{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T14:26:21Z","timestamp":1750343181256,"version":"3.37.3"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T00:00:00Z","timestamp":1609977600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T00:00:00Z","timestamp":1609977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2021,10]]},"DOI":"10.1007\/s12190-020-01488-4","type":"journal-article","created":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T13:04:53Z","timestamp":1610024693000},"page":"175-206","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Well-posedness and ill-posedness results for backward problem for fractional pseudo-parabolic equation"],"prefix":"10.1007","volume":"67","author":[{"given":"Le Dinh","family":"Long","sequence":"first","affiliation":[]},{"given":"Yong","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Rathinasamy","family":"Sakthivel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6962-1898","authenticated-orcid":false,"given":"Nguyen Huy","family":"Tuan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,1,7]]},"reference":[{"key":"1488_CR1","volume-title":"Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies","author":"AA Kilbas","year":"2006","unstructured":"Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V, Amsterdam (2006)"},{"key":"1488_CR2","volume-title":"Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics 301","author":"V Kiryakova","year":"1994","unstructured":"Kiryakova, V.: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics 301. Longman, Harlow (1994)"},{"key":"1488_CR3","volume-title":"Fractional Differential Equations, Mathematics in Science and Engineering","author":"I Podlubny","year":"1990","unstructured":"Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press Inc, San Diego, CA (1990)"},{"issue":"3","key":"1488_CR4","doi-asserted-by":"publisher","first-page":"2867","DOI":"10.1137\/17M1160318","volume":"50","author":"L Li","year":"2018","unstructured":"Li, L., Liu, G.J.: A generalized definition of Caputo derivatives and its application to fractional ODEs. SIAM J. Math. Anal. 50(3), 2867\u20132900 (2018)","journal-title":"SIAM J. Math. Anal."},{"issue":"5","key":"1488_CR5","doi-asserted-by":"publisher","first-page":"1286","DOI":"10.1016\/0021-8928(60)90107-6","volume":"24","author":"GI Barenblat","year":"1960","unstructured":"Barenblat, G.I., Kochiva, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rock. J. Appl. Math. Mech. 24(5), 1286\u20131303 (1960)","journal-title":"J. Appl. Math. Mech."},{"key":"1488_CR6","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1098\/rsta.1972.0032","volume":"272","author":"TB Benjamin","year":"1972","unstructured":"Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272, 47\u201378 (1972)","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"1488_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF00250690","volume":"14","author":"WT Ting","year":"1963","unstructured":"Ting, W.T.: Certain non-steady flows of second-order fluids Arch. Ration. Mech. Anal. 14, 1\u201326 (1963)","journal-title":"Ration. Mech. Anal."},{"key":"1488_CR8","doi-asserted-by":"publisher","first-page":"2739","DOI":"10.1090\/S0002-9947-03-03340-3","volume":"356","author":"V Padron","year":"2004","unstructured":"Padron, V.: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation Trans. Am. Math. Soc. 356, 2739\u20132756 (2004)","journal-title":"Am. Math. Soc."},{"key":"1488_CR9","doi-asserted-by":"publisher","first-page":"4568","DOI":"10.1016\/j.jde.2009.03.021","volume":"246","author":"Y Cao","year":"2009","unstructured":"Cao, Y., Yin, J., Wang, C.: Cauchy problems of semilinear pseudo-parabolic equations. J. Differ. Equ. 246, 4568\u20134590 (2009)","journal-title":"J. Differ. Equ."},{"issue":"2","key":"1488_CR10","doi-asserted-by":"publisher","first-page":"1185","DOI":"10.3934\/dcds.2019051","volume":"39","author":"H Chen","year":"2019","unstructured":"Chen, H., Xu, H.: Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete Contin. Dyn. Syst. 39(2), 1185\u20131203 (2019)","journal-title":"Discrete Contin. Dyn. Syst."},{"issue":"12","key":"1488_CR11","doi-asserted-by":"publisher","first-page":"4424","DOI":"10.1016\/j.jde.2015.01.038","volume":"258","author":"H Chen","year":"2015","unstructured":"Chen, H., Tian, S.: Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differ. Equ. 258(12), 4424\u20134442 (2015)","journal-title":"J. Differ. Equ."},{"key":"1488_CR12","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1016\/j.jmaa.2019.05.018","volume":"478","author":"H Ding","year":"2019","unstructured":"Ding, H., Zhou, J.: Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity. J. Math. Anal. Appl. 478, 393\u2013420 (2019)","journal-title":"J. Math. Anal. Appl."},{"issue":"10","key":"1488_CR13","doi-asserted-by":"publisher","first-page":"2221","DOI":"10.1016\/j.camwa.2017.03.005","volume":"73","author":"L Jin","year":"2017","unstructured":"Jin, L., Li, L., Fang, S.: The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation. Comput. Math. Appl. 73(10), 2221\u20132232 (2017)","journal-title":"Comput. Math. Appl."},{"key":"1488_CR14","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.amc.2018.02.003","volume":"329","author":"X Zhu","year":"2018","unstructured":"Zhu, X., Li, F., Li, Y.: Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term. Appl. Math. Comput. 329, 38\u201351 (2018)","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"1488_CR15","doi-asserted-by":"publisher","first-page":"1531","DOI":"10.1016\/j.camwa.2019.03.040","volume":"78","author":"M Abbaszadeh","year":"2019","unstructured":"Abbaszadeh, M., Dehghan, M.: Analysis of mixed finite element method (MFEM) for solving the generalized fractional reaction-diffusion equation on nonrectangular domains. Comput. Math. Appl. 78(5), 1531\u20131547 (2019)","journal-title":"Comput. Math. Appl."},{"key":"1488_CR16","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1016\/j.apnum.2019.05.005","volume":"145","author":"M Abbaszadeh","year":"2019","unstructured":"Abbaszadeh, M., Dehghan, M.: Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method. Appl. Numer. Math. 145, 488\u2013506 (2019)","journal-title":"Appl. Numer. Math."},{"key":"1488_CR17","doi-asserted-by":"publisher","first-page":"620","DOI":"10.1140\/epjp\/i2019-12951-0","volume":"134","author":"M Abbaszadeh","year":"2019","unstructured":"Abbaszadeh, M., Dehghan, M.: A reduced order finite difference method for solving space-fractional reaction-diffusion systems: the Gray\u2013Scott model. Eur. Phys. J. Plus 134, 620 (2019)","journal-title":"Eur. Phys. J. Plus"},{"issue":"1","key":"1488_CR18","first-page":"18","volume":"3","author":"K Aissania","year":"2019","unstructured":"Aissania, K., Benchohrab, M., Nieto, J.J.: Controllability for impulsive fractional evolution inclusions with state-dependent delay. Adv. Theory Nonlinear Anal. Appl. 3(1), 18\u201334 (2019)","journal-title":"Adv. Theory Nonlinear Anal. Appl."},{"issue":"4","key":"1488_CR19","first-page":"224","volume":"2","author":"TH Kaddoura","year":"2018","unstructured":"Kaddoura, T.H., Hakema, A.: Sufficient conditions of non global solution for fractional damped wave equations with non-linear memory. Adv. Theory Nonlinear Anal. Appl. 2(4), 224\u2013237 (2018)","journal-title":"Adv. Theory Nonlinear Anal. Appl."},{"key":"1488_CR20","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1016\/j.jmaa.2011.04.058","volume":"382","author":"K Sakamoto","year":"2011","unstructured":"Sakamoto, K., Yamamoto, M.: Initial value\/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426\u2013447 (2011)","journal-title":"J. Math. Anal. Appl."},{"key":"1488_CR21","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.aml.2018.11.015","volume":"92","author":"NH Tuan","year":"2019","unstructured":"Tuan, N.H., Huynh, L.N., Ngoc, T.B., Zhou, Y.: On a backward problem for nonlinear fractional diffusion equations. Appl. Math. Lett. 92, 76\u201384 (2019)","journal-title":"Appl. Math. Lett."},{"issue":"18","key":"1488_CR22","doi-asserted-by":"publisher","first-page":"5945","DOI":"10.1002\/mma.5165","volume":"42","author":"Y Zhou","year":"2019","unstructured":"Zhou, Y., He, J.W., Ahmad, B., Tuan, N.H.: Existence and regularity results of a backward problem for fractional diffusion equations. Math. Meth. Appl. Sci. 42(18), 5945\u20137545 (2019)","journal-title":"Math. Meth. Appl. Sci."},{"issue":"5","key":"1488_CR23","doi-asserted-by":"publisher","first-page":"1396","DOI":"10.1016\/j.camwa.2018.11.036","volume":"78","author":"NH Tuan","year":"2019","unstructured":"Tuan, N.H., Debbouche, A., Ngoc, T.B.: Existence and regularity of final value problems for time fractional wave equations. Comput. Math. Appl. 78(5), 1396\u20131414 (2019)","journal-title":"Comput. Math. Appl."},{"key":"1488_CR24","first-page":"3","volume":"10","author":"MK Beshtokov","year":"2018","unstructured":"Beshtokov, M.K.: To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov\u2013Caputo fractional derivative. Izv. Vyssh. Uchebn. Zaved. Mat. 10, 3\u201316 (2018)","journal-title":"Izv. Vyssh. Uchebn. Zaved. Mat."},{"key":"1488_CR25","volume-title":"Boundary-value Problems for Loaded Pseudoparabolic Equations of Fractional Order and Difference Methods of Their Solving, Russian Mathematics","author":"MK Beshtokov","year":"2019","unstructured":"Beshtokov, M.K.: Boundary-value Problems for Loaded Pseudoparabolic Equations of Fractional Order and Difference Methods of Their Solving, Russian Mathematics. Springer, Berlin (2019)"},{"issue":"7","key":"1488_CR26","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1134\/S0374064119070021","volume":"55","author":"MK Beshtokov","year":"2019","unstructured":"Beshtokov, M.K.: Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative. Differ. Uravn. 55(7), 919\u2013928 (2019)","journal-title":"Differ. Uravn."},{"issue":"7","key":"1488_CR27","doi-asserted-by":"publisher","first-page":"884","DOI":"10.1134\/S0012266119070024","volume":"55","author":"MK Beshtokov","year":"2019","unstructured":"Beshtokov, M.K.: Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative. Differ. Equ. 55(7), 884\u2013893 (2019)","journal-title":"Differ. Equ."},{"issue":"2","key":"1488_CR28","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1007\/s00574-018-0112-x","volume":"50","author":"CVJ Sousa","year":"2019","unstructured":"Sousa, C.V.J., de Oliveira, C.E.: Fractional order pseudoparabolic partial differential equation: Ulam\u2013Hyers stability Bull. Braz. Math. Soc. (N.S.) 50(2), 481\u2013496 (2019)","journal-title":"Braz. Math. Soc. (N.S.)"},{"key":"1488_CR29","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2014.979808","author":"S Tatara","year":"2014","unstructured":"Tatara, S., Ulusoy, S.: An inverse source problem for a one-dimensional space-time fractional diffusion equation. Appl. Anal. (2014). https:\/\/doi.org\/10.1080\/00036811.2014.979808","journal-title":"Appl. Anal."},{"issue":"1","key":"1488_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/03605302.2017.1363229","volume":"43","author":"W Arendt","year":"2018","unstructured":"Arendt, W., ter Elst, A.F.M., Warma, M.: Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Commun. Partial Differ. Equ. 43(1), 1\u201324 (2018)","journal-title":"Commun. Partial Differ. Equ."},{"issue":"1","key":"1488_CR31","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1090\/S0002-9947-99-02528-3","volume":"352","author":"JM Arrieta","year":"1999","unstructured":"Arrieta, J.M., Carvalho, A.N.: Abstract parabolic problems with critical nonlinearities and applications to Navier\u2013Stokes and heat equations. Trans. Am. Math. Soc. 352(1), 285\u2013310 (1999)","journal-title":"Trans. Am. Math. Soc."},{"key":"1488_CR32","doi-asserted-by":"publisher","DOI":"10.1080\/17415977.2016.1259316","author":"NH Tuan","year":"2016","unstructured":"Tuan, N.H., Long, L.D., Thinh, V.N., Thanh, T.: On a final value problem for the time-fractional diffusion equation with inhomogeneous source. Inverse Probl. Sci. Eng. (2016). https:\/\/doi.org\/10.1080\/17415977.2016.1259316","journal-title":"Inverse Probl. Sci. Eng."},{"key":"1488_CR33","volume-title":"Mehtod of Mathematical Physics","author":"R Courant","year":"1953","unstructured":"Courant, R., Hilbert, D.: Mehtod of Mathematical Physics. Interscience, New York (NY) (1953)"},{"key":"1488_CR34","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1515\/jiip.2010.028","volume":"6","author":"XL Feng","year":"2010","unstructured":"Feng, X.L., Elden, L., Fu, C.L.: A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data. J. Inverse Ill-Posed Probl. 6, 617\u2013645 (2010)","journal-title":"J. Inverse Ill-Posed Probl."},{"key":"1488_CR35","doi-asserted-by":"publisher","first-page":"1045","DOI":"10.1080\/17415977.2013.850683","volume":"7","author":"XL Feng","year":"2014","unstructured":"Feng, X.L., Ning, W., Qian, Z.: A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions. Inverse Probl. Sci. Eng. 7, 1045\u20131061 (2014)","journal-title":"Inverse Probl. Sci. Eng."},{"issue":"12","key":"1488_CR36","doi-asserted-by":"publisher","first-page":"1689","DOI":"10.1080\/00207160.2012.693174","volume":"89","author":"HW Zhang","year":"2012","unstructured":"Zhang, H.W.: Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation. Int. J. Comput. Math. 89(12), 1689\u20131703 (2012)","journal-title":"Int. J. Comput. Math."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01488-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-020-01488-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01488-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T01:14:05Z","timestamp":1724289245000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-020-01488-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,7]]},"references-count":36,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["1488"],"URL":"https:\/\/doi.org\/10.1007\/s12190-020-01488-4","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2021,1,7]]},"assertion":[{"value":"21 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 December 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 December 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}