{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T04:40:42Z","timestamp":1769834442383,"version":"3.49.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s00500-025-10942-z","type":"journal-article","created":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T04:12:53Z","timestamp":1762575173000},"page":"473-490","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Numerical solution of a multi term time-fractional convection diffusion reaction equation with variable coefficients subjected to weakly singular solution"],"prefix":"10.1007","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-3343-8672","authenticated-orcid":false,"given":"Jyoti","family":"Yadav","sequence":"first","affiliation":[]},{"given":"Pradip","family":"Roul","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,11,8]]},"reference":[{"key":"10942_CR1","doi-asserted-by":"crossref","unstructured":"Alhasan ASH, Saranya S, Mdallal QM (2024) Fractional derivative modeling of heat transfer and fluid flow around a contracting permeable infinite cylinder: Computational study. 11, 100794","DOI":"10.1016\/j.padiff.2024.100794"},{"key":"10942_CR2","first-page":"46","volume":"101","author":"A Atangana","year":"2017","unstructured":"Atangana A, Baleanu D (2017) Caputo-fabrizio fractional derivatives applied to groundwater flow within confined aquifers. XJ Engg Mech 101:46\u201353","journal-title":"XJ Engg Mech"},{"key":"10942_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/s11704-023-2640-9","author":"W Bao","year":"2024","unstructured":"Bao W, Yang B (2024) Protein acetylation sites with complex-valued polynomial model. Front Comput Sci. https:\/\/doi.org\/10.1007\/s11704-023-2640-9","journal-title":"Front Comput Sci"},{"key":"10942_CR4","doi-asserted-by":"publisher","DOI":"10.3389\/fmicb.2023.1277121","volume":"14","author":"W Bao","year":"2024","unstructured":"Bao W, Liu Y, Chen B (2024) Oral. Front Microbiol 14:1277121. https:\/\/doi.org\/10.3389\/fmicb.2023.1277121","journal-title":"Front Microbiol"},{"key":"10942_CR5","doi-asserted-by":"publisher","DOI":"10.2174\/0115734099277249240129114123","author":"W Bao","year":"2024","unstructured":"Bao W, Chen B, Zhang Y (2024) A weakly supervised hybrid neural network for the identification of DNA-protein binding sites. Curr Comput Aided Drug Des. https:\/\/doi.org\/10.2174\/0115734099277249240129114123","journal-title":"Curr Comput Aided Drug Des"},{"key":"10942_CR6","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1016\/j.cam.2017.09.011","volume":"330","author":"H Chen","year":"2018","unstructured":"Chen H, L\u00fc S, Chen W (2018) A unified numerical scheme for the multi-term time fractional diffusion and diffusion wave equations with variable coefficients. J Comput Appl Math 330:380\u2013397","journal-title":"J Comput Appl Math"},{"key":"10942_CR7","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/j.cam.2015.04.037","volume":"290","author":"M Dehghan","year":"2015","unstructured":"Dehghan M, Safarpoor M, Abbaszadeh M (2015) Two high order numerical algorithms for solving the multi term time fractional diffusion wave equations. J Comput Appl Math 290:174\u2013195","journal-title":"J Comput Appl Math"},{"key":"10942_CR8","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1016\/j.cnsns.2018.01.020","volume":"61","author":"CJM Duarte","year":"2018","unstructured":"Duarte CJM, Garcia JR, Correa CR, Perez AG, Cervantes AJG (2018) A closed form expression for the gaussian-based caputo-fabrizio fractional derivative for signal processing applications. Commun Nonlinear Sci 61:138\u2013148","journal-title":"Commun Nonlinear Sci"},{"issue":"1","key":"10942_CR9","doi-asserted-by":"publisher","DOI":"10.1088\/0004-637X\/783\/1\/15","volume":"783","author":"F Effenberger","year":"2014","unstructured":"Effenberger F, Litvinenko Y (2014) The diffusion approximation versus the telegraph equation for modeling solar energetic particle transport with adiabatic focusing. Astrophys J 783(1):15","journal-title":"Astrophys J"},{"key":"10942_CR10","doi-asserted-by":"publisher","first-page":"3698","DOI":"10.1007\/s10773-014-2123-8","volume":"53","author":"JH He","year":"2014","unstructured":"He JH (2014) A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 53:3698\u20133718","journal-title":"Int J Theor Phys"},{"key":"10942_CR11","doi-asserted-by":"publisher","first-page":"272","DOI":"10.1016\/j.rinp.2018.06.011","volume":"10","author":"JH He","year":"2018","unstructured":"He JH (2018) Fractal calculus and its geometrical explanation. Results Phys 10:272\u2013276","journal-title":"Results Phys"},{"key":"10942_CR12","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1515\/cmam-2019-0042","volume":"20","author":"C Huang","year":"2020","unstructured":"Huang C, Liu X, Meng X, Stynes M (2020) Error analysis of a finite difference method on graded meshes for a multi term time fractional initial boundary value problem. Comput Methods Appl Math 20:815\u2013825","journal-title":"Comput Methods Appl Math"},{"key":"10942_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jhydrol.2023.129059","volume":"617","author":"AK Kwaw","year":"2023","unstructured":"Kwaw AK, Dou Z, Wang J, Zhang X, Chen Y (2023) Advancing the knowledge of solute transport in the presence of bound water in mixed porous media based on low-field nuclear magnetic resonance. J Hydrol 617:129059","journal-title":"J Hydrol"},{"key":"10942_CR14","doi-asserted-by":"publisher","first-page":"1115","DOI":"10.1080\/01630563.2021.1936019","volume":"42","author":"C Li","year":"2021","unstructured":"Li C, Wang Z (2021) Numerical methods for the time fractional convection diffusion reaction equation. Numer Funct Anal Optim 42:1115\u20131153","journal-title":"Numer Funct Anal Optim"},{"key":"10942_CR15","first-page":"381","volume":"257","author":"Z Li","year":"2015","unstructured":"Li Z, Liu Y, Yamamoto M (2015) Initial-boundary value problems for multi term time fractional diffusion equations with positive constant coefficients. Appl Math Comput 257:381\u2013397","journal-title":"Appl Math Comput"},{"key":"10942_CR16","doi-asserted-by":"publisher","first-page":"1867","DOI":"10.2298\/TSCI160415101L","volume":"21","author":"FJ Liu","year":"2017","unstructured":"Liu FJ, Liu HY, Li ZB, He JH (2017) A delayed fractional model for cocoon heat-proof property. Therm Sci 21:1867\u20131871","journal-title":"Therm Sci"},{"key":"10942_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-024-01830-y","author":"Z Lu","year":"2024","unstructured":"Lu Z, Fan W (2024) A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition. Numer Algorithms. https:\/\/doi.org\/10.1007\/s11075-024-01830-y","journal-title":"Numer Algorithms"},{"key":"10942_CR18","doi-asserted-by":"publisher","first-page":"538","DOI":"10.1016\/j.jmaa.2010.08.048","volume":"374","author":"Y Luchko","year":"2011","unstructured":"Luchko Y (2011) Initial boundary value problems for the generalized multi-term time-fractional diffusion equation. J Math Anal Appl 374:538\u2013548","journal-title":"J Math Anal Appl"},{"issue":"4","key":"10942_CR19","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1016\/j.cnsns.2009.05.004","volume":"15","author":"FC Meral","year":"2010","unstructured":"Meral FC, Royston TJ, Magin RL (2010) Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul 15(4):939\u2013945","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"10942_CR21","first-page":"357","volume":"03","author":"YF Pu","year":"2007","unstructured":"Pu YF (2007) Fractional differential analysis for texture of digital image. J Algorithms Comput Technol 03:357\u2013380","journal-title":"J Algorithms Comput Technol"},{"key":"10942_CR22","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.cjph.2024.02.051","volume":"89","author":"Dwivedi HK Rajeev","year":"2024","unstructured":"Rajeev Dwivedi HK (2024) A fast difference scheme for the multi term time fractional advection diffusion equation with a non-linear source term. Chinese J Phys 89:86\u2013103","journal-title":"Chinese J Phys"},{"key":"10942_CR23","doi-asserted-by":"publisher","first-page":"242","DOI":"10.4208\/eajam.181113.280514a","volume":"4","author":"J Ren","year":"2014","unstructured":"Ren J, Sun ZZ (2014) Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations. East Asian J Appl Math 4:242\u2013266","journal-title":"East Asian J Appl Math"},{"issue":"18","key":"10942_CR24","doi-asserted-by":"publisher","first-page":"7396","DOI":"10.1002\/mma.4536","volume":"40","author":"P Roul","year":"2017","unstructured":"Roul P (2017) On the numerical solution of singular two-point boundary value problems: a domain decomposition homotopy perturbation approach. Math Methods Appl Sci 40(18):7396\u20137409","journal-title":"Math Methods Appl Sci"},{"issue":"3","key":"10942_CR25","doi-asserted-by":"publisher","first-page":"504","DOI":"10.1016\/j.camwa.2020.04.001","volume":"80","author":"P Roul","year":"2020","unstructured":"Roul P (2020) A fourth order numerical method based on B-spline functions for pricing asian options. Comput Math Appl 80(3):504\u2013521","journal-title":"Comput Math Appl"},{"key":"10942_CR26","doi-asserted-by":"crossref","unstructured":"Roul P (2024) A high-order numerical scheme and its analysis for Caputo temporal-fractional Black-Scholes model: European double barrier knock-out option. Numer Algorithms 1-36","DOI":"10.1007\/s11075-024-01802-2"},{"key":"10942_CR27","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.apnum.2021.03.017","volume":"166","author":"P Roul","year":"2021","unstructured":"Roul P, Goura VMKP (2021) Compact finite difference scheme for fractional Black-Scholes option pricing model. Appl Numer Math 166:40\u201360","journal-title":"Appl Numer Math"},{"issue":"2","key":"10942_CR28","doi-asserted-by":"publisher","first-page":"1506","DOI":"10.1002\/num.22594","volume":"37","author":"P Roul","year":"2021","unstructured":"Roul P, Goura VMKP (2021) A high order numerical scheme for solving a class of non-homogeneous time-fractional reaction diffusion equation. Numer Methods Partial Differ Equ 37(2):1506\u20131534","journal-title":"Numer Methods Partial Differ Equ"},{"key":"10942_CR29","doi-asserted-by":"publisher","first-page":"16857","DOI":"10.1002\/mma.9478","volume":"46","author":"P Roul","year":"2023","unstructured":"Roul P, Rohil V (2023) An efficient numerical scheme and its analysis for the multiterm time fractional convection diffusion reaction equation. Math Methods Appl Sci 46:16857\u201316875","journal-title":"Math Methods Appl Sci"},{"key":"10942_CR30","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/j.apnum.2019.05.029","volume":"145","author":"P Roul","year":"2019","unstructured":"Roul P, Goura VMKP, Madduri H, Obaidurrahman K (2019) Design and stability analysis of an implicit non-standard finite difference scheme for fractional neutron point kinetic equation. Appl Numer Math 145:201\u2013226","journal-title":"Appl Numer Math"},{"key":"10942_CR31","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/j.pnucene.2024.105144","volume":"170","author":"P Roul","year":"2024","unstructured":"Roul P, Rohil V, Espinosa-Paredes G, Obaidurrahman K (2024) Numerical approximation of a two-dimensional fractional neutron diffusion model describing dynamics of neutron flux in a nuclear reactor. Prog Nucl Energy 170:105\u2013144","journal-title":"Prog Nucl Energy"},{"key":"10942_CR32","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s11071-015-2326-4","volume":"83","author":"SC Shiralashetti","year":"2016","unstructured":"Shiralashetti SC, Deshi AB (2016) An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dyn 83:293\u2013303","journal-title":"Nonlinear Dyn"},{"key":"10942_CR33","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1016\/j.mcm.2009.11.002","volume":"51","author":"V Srivastava","year":"2010","unstructured":"Srivastava V, Rai KN (2010) A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math Comput Model 51:616\u2013624","journal-title":"Math Comput Model"},{"key":"10942_CR34","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1137\/16M1082329","volume":"55","author":"M Stynes","year":"2017","unstructured":"Stynes M, O\u2019Riordan E, Gracia JL (2017) Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J Numer Anal 55:1057\u20131079","journal-title":"SIAM J Numer Anal"},{"key":"10942_CR35","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1016\/j.chaos.2017.03.060","volume":"102","author":"HG Sun","year":"2017","unstructured":"Sun HG, Li ZP, Zhang Y, Chen W (2017) Fractional and fractal derivative models for transient anomalous diffusion: model comparison. Chaos Solitons Fractals 102:346\u2013353","journal-title":"Chaos Solitons Fractals"},{"key":"10942_CR36","doi-asserted-by":"publisher","DOI":"10.3389\/fphy.2017.00052","volume":"5","author":"AA Tateishi","year":"2017","unstructured":"Tateishi AA, Ribeiro HV, Lenzi EK (2017) The role of fractional time-derivative operators on anomalous diffusion. Front Phys 5:52","journal-title":"Front Phys"},{"key":"10942_CR37","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/j.jmbbm.2018.09.029","volume":"89","author":"GZ Voyiadjis","year":"2018","unstructured":"Voyiadjis GZ, Sumelka W (2018) Brain modelling in the framework of anisotropic hyperelasticity with time fractional damage evolution governed by the Caputo-Almeida fractional derivative. J Mech Behav Biomed 89:209\u2013216","journal-title":"J Mech Behav Biomed"},{"key":"10942_CR38","volume":"381","author":"YM Wang","year":"2020","unstructured":"Wang YM, Wen X (2020) A compact exponential difference method for multi term time fractional convection reaction diffusion problems withnon smooth solutions. Appl Math Comput 381:125316","journal-title":"Appl Math Comput"},{"issue":"2","key":"10942_CR39","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-021-01455-0","volume":"40","author":"L Wu","year":"2021","unstructured":"Wu L, Pan Y, Yang X (2021) An efficient alternating segment parallel finite difference method for multi-term time fractional diffusion-wave equation. Comput Appl Math 40(2):67","journal-title":"Comput Appl Math"},{"key":"10942_CR40","doi-asserted-by":"publisher","first-page":"1965","DOI":"10.1007\/s11075-021-01102-z","volume":"88","author":"B Zhang","year":"2010","unstructured":"Zhang B, Bu W, Xiao A (2010) Efficient difference method for time-space fractional diffusion equation with robin fractional derivative boundary condition. Numer Algorithms 88:1965\u20131988","journal-title":"Numer Algorithms"},{"key":"10942_CR41","doi-asserted-by":"publisher","first-page":"1087","DOI":"10.1016\/j.camwa.2016.05.005","volume":"73","author":"Y Zhao","year":"2017","unstructured":"Zhao Y, Zhang Y, Liu F, Turner I, Tang Y, Anh V (2017) Convergence and superconvergence of a fully discrete scheme for multi term time fractional diffusion equations. Comput Math Appl 73:1087\u20131099","journal-title":"Comput Math Appl"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-025-10942-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-025-10942-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-025-10942-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T15:54:42Z","timestamp":1769788482000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-025-10942-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,8]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["10942"],"URL":"https:\/\/doi.org\/10.1007\/s00500-025-10942-z","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,8]]},"assertion":[{"value":"8 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 October 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There are no Conflicts.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}}]}}