4 JH Ahlberg (331_CR40) 1975; 29 RK Mohanty (331_CR17) 2011; 2011 M Dosti (331_CR37) 2012; 7 RC Mittal (331_CR38) 2012; 2012 H Nojavana (331_CR34) 2018; 330 H Ding (331_CR26) 2009; 230 331_CR33 J Rashidinia (331_CR36) 2014; 2014 331_CR32 M Dehghan (331_CR13) 2012; 35 331_CR27 M Dehghan (331_CR12) 2010; 34 J Rashidinia (331_CR7) 2010; 181 C Boor (331_CR39) 1962; 41 MM Chawla (331_CR10) 2003; 80 RK Mohanty (331_CR18) 2007; 190 S Bouhiri (331_CR31) 2019; 164 J Rashidinia (331_CR8) 2010; 233 RK Mohanty (331_CR20) 2001; 17 R Jiwari (331_CR25) 2012; 218 AB Parsa (331_CR4) 2013; 43 GD Smith (331_CR43) 1978 RC Mittal (331_CR35) 2014; 244 331_CR41 R Spiteri (331_CR42) 2002; 40 B Bülbül (331_CR24) 2004; 17 RK Mohanty (331_CR19) 2004; 152 RK Mohanty (331_CR15) 2009; 86 331_CR23 MM Rashidi (331_CR1) 2011; 21 RK Mohanty (331_CR22) 1996; 70 MM Rashidi (331_CR2) 2013; 42 M Dehghan (331_CR11) 2009; 25 331_CR14 WG Bickley (331_CR28) 1941; 32 RZ Panton (331_CR29) 1975; 3 M Zarebnia (331_CR30) 2013; 7 OA Bég (331_CR3) 2012; 12 RK Mohanty (331_CR21) 2001; 17 MM Rashidi (331_CR5) 2009; 25 J Rashidinia (331_CR6) 2011; 56 |
| References_xml | – volume: 17 start-page: 684 year: 2001 end-page: 688 ident: CR20 article-title: An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.1034 – volume: 17 start-page: 101 year: 2004 end-page: 105 ident: CR24 article-title: A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation publication-title: Appl. Math. Lett. doi: 10.1016/S0893-9659(04)90019-5 – volume: 2012 start-page: 1 year: 2012 end-page: 16 ident: CR38 article-title: Numerical solution of general Rosenau-RLW equation using quintic B-splines collocation method publication-title: Commun. Numer. Anal. doi: 10.5899/2012/cna-00129 – ident: CR14 – year: 1978 ident: CR43 publication-title: Numerical Solution of Partial Differential Equations: Finite Difference Methods – volume: 12 start-page: 1 issue: 3 year: 2012 end-page: 21 ident: CR3 article-title: Homotopy analysis of transient magneto-bio-fluid dynamics of micropolar squeeze film in a porous medium: A model for magneto-bio-rheological lubrication publication-title: J. Mech. Med. Biol. – volume: 17 start-page: 607 year: 2001 end-page: 618 ident: CR21 article-title: Linear stability analysis and fourth-order approximations at first time level for the two space dimensional mildly quasi-linear hyperbolic equations publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.1029 – volume: 86 start-page: 2061 issue: 12 year: 2009 end-page: 2071 ident: CR15 article-title: New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations publication-title: Int. J. Comput. Math. doi: 10.1080/00207160801965271 – volume: 152 start-page: 799 issue: 3 year: 2004 end-page: 806 ident: CR19 article-title: An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions publication-title: Appl. Math. Comput. – ident: CR33 – volume: 25 start-page: 232 issue: 1 year: 2009 end-page: 243 ident: CR9 article-title: High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.20341 – volume: 3 start-page: 257 year: 1975 end-page: 269 ident: CR29 publication-title: Comput. Fluids doi: 10.1016/0045-7930(75)90022-5 – volume: 7 start-page: 83 issue: 2 year: 2012 end-page: 90 ident: CR37 article-title: Quartic B-spline collocation method for solving one-dimensional hyperbolic telegraph equation publication-title: J. Inf. Comput. Sci. – volume: 330 start-page: 23 year: 2018 end-page: 41 ident: CR34 article-title: Local variably scaled Newton basis functions collocation method for solving Burgers’ equation publication-title: Appl. Math. Comput. – volume: 42 start-page: 31 issue: 1 year: 2013 end-page: 45 ident: CR2 article-title: A study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method publication-title: Heat Transf. Asian Res. doi: 10.1002/htj.21030 – volume: 40 start-page: 469 year: 2002 end-page: 491 ident: CR42 article-title: A new class of optimal high-order strong-stability-preserving time discretization methods publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142901389025 – volume: 35 start-page: 1220 issue: 10 year: 2012 end-page: 1233 ident: CR13 article-title: A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraph equation publication-title: Math. Methods Appl. Sci. doi: 10.1002/mma.2517 – volume: 4 start-page: 156 year: 2014 end-page: 163 ident: CR16 article-title: New high accuracy super stable alternating direction implicit methods for two and three dimensional hyperbolic damped wave equations publication-title: Results Phys. doi: 10.1016/j.rinp.2014.08.009 – ident: CR27 – ident: CR23 – volume: 218 start-page: 7279 year: 2012 end-page: 7294 ident: CR25 article-title: A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions publication-title: Appl. Math. Comput. – volume: 41 start-page: 212 year: 1962 end-page: 218 ident: CR39 article-title: Bicubic spline interpolation publication-title: J. Math. Phys. doi: 10.1002/sapm1962411212 – volume: 21 start-page: 864 issue: 7 year: 2011 end-page: 883 ident: CR1 article-title: The modified differential transform method for investigating nano boundary-layers over stretching surfaces publication-title: Int. J. Numer. Meth. Heat Fluid Flow doi: 10.1108/09615531111162837 – volume: 80 start-page: 357 issue: 3 year: 2003 end-page: 365 ident: CR10 article-title: A linearly implicit one-step time integration scheme for nonlinear hyperbolic equations in two space dimensions publication-title: Int. J. Comput. Math. doi: 10.1080/0020716022000002800 – volume: 190 start-page: 1683 year: 2007 end-page: 1690 ident: CR18 article-title: Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms publication-title: Appl. Math. Comput. – volume: 2011 start-page: 1 year: 2011 end-page: 22 ident: CR17 article-title: A new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic equations publication-title: Adv. Math. Phys. doi: 10.1155/2011/420608 – volume: 43 start-page: 1142 issue: 9 year: 2013 end-page: 1153 ident: CR4 article-title: Semi-computational simulation of magneto-hemodynamic flow in a semi-porous channel using optimal homotopy and differential transform methods publication-title: Comput. Biol. Med. doi: 10.1016/j.compbiomed.2013.05.019 – volume: 70 start-page: 231 year: 1996 end-page: 243 ident: CR22 article-title: High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients publication-title: J. Comput. Appl. Math. doi: 10.1016/0377-0427(95)00201-4 – volume: 7 start-page: 350 issue: 3 year: 2013 end-page: 353 ident: CR30 article-title: Cubic B-spline collocation method for numerical solution of the Benjamin–Bona–Mohany–Burgers equation publication-title: Int. J. Math. Comput. Nat. Phys. Eng. – volume: 56 start-page: 129 year: 2011 end-page: 142 ident: CR6 article-title: Tension spline solution of nonlinear sine-Gordon equation publication-title: Numer. Algoritm. doi: 10.1007/s11075-010-9377-x – volume: 233 start-page: 1866 issue: 8 year: 2010 end-page: 1878 ident: CR8 article-title: Numerical solution of the nonlinear Klein- Gordon equation publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2009.09.023 – volume: 181 start-page: 78 year: 2010 end-page: 91 ident: CR7 article-title: Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation publication-title: Comput. Phys. Commun. doi: 10.1016/j.cpc.2009.09.001 – volume: 2014 start-page: 1 year: 2014 end-page: 8 ident: CR36 article-title: Numerical solution of one-dimensional telegraph equation using cubic B-spline collocation method publication-title: Int. Sci. Publ. Consult. Serv. – volume: 25 start-page: 494 year: 2009 end-page: 506 ident: CR11 article-title: A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.20357 – ident: CR32 – volume: 244 start-page: 976 year: 2014 end-page: 997 ident: CR35 article-title: A numerical study of two-dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method publication-title: Appl. Math. Comput. – volume: 164 start-page: 33 year: 2019 end-page: 45 ident: CR31 article-title: Cubic quasi-interpolation spline collocation method for solving convection–diffusion equations publication-title: Math. Comput. Simul. doi: 10.1016/j.matcom.2018.11.003 – volume: 34 start-page: 324 issue: 4 year: 2010 end-page: 336 ident: CR12 article-title: Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation publication-title: Eng. Anal. Bound. Elem. doi: 10.1016/j.enganabound.2009.10.010 – volume: 32 start-page: 50 issue: 7 year: 1941 end-page: 66 ident: CR28 article-title: Experiments in approximating to solutions of a partial differential equation publication-title: Philos. Mag. doi: 10.1080/14786444108521297 – ident: CR41 – volume: 230 start-page: 626 issue: 2 year: 2009 end-page: 632 ident: CR26 article-title: A new fourth-order compact difference scheme for the two-dimensional second-order hyperbolic equation publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2009.01.001 – volume: 29 start-page: 761 issue: 131 year: 1975 end-page: 776 ident: CR40 article-title: A collocation method for two-point boundary value problems publication-title: Math. Comput. doi: 10.1090/S0025-5718-1975-0375785-7 – volume: 25 start-page: 409 issue: 2 year: 2009 end-page: 417 ident: CR5 article-title: Explicit analytical solutions of the generalized Burger and Burger-Fisher equations by homotopy perturbation method publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.20350 – volume: 25 start-page: 409 issue: 2 year: 2009 ident: 331_CR5 publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.20350 – volume: 41 start-page: 212 year: 1962 ident: 331_CR39 publication-title: J. Math. Phys. doi: 10.1002/sapm1962411212 – volume: 230 start-page: 626 issue: 2 year: 2009 ident: 331_CR26 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2009.01.001 – volume: 2012 start-page: 1 year: 2012 ident: 331_CR38 publication-title: Commun. Numer. Anal. doi: 10.5899/2012/cna-00129 – volume: 2011 start-page: 1 year: 2011 ident: 331_CR17 publication-title: Adv. Math. Phys. doi: 10.1155/2011/420608 – volume: 70 start-page: 231 year: 1996 ident: 331_CR22 publication-title: J. Comput. Appl. Math. doi: 10.1016/0377-0427(95)00201-4 – volume: 42 start-page: 31 issue: 1 year: 2013 ident: 331_CR2 publication-title: Heat Transf. Asian Res. doi: 10.1002/htj.21030 – volume: 244 start-page: 976 year: 2014 ident: 331_CR35 publication-title: Appl. Math. Comput. – volume: 86 start-page: 2061 issue: 12 year: 2009 ident: 331_CR15 publication-title: Int. J. Comput. Math. doi: 10.1080/00207160801965271 – volume: 3 start-page: 257 year: 1975 ident: 331_CR29 publication-title: Comput. Fluids doi: 10.1016/0045-7930(75)90022-5 – ident: 331_CR23 doi: 10.1002/num.1029 – volume: 2014 start-page: 1 year: 2014 ident: 331_CR36 publication-title: Int. Sci. Publ. Consult. Serv. – volume: 7 start-page: 83 issue: 2 year: 2012 ident: 331_CR37 publication-title: J. Inf. Comput. Sci. – ident: 331_CR41 – volume: 233 start-page: 1866 issue: 8 year: 2010 ident: 331_CR8 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2009.09.023 – ident: 331_CR33 doi: 10.1007/s40314-019-0838-0 – volume-title: Numerical Solution of Partial Differential Equations: Finite Difference Methods year: 1978 ident: 331_CR43 – volume: 43 start-page: 1142 issue: 9 year: 2013 ident: 331_CR4 publication-title: Comput. Biol. Med. doi: 10.1016/j.compbiomed.2013.05.019 – volume: 17 start-page: 684 year: 2001 ident: 331_CR20 publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.1034 – volume: 25 start-page: 232 issue: 1 year: 2009 ident: 331_CR9 publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.20341 – volume: 29 start-page: 761 issue: 131 year: 1975 ident: 331_CR40 publication-title: Math. Comput. doi: 10.1090/S0025-5718-1975-0375785-7 – volume: 164 start-page: 33 year: 2019 ident: 331_CR31 publication-title: Math. Comput. Simul. doi: 10.1016/j.matcom.2018.11.003 – volume: 4 start-page: 156 year: 2014 ident: 331_CR16 publication-title: Results Phys. doi: 10.1016/j.rinp.2014.08.009 – volume: 17 start-page: 101 year: 2004 ident: 331_CR24 publication-title: Appl. Math. Lett. doi: 10.1016/S0893-9659(04)90019-5 – volume: 181 start-page: 78 year: 2010 ident: 331_CR7 publication-title: Comput. Phys. Commun. doi: 10.1016/j.cpc.2009.09.001 – volume: 12 start-page: 1 issue: 3 year: 2012 ident: 331_CR3 publication-title: J. Mech. Med. Biol. – volume: 35 start-page: 1220 issue: 10 year: 2012 ident: 331_CR13 publication-title: Math. Methods Appl. Sci. doi: 10.1002/mma.2517 – ident: 331_CR27 – volume: 152 start-page: 799 issue: 3 year: 2004 ident: 331_CR19 publication-title: Appl. Math. Comput. – ident: 331_CR14 doi: 10.1080/00036811.2014.998654 – volume: 330 start-page: 23 year: 2018 ident: 331_CR34 publication-title: Appl. Math. Comput. – volume: 21 start-page: 864 issue: 7 year: 2011 ident: 331_CR1 publication-title: Int. J. Numer. Meth. Heat Fluid Flow doi: 10.1108/09615531111162837 – volume: 218 start-page: 7279 year: 2012 ident: 331_CR25 publication-title: Appl. Math. Comput. – volume: 7 start-page: 350 issue: 3 year: 2013 ident: 331_CR30 publication-title: Int. J. Math. Comput. Nat. Phys. Eng. – volume: 80 start-page: 357 issue: 3 year: 2003 ident: 331_CR10 publication-title: Int. J. Comput. Math. doi: 10.1080/0020716022000002800 – volume: 25 start-page: 494 year: 2009 ident: 331_CR11 publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.20357 – ident: 331_CR32 doi: 10.1007/s40096-019-00295-8 – volume: 190 start-page: 1683 year: 2007 ident: 331_CR18 publication-title: Appl. Math. Comput. – volume: 17 start-page: 607 year: 2001 ident: 331_CR21 publication-title: Numer. Methods Part. Differ. Equ. doi: 10.1002/num.1029 – volume: 56 start-page: 129 year: 2011 ident: 331_CR6 publication-title: Numer. Algoritm. doi: 10.1007/s11075-010-9377-x – volume: 32 start-page: 50 issue: 7 year: 1941 ident: 331_CR28 publication-title: Philos. Mag. doi: 10.1080/14786444108521297 – volume: 40 start-page: 469 year: 2002 ident: 331_CR42 publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142901389025 – volume: 34 start-page: 324 issue: 4 year: 2010 ident: 331_CR12 publication-title: Eng. Anal. Bound. Elem. doi: 10.1016/j.enganabound.2009.10.010 |
| SSID | ssj0000779045 |
| Score | 2.1675606 |
| Snippet | A method based on B-splines has been introduced for the solution of second-order nonlinear hyperbolic equation in 2-dimensions subject to appropriate initial... |
| SourceID | proquest crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 201 |
| SubjectTerms | Applications of Mathematics Boundary conditions Collocation methods Dirichlet problem Exact solutions Mathematics Mathematics and Statistics Numerical methods Ordinary differential equations Original Research Partial differential equations |
| SummonAdditionalLinks | – databaseName: Springer Open Access Hybrid - NESLI2 2011-2012 dbid: 40G link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV07T8MwELagLDAgnqJQkAc2sBTHcZyMgIAKqZ2o1C2KXwKpSqFJhfrv8TmPCgRIrInj4R6573y-7xC6NHFiXGLl_FsEnERGGJLkgSI0d54XU6Othubk0TgeTqKnKZ82TWFle9u9LUn6P3XX7BYB3CaQ7sAAMkpWm2gL6MRgbkEUPHYnKwFQ6PnpxL62TxlPm26Zn7f5GpHWMPNbZdQHnIc9tNsgRXxTq3YfbZjiAO2MOprV8hAV42VdcJnh1oTw3OISklxNPKsmrj7mRAOFf02_gV9c4rmQwAaMzXvN843lCstXopbSPbwlJTTpGgwWMq8P9HA9Z_oITR7un--GpBmgQBTjSUXyKHVwKI9znQrloIoSOjcRzY2CPMkmwmE3nupYaaoN00ry0OE9ri3TochVzI5Rr5gX5gRho5mlljIrNI2ENImymlIVMOkCWqKDPqKtEDPVsIvDkItZ1vEie8FnTvCZF3y26qOr7pu3mlvjz9WDVjdZ42dlFkLGxxMXcfvoutXX-vXvu53-b_kZ2g69ycDlsgHqVYulOXdopJIX3vg-AROk15k priority: 102 providerName: Springer Nature |
| Title | Numerical solution of second-order two-dimensional hyperbolic equation by bi-cubic B-spline collocation method |
| URI | https://link.springer.com/article/10.1007/s40096-020-00331-y https://www.proquest.com/docview/2428758678 |
| Volume | 14 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1LS8NAEB5se9GD-MRqLXvwpovZ5tmTWOkDoUXEQm8h2dmgUJq2aZH-e3c2SYuCveaxh9mZnW9mdr4BuFNeoHRgpe3bt1zuKF_xILIkF5G2PE8oTJCak4cjbzB2XifupEi4ZcW1yvJMNAc1ppJy5I8twvZuoM_Wp_mC09Qoqq4WIzQqUBPa05CeB73-NsdiEZmemVNsqvzCdttF34zpnnMIv3OKn2iimeCb375pBzj_1EiN6-mdwHGBGdlzvsmncKBmZ3A03BKuZucwG63z0suUlcrE0oRlFO4iN_yabPWdciQy_5yIg33qEHQZEy8wU4uc8ZvFGxZ_cbmO9cMOz6hdVzHSlTRP7bF84vQFjHvdj5cBL0YpcGm7wYpHTlsDo8iLsO1LDVqkj5FyRKQkRUxJ4GsU57bRkyhQ2Shjt6WRn4uJjS0_kp59CdVZOlNXwBTaiUiEnfgoHD9WgUxQCGnZsXZtAVp1EKUQQ1nwjNO4i2m4ZUg2gg-14EMj-HBTh_vtP_OcZWPv141yb8LC4rJwpx91eCj3a_f6_9Wu9692A4ctoyJ0rawB1dVyrW41DlnFTag4Vr9pVK4JtU539Pb-A-IP3PA |
| linkProvider | ProQuest |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LT-MwEB5BOSwcVrAP0V0ePuyeFmvjvHtAiKfKoxVCIHHLJh5HIKEGmlZV_9T-RmacpBVIcOOa2D6MP3tmbM_3AfwyYWwosaL1HTmB9E1kZJw6WqqUVl6oDObIxcm9fti98c9ug9sF-N_UwvCzymZPtBs1FprPyP-6HNsHMe2te49PklWj-Ha1kdCoYHFuphNK2crd0yOa39-ue3J8fdiVtaqA1F4Qj2TqdyhGSMMUO5Em_60jTI2vUqM5ecjjiAKaoIOhRoXGQ50FLgVBAeYeulGqQ4_GXYQlnytaW7B0cNy_vJqd6jhM32eVke27AkWN6kodW6_nc8YgOWNjDTUlpy-94TzEfXUra53dySp8rqNUsV_Bag0WzOALrPRmFK_lVxj0x9Vlz4No4CuKXJScYKO0jJ5iNCkksnxARf0h7ijpHWbMRCzMU8UxLrKpyO6lHmf08UCWXCBsBKOzqA4TRaVx_Q1uPsTM36E1KAZmHYRBL1e58vIIlR9lJtY5KqUdLyNnGqPTBtUYMdE1szkLbDwkM05ma_iEDJ9YwyfTNvyZ9XmseD3ebb3RzE1Sr_EymSOyDTvNfM1_vz3aj_dH24ZP3eveRXJx2j__CcuuhQs_atuA1mg4NpsUBY2yrRp6Av59NNqfARDlGYQ |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LT9wwEB7RRULtoSqlFdvy8KE9FYs4L2cPCEFhBaWsECoStzTx2GoltAGyK7R_rb-uM06yKyqVG9c8fBh_9szYM98H8MmmmaXEita3DhIZW21lVgRGqoJWXqosOuTm5PNRenIVf7tOrpfgT9cLw2WV3Z7oN2qsDJ-R74Yc2ycZ7a27ri2LuDga7t_eSVaQ4pvWTk6jgciZnT1Q-lbvnR7RXH8Ow-Hxj68nslUYkCZKsoks4gHFC0Va4EAb8uVGY2FjVVjDiYTLNAU3yQBTgwpthKZMQgqIEnQRhrowaUTjvoBlTV4x7sHy4fHo4nJ-whMwlZ9XSfY1BipKBm3Xju_dizl7kJy9sZ6akrPHnnER7v5zQ-sd3_ANvG4jVnHQQGwVluz4Lbw6n9O91mswHk2bi58b0UFZVE7UnGyj9OyeYvJQSWQpgYYGRPyiBPi-ZFZiYe8avnFRzkT5W5ppSQ8PZc3NwlYwUqvmYFE0etfv4OpZzPweeuNqbNdBWIyccipyGlWsS5sZh0qZICrJsWYY9EF1RsxNy3LOYhs3-Zyf2Rs-J8Pn3vD5rA9f5v_cNhwfT3690c1N3q73Ol-gsw873XwtXv9_tA9Pj7YNK4Ty_Pvp6OwjvAw9Wri-bQN6k_up3aSAaFJutcgT8PO5wf4X7uEdsA |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Numerical+solution+of+second-order+two-dimensional+hyperbolic+equation+by+bi-cubic+B-spline+collocation+method&rft.jtitle=Mathematical+sciences+%28Karaj%2C+Iran%29&rft.au=Arora%2C+Rajni&rft.au=Singh%2C+Swarn&rft.au=Singh%2C+Suruchi&rft.date=2020-09-01&rft.issn=2008-1359&rft.eissn=2251-7456&rft.volume=14&rft.issue=3&rft.spage=201&rft.epage=213&rft_id=info:doi/10.1007%2Fs40096-020-00331-y&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s40096_020_00331_y |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2008-1359&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2008-1359&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2008-1359&client=summon |