{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T12:58:53Z","timestamp":1774357133336,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T00:00:00Z","timestamp":1606089600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T00:00:00Z","timestamp":1606089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11771162"],"award-info":[{"award-number":["11771162"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s12190-020-01467-9","type":"journal-article","created":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T18:04:23Z","timestamp":1606154663000},"page":"853-870","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A novel numerical scheme for a time fractional Black\u2013Scholes equation"],"prefix":"10.1007","volume":"66","author":[{"given":"Mianfu","family":"She","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lili","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Renxuan","family":"Tang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dongfang","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,11,23]]},"reference":[{"key":"1467_CR1","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1086\/260062","volume":"81","author":"F Black","year":"1973","unstructured":"Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637\u2013654 (1973)","journal-title":"J. Polit. Econ."},{"key":"1467_CR2","doi-asserted-by":"crossref","first-page":"141","DOI":"10.2307\/3003143","volume":"4","author":"RC Merton","year":"1973","unstructured":"Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141\u2013183 (1973)","journal-title":"Bell J. Econ. Manag. Sci."},{"key":"1467_CR3","first-page":"51","volume":"3","author":"W Wyss","year":"2000","unstructured":"Wyss, W.: The fractional Black\u2013Scholes equation. Fract. Calc. Appl. Anal. 3, 51\u201361 (2000)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"1467_CR4","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1016\/j.aml.2010.03.022","volume":"23","author":"J Liang","year":"2010","unstructured":"Liang, J., Wang, J., Zhang, W., Qiu, W., Ren, F.: Option pricing of a bi-fractional Black\u2013Merton\u2013Scholes model with the Hurst exponent H in $$[\\frac{1}{2},1]$$. Appl. Math. Lett. 23, 859\u2013863 (2010)","journal-title":"Appl. Math. Lett."},{"key":"1467_CR5","doi-asserted-by":"crossref","first-page":"124380","DOI":"10.1016\/j.physa.2020.124380","volume":"550","author":"RM Jena","year":"2020","unstructured":"Jena, R.M., Chakraverty, S., Baleanu, D.: A novel analytical technique for the solution of time-fractional Ivancevic option pricing model. Physica A. 550, 124380 (2020)","journal-title":"Physica A."},{"key":"1467_CR6","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1016\/j.camwa.2015.03.025","volume":"69","author":"W Chen","year":"2015","unstructured":"Chen, W., Xu, X., Zhu, S.P.: Analytically pricing double barrier options based on a time-fractional Black\u2013Scholes equation. Comput. Math. Appl. 69, 1407\u20131419 (2015)","journal-title":"Comput. Math. Appl."},{"key":"1467_CR7","doi-asserted-by":"crossref","first-page":"1772","DOI":"10.1016\/j.camwa.2016.02.007","volume":"71","author":"H Zhang","year":"2016","unstructured":"Zhang, H., Liu, F., Turner, I., Yang, Q.: Numerical solution of the time fractional Black\u2013Scholes model governing European options. Comput. Math. Appl. 71, 1772\u20131783 (2016)","journal-title":"Comput. Math. Appl."},{"key":"1467_CR8","doi-asserted-by":"crossref","first-page":"122040","DOI":"10.1016\/j.physa.2019.122040","volume":"533","author":"VP Dubey","year":"2019","unstructured":"Dubey, V.P., Kumar, R., Kumar, D.: A reliable treatment of residual power series method for time-fractional Black\u2013Scholes European option pricing equation. Physica A. 533, 122040 (2019)","journal-title":"Physica A."},{"key":"1467_CR9","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.apnum.2019.11.004","volume":"151","author":"P Roul","year":"2020","unstructured":"Roul, P.: A high accuracy numerical method and its convergence for time-fractional Black\u2013Scholes equation governing European options. Appl. Numer. Math. 151, 472\u2013493 (2020)","journal-title":"Appl. Numer. Math."},{"key":"1467_CR10","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1016\/j.camwa.2017.06.005","volume":"74","author":"RH De Staelen","year":"2017","unstructured":"De Staelen, R.H., Hendy, A.S.: Numerically pricing double barrier options in a time-fractional Black\u2013Scholes model. Comput. Math. Appl. 74, 1166\u20131175 (2017)","journal-title":"Comput. Math. Appl."},{"key":"1467_CR11","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.ejbas.2014.10.003","volume":"1","author":"S Kumar","year":"2014","unstructured":"Kumar, S., Kumar, D., Singh, J.: Numerical computation of fractional Black\u2013Scholes equation arising in financial market. Egyptian J. Basic Appl. Sci. 1, 177\u2013193 (2014)","journal-title":"Egyptian J. Basic Appl. Sci."},{"key":"1467_CR12","first-page":"1699","volume":"36","author":"MN Koleva","year":"2017","unstructured":"Koleva, M.N., Vulkov, L.G.: Numerical solution of time-fractional Black\u2013Scholes equation. J. Comput. Appl. Math. 36, 1699\u20131715 (2017)","journal-title":"J. Comput. Appl. Math."},{"key":"1467_CR13","first-page":"1","volume":"71","author":"X Yang","year":"2016","unstructured":"Yang, X., Wu, L., Sun, S., Zhang, X.: A universal difference method for time-space fractional Black\u2013Scholes equation. Adv. Differ. Equ. 71, 1\u201314 (2016)","journal-title":"Adv. Differ. Equ."},{"key":"1467_CR14","first-page":"86","volume":"24","author":"D Li","year":"2018","unstructured":"Li, D., Liao, H., Sun, W., Wang, J., Zhang, J.: Analysis of L1-Galerkin FEMs for time fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86\u2013103 (2018)","journal-title":"Commun. Comput. Phys."},{"key":"1467_CR15","doi-asserted-by":"crossref","first-page":"3067","DOI":"10.1137\/16M1105700","volume":"39","author":"D Li","year":"2017","unstructured":"Li, D., Wang, J., Zhang, J.: Unconditionally convergent $$L1$$-Galerkin FEMs for nonlinear time-fractional Schr\u00f6dinger equations. SIAM J. Sci. Comput. 39, 3067\u20133088 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"1467_CR16","doi-asserted-by":"crossref","first-page":"1867","DOI":"10.1080\/00036811.2016.1197914","volume":"96","author":"Q Zhang","year":"2017","unstructured":"Zhang, Q., Ran, M., Xu, D.: Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay. Appl. Anal. 96, 1867\u20131884 (2017)","journal-title":"Appl. Anal."},{"key":"1467_CR17","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"ZZ Sun","year":"2006","unstructured":"Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193\u2013209 (2006)","journal-title":"Appl. Numer. Math."},{"key":"1467_CR18","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1088\/0253-6102\/60\/3\/03","volume":"60","author":"P Roul","year":"2013","unstructured":"Roul, P.: Analytical approach for the nonlinear partial differential equations of fractional order. Commun. Theor. Phys. 60, 269\u2013277 (2013)","journal-title":"Commun. Theor. Phys."},{"key":"1467_CR19","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1137\/16M1103622","volume":"56","author":"Z Mao","year":"2018","unstructured":"Mao, Z., Karniadakis, G.E.: A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56, 24\u201349 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1467_CR20","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.aml.2018.05.007","volume":"84","author":"X Chen","year":"2018","unstructured":"Chen, X., Di, Y., Duan, J., Li, D.: Linearized compact ADI schems for nonlinear time-fractional Schr\u00f6dinger equations. Appl. Math. Lett. 84, 160\u2013167 (2018)","journal-title":"Appl. Math. Lett."},{"key":"1467_CR21","doi-asserted-by":"crossref","first-page":"854","DOI":"10.4208\/nmtma.2018.s10","volume":"11","author":"C Sheng","year":"2018","unstructured":"Sheng, C., Shen, J.: A space-time Petrov\u2013Galerkin spectral method for time fractional diffusion equation. Numer. Math. Theor. Meth. Appl. 11, 854\u2013876 (2018)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"1467_CR22","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.4208\/nmtma.OA-2017-0144","volume":"12","author":"H Sun","year":"2019","unstructured":"Sun, H., Sun, Z., Du, R.: A linearized second-order difference scheme for the nonlinear time-fractional fourth-order reaction-diffusion equation. Numer. Math. Theor. Meth. Appl. 12, 1168\u20131190 (2019)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"1467_CR23","first-page":"1004","volume":"12","author":"M Cui","year":"2019","unstructured":"Cui, M.: Finite difference schemes for the variable coefficients single and multi-term time-fractional diffusion equations with non-smooth solutions on graded and uniform meshes. Numer. Math. Theor. Meth. Appl. 12, 1004\u20138979 (2019)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"1467_CR24","doi-asserted-by":"crossref","first-page":"239","DOI":"10.4208\/jcm.1805-m2017-0081","volume":"38","author":"M Ran","year":"2020","unstructured":"Ran, M., Zhang, C.: A high-order accuracy method for solving the fractional diffusion equations. J. Comput. Math. 38, 239\u2013253 (2020)","journal-title":"J. Comput. Math."},{"key":"1467_CR25","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1007\/s10543-014-0539-4","volume":"55","author":"N Kopteva","year":"2015","unstructured":"Kopteva, N., Stynes, M.: An efficient collocation method for a Caputo two-point boundary value problem. BIT. 55, 1105\u20131123 (2015)","journal-title":"BIT."},{"key":"1467_CR26","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1093\/imanum\/dru011","volume":"35","author":"M Stynes","year":"2015","unstructured":"Stynes, M., Gracia, J.L.: A finite difference method for a two- point boundary value problem with a Caputo fractional derivative. IMA J. Numer. Anal. 35, 689\u2013721 (2015)","journal-title":"IMA J. Numer. Anal."},{"key":"1467_CR27","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.cam.2014.05.025","volume":"273","author":"JL Gracia","year":"2015","unstructured":"Gracia, J.L., Stynes, M.: Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. J. Comput. Appl. Math. 273, 103\u2013115 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"1467_CR28","doi-asserted-by":"crossref","first-page":"106011","DOI":"10.1016\/j.aml.2019.106011","volume":"100","author":"L Li","year":"2020","unstructured":"Li, L., Li, D.: Exact solutions and numerical study of time fractional Burgers\u2019 equations. Appl. Math. Lett. 100, 106011 (2020)","journal-title":"Appl. Math. Lett."},{"key":"1467_CR29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/16M1089320","volume":"56","author":"B Jin","year":"2018","unstructured":"Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56, 1\u201323 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1467_CR30","doi-asserted-by":"publisher","DOI":"10.4208\/nmtma.OA-2020-0129","author":"D Li","year":"2021","unstructured":"Li, D., Sun, W., Wu, C.: A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions. Numer. Math. Theor. Meth. Appl. (2021). https:\/\/doi.org\/10.4208\/nmtma.OA-2020-0129","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"1467_CR31","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1137\/16M1082329","volume":"55","author":"M Stynes","year":"2017","unstructured":"Stynes, M., O\u2019Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057\u20131079 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"1467_CR32","doi-asserted-by":"crossref","first-page":"2874","DOI":"10.1016\/j.camwa.2018.01.016","volume":"75","author":"Z Cen","year":"2018","unstructured":"Cen, Z., Huang, J., Xu, A., Le, A.: Numerical approximation of a time-fractional Black\u2013Scholes equation. Comput. Math. Appl. 75, 2874\u20132887 (2018)","journal-title":"Comput. Math. Appl."},{"key":"1467_CR33","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s10915-019-00943-0","volume":"80","author":"D Li","year":"2019","unstructured":"Li, D., Wu, C., Zhang, J.: Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction. J. Sci. Comput. 80, 403\u2013419 (2019)","journal-title":"J. Sci. Comput."},{"key":"1467_CR34","doi-asserted-by":"crossref","first-page":"2135","DOI":"10.1090\/mcom\/3410","volume":"88","author":"N Kopteva","year":"2019","unstructured":"Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88, 2135\u20132155 (2019)","journal-title":"Math. Comput."},{"key":"1467_CR35","doi-asserted-by":"crossref","first-page":"3070","DOI":"10.1137\/16M1070323","volume":"38","author":"W Cao","year":"2016","unstructured":"Cao, W., Zeng, F., Zhang, Z., Karniadakis, G.E.: Implicit-Explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput. 38, 3070\u20133093 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"1467_CR36","doi-asserted-by":"crossref","first-page":"3129","DOI":"10.1137\/17M1118816","volume":"39","author":"B Jin","year":"2017","unstructured":"Jin, B., Li, B., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39, 3129\u20133152 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"1467_CR37","volume-title":"Methods of Mathematical Physics","author":"R Courant","year":"1953","unstructured":"Courant, R., Hilbert, D., Bergmann, P.: Methods of Mathematical Physics. Interscience Publishers Inc, New York (1953)"},{"key":"1467_CR38","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)"},{"key":"1467_CR39","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral Methods","author":"J Shen","year":"2011","unstructured":"Shen, J., Tang, T., Wang, L.L.: Spectral Methods. Springer, Heidelberg (2011)"},{"key":"1467_CR40","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1090\/S0025-5718-1982-0637287-3","volume":"38","author":"C Canuto","year":"1982","unstructured":"Canuto, C., Quarteroni, A.: Approximation results for orthognal polynomials in Sobolev Spaces. Math. Comput. 38, 67\u201386 (1982)","journal-title":"Math. Comput."},{"key":"1467_CR41","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1137\/0723077","volume":"23","author":"N Bressan","year":"1986","unstructured":"Bressan, N., Quarteroni, A.: Analysis of Chebyshev collocation methods for parabolic equations. SIAM J. Numer. Anal. 23, 1138\u20131154 (1986)","journal-title":"SIAM J. Numer. Anal."},{"key":"1467_CR42","volume-title":"Galerkin finite element methods for parabolic problems","author":"V Thom\u00e9e","year":"2006","unstructured":"Thom\u00e9e, V.: Galerkin finite element methods for parabolic problems. Springer, Berlin (2006)"},{"key":"1467_CR43","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s10915-020-01350-6","volume":"85","author":"B Zhou","year":"2020","unstructured":"Zhou, B., Chen, X., Li, D.: Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations. J. Sci. Comput. 85, 39 (2020)","journal-title":"J. Sci. Comput."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01467-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-020-01467-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01467-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,11]],"date-time":"2021-05-11T17:49:01Z","timestamp":1620755341000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-020-01467-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,23]]},"references-count":43,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["1467"],"URL":"https:\/\/doi.org\/10.1007\/s12190-020-01467-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,23]]},"assertion":[{"value":"22 September 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 November 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}