{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T03:42:09Z","timestamp":1776397329754,"version":"3.51.2"},"reference-count":67,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T00:00:00Z","timestamp":1747267200000},"content-version":"vor","delay-in-days":134,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>The study examines the convergence analysis of a higher order approximation for a transport equation with both nonhomogeneous and homogeneous boundary conditions by using the Crank\u2013Nicolson and their modified schemes. Using the Taylor series expressions, the suggested numerical schemes are created. Von Neumann stability analysis combined with the error\u2010boundedness criterion provides a complete examination of stability and convergence. The analysis\u2019s findings show that both Crank\u2013Nicolson systems exhibit unconditional stability and are convergent with second\u2010order accuracy in both directions. Using the current two approaches, extensive numerical experiments are carried out. The outcomes of carried\u2010out trials are compared to the accuracy of the previous schemes in the literature. The comparative analysis demonstrates that the current approaches yield higher accuracy than the earlier techniques. In addition, we contrast the results of the two approaches and compare them. Based on these findings, the accuracy of the modified Crank\u2013Nicolson scheme is better than that of the original Crank\u2013Nicolson scheme. Overall, this research demonstrates that current methods effectively solve partial differential equations.<\/jats:p>","DOI":"10.1155\/jama\/5661721","type":"journal-article","created":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T04:35:37Z","timestamp":1747283737000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Convergence Analysis of Higher Order Approximation for the Transport Equation With Nonhomogeneous and Homogeneous Boundary Conditions"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3769-4870","authenticated-orcid":false,"given":"Kedir Aliyi","family":"Koroche","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-0964-7951","authenticated-orcid":false,"given":"Dase Fayera","family":"Hule","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,5,15]]},"reference":[{"key":"e_1_2_12_1_2","doi-asserted-by":"publisher","DOI":"10.11648\/j.ijamtp.20210702.11"},{"key":"e_1_2_12_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00020-012-1963-y"},{"key":"e_1_2_12_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10958-015-2633-9"},{"key":"e_1_2_12_4_2","doi-asserted-by":"publisher","DOI":"10.11648\/j.ajmcm.20210602.12"},{"key":"e_1_2_12_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/jmri.25334"},{"key":"e_1_2_12_6_2","doi-asserted-by":"publisher","DOI":"10.1155\/2022\/8613490"},{"key":"e_1_2_12_7_2","article-title":"Numerical Solution of Second Numerical Solution of Second-Order One Dimensional Linear Hyperbolic Telegraph Equation","volume":"4","author":"Dingeta M.","year":"2018","journal-title":"Ethiopian Journal of Education and Science"},{"key":"e_1_2_12_8_2","doi-asserted-by":"publisher","DOI":"10.5194\/hess-21-3635-2017"},{"key":"e_1_2_12_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2024.100798"},{"key":"e_1_2_12_10_2","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:1999125"},{"key":"e_1_2_12_11_2","doi-asserted-by":"publisher","DOI":"10.1115\/1.4042687"},{"key":"e_1_2_12_12_2","first-page":"716","article-title":"Mathematical Modeling of Parameter Identification Process of Convection-Diffusion Transport Models Using the SVD-Based Kalman Filter","volume":"25","author":"Kuvshinova A. N.","year":"2021","journal-title":"Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences"},{"key":"e_1_2_12_13_2","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3207"},{"key":"e_1_2_12_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2010.01.013"},{"key":"e_1_2_12_15_2","volume-title":"Partial Differential Equations for Scientists and Engineers","author":"Farlow S. J.","year":"1993"},{"key":"e_1_2_12_16_2","first-page":"211","article-title":"Calculation and Experimental Study of Heat Exchange in a System of Plane-Parallel Channels with Surface Intensifiers","volume":"17","author":"Kaskov S. I.","year":"2021","journal-title":"Russian Journal of Nonlinear Dynamics"},{"key":"e_1_2_12_17_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.77.021122"},{"key":"e_1_2_12_18_2","doi-asserted-by":"publisher","DOI":"10.1142\/S0219887817500840"},{"key":"e_1_2_12_19_2","unstructured":"WastonD. Radial Basis Function Differential Quadrature Method for the Numerical Solution of Partial Differential Equations 2017 Diss. The University of Southern Mississippi."},{"key":"e_1_2_12_20_2","first-page":"433","volume-title":"Proceedings of the CEUR Workshop Proceedings","author":"Matveev M. G.","year":"2017"},{"key":"e_1_2_12_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.advengsoft.2006.10.006"},{"key":"e_1_2_12_22_2","doi-asserted-by":"publisher","DOI":"10.29121\/ijoest.v6.i2.2022.299"},{"key":"e_1_2_12_23_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-56103-0_5"},{"key":"e_1_2_12_24_2","first-page":"49","article-title":"A Non-Iterative Domain Decomposition Method for the Forward-Backward Heat Equation","volume":"10","author":"Banei S.","year":"2020","journal-title":"Iranian Journal of Numerical Analysis and Optimization"},{"key":"e_1_2_12_25_2","article-title":"Numerical Simulation of Three-Dimensional Fractional-Order Convection-Diffusion PDES by a Local Meshless Method","volume":"210","author":"Srivastava M. H.","year":"2020","journal-title":"Thermal Science"},{"key":"e_1_2_12_26_2","first-page":"347","article-title":"A Numerical ssingSolution of the Advection-Diffusion Equation by Using Extended Cubic B-Spline Functions","volume":"19","author":"G\u00f6rg\u00fcl\u00fc M. Z.","year":"2018","journal-title":"Anadolu University Journal of Science and Technology A-Applied Sciences and Engineering"},{"key":"e_1_2_12_27_2","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/1745\/1\/012110"},{"key":"e_1_2_12_28_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2015.11.041"},{"key":"e_1_2_12_29_2","doi-asserted-by":"publisher","DOI":"10.1515\/phys-2021-0011"},{"key":"e_1_2_12_30_2","doi-asserted-by":"publisher","DOI":"10.1080\/10407790701443859"},{"key":"e_1_2_12_31_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.12.015"},{"key":"e_1_2_12_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.advengsoft.2006.08.001"},{"key":"e_1_2_12_33_2","doi-asserted-by":"publisher","DOI":"10.1137\/09076756X"},{"key":"e_1_2_12_34_2","doi-asserted-by":"publisher","DOI":"10.1142\/S021989161750014X"},{"key":"e_1_2_12_35_2","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/8841718"},{"key":"e_1_2_12_36_2","first-page":"267","article-title":"Thin-Plate Spline Radial Basis Function Scheme for Advection-Diffusion Problems","volume":"2","author":"Boztosun I.","year":"2002","journal-title":"Electronic Journal of Boundary Elements"},{"key":"e_1_2_12_37_2","doi-asserted-by":"publisher","DOI":"10.3390\/fractalfract8060313"},{"key":"e_1_2_12_38_2","doi-asserted-by":"publisher","DOI":"10.2514\/3.5741"},{"key":"e_1_2_12_39_2","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/hxad025"},{"key":"e_1_2_12_40_2","doi-asserted-by":"publisher","DOI":"10.1155\/2014\/560567"},{"key":"e_1_2_12_41_2","doi-asserted-by":"publisher","DOI":"10.4236\/jamp.2024.124068"},{"key":"e_1_2_12_42_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-021-06565-9"},{"key":"e_1_2_12_43_2","first-page":"124","article-title":"Crank-Nicholson Method for Uncertain Heat Equations: Stability and Convergence Analysis","volume":"367","author":"Chen S.","year":"2020","journal-title":"Applied Mathematics and Computation"},{"key":"e_1_2_12_44_2","doi-asserted-by":"publisher","DOI":"10.11648\/j.mcs.20200504.12"},{"key":"e_1_2_12_45_2","doi-asserted-by":"publisher","DOI":"10.1002\/mop.26349"},{"key":"e_1_2_12_46_2","doi-asserted-by":"publisher","DOI":"10.3390\/computation9080092"},{"key":"e_1_2_12_47_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2014.02.063"},{"key":"e_1_2_12_48_2","first-page":"167","article-title":"High-Order Exponential Time Differencing Methods for Solving One-Dimensional Burgers\u2019 Equation","volume":"43","author":"Deng D.","year":"2016","journal-title":"IAENG International Journal of Computer Science"},{"key":"e_1_2_12_49_2","volume-title":"An Exponential Time Differencing Scheme With a Real Distinct Poles Rational Function for Advection-Diffusion Reaction Equations","author":"Asante-Asamani E. O.","year":"2016"},{"key":"e_1_2_12_50_2","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.OA-2021-0074"},{"key":"e_1_2_12_51_2","unstructured":"HiguerasI.andRold\u00e1nT. A New Insight on Positivity and Contractivity of the Crank-Nicolson Scheme for the Heat Equation 2023 https:\/\/arxiv.org\/abs\/2301.01066."},{"key":"e_1_2_12_52_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-017-0390-2"},{"key":"e_1_2_12_53_2","doi-asserted-by":"publisher","DOI":"10.1002\/mma.5871"},{"key":"e_1_2_12_54_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-020-09710-2"},{"key":"e_1_2_12_55_2","doi-asserted-by":"publisher","DOI":"10.1109\/ICVR57957.2023.10169570"},{"key":"e_1_2_12_56_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40995-021-01141-0"},{"key":"e_1_2_12_57_2","doi-asserted-by":"publisher","DOI":"10.4314\/ejst.v9i1.5"},{"key":"e_1_2_12_58_2","doi-asserted-by":"publisher","DOI":"10.2478\/s13540-013-0039-2"},{"key":"e_1_2_12_59_2","first-page":"25","article-title":"B-spline Collocation Approach for Solution of Klein- Gordon Equation","volume":"3","author":"Rashidinia J.","year":"2013","journal-title":"International Journal of Mathematical Modeling and Computations"},{"key":"e_1_2_12_60_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2016.01.049"},{"key":"e_1_2_12_61_2","unstructured":"NeusserK. Difference Equations for Economists 2016 Unpublished Manuscript."},{"key":"e_1_2_12_62_2","doi-asserted-by":"publisher","DOI":"10.1080\/1023619031000078315"},{"key":"e_1_2_12_63_2","volume-title":"Using Eigenvalues and Eigenvectors to Find Stability and Solve ODEs","author":"Katzman D.","year":"2024"},{"key":"e_1_2_12_64_2","unstructured":"KellettC. M. Classical Converse Theorems in Lyapunov\u2032s Second Method 2015 https:\/\/arxiv.org\/abs\/1502.04809."},{"key":"e_1_2_12_65_2","first-page":"423","article-title":"Sensitivity of Hurwitz Stability of Linear Differential Equation Systems With Periodic Coefficients","volume":"8","author":"Keten A.","year":"2020","journal-title":"Konuralp Journal of Mathematics"},{"key":"e_1_2_12_66_2","first-page":"188","volume-title":"Proceedings of the II International Scientific and Practical Conference","author":"Tsyganov A. V.","year":"2018"},{"key":"e_1_2_12_67_2","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2017.1408084"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/5661721","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/jama\/5661721","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/5661721","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T02:44:44Z","timestamp":1776393884000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/jama\/5661721"}},"subtitle":[],"editor":[{"given":"Manzoor","family":"Hussain","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":67,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1155\/jama\/5661721"],"URL":"https:\/\/doi.org\/10.1155\/jama\/5661721","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2024-09-12","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-04-05","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-05-15","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"5661721"}}