{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T02:28:05Z","timestamp":1783132085564,"version":"3.54.6"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,21]],"date-time":"2021-08-21T00:00:00Z","timestamp":1629504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations.<\/jats:p>","DOI":"10.3390\/e23081086","type":"journal-article","created":{"date-parts":[[2021,8,22]],"date-time":"2021-08-22T21:47:52Z","timestamp":1629668872000},"page":"1086","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":42,"title":["An Analytical Technique, Based on Natural Transform to Solve Fractional-Order Parabolic Equations"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0634-2370","authenticated-orcid":false,"given":"Ravi P.","family":"Agarwal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 78363, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-7382","authenticated-orcid":false,"given":"Fatemah","family":"Mofarreh","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences College of Sciences, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4306-8489","authenticated-orcid":false,"given":"Rasool","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali khan University, Mardan 23200, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3636-4880","authenticated-orcid":false,"given":"Waewta","family":"Luangboon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1080\/00207168708803608","article-title":"A family of second order methods for variable coefficient fourth order parabolic partial differential equations","volume":"23","author":"Khaliq","year":"1987","journal-title":"Int. J. Comput. Math."},{"key":"ref_2","unstructured":"Gorman, D.J. (1975). Free Vibration Analysisi of Beams and Shafts(Book), Wiley-Interscience. Research supported by the National Research Council of Canada."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0771-050X(77)90019-5","article-title":"High accuracy ADI methods for fourth order parabolic equations with variable coefficients","volume":"3","author":"Andrade","year":"1977","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1145\/320856.320861","article-title":"A stable implicit finite difference approximation to a fourth order parabolic equation","volume":"4","author":"Conte","year":"1957","journal-title":"J. ACM"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1090\/S0002-9939-1956-0078569-2","article-title":"Convergence of finite difference solutions to a solution of the equation of the vibrating rod","volume":"7","author":"Royster","year":"1956","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1093\/comjnl\/8.3.280","article-title":"A stable explicit method for the finite-difference solution of a fourth-order parabolic partial differential equation","volume":"8","author":"Evans","year":"1965","journal-title":"Comput. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1080\/00207169108804004","article-title":"A note on solving the fourth order parabolic equation by the AGE method","volume":"40","author":"Evans","year":"1991","journal-title":"Int. J. Comput. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1080\/00207169508804424","article-title":"On the solution of the fourth order parabolic equation by the decomposition method","volume":"57","author":"Wazwaz","year":"1995","journal-title":"Int. J. Comput. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1002\/num.10012","article-title":"An efficient high-order algorithm for solving systems of reaction-diffusion equations","volume":"18","author":"Liao","year":"2002","journal-title":"Numer. Methods Part. Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1002\/nme.1620100614","article-title":"Higher order difference formulas for a fourth order parabolic partial differential equation","volume":"10","author":"Jain","year":"1976","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_11","first-page":"597","article-title":"Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations","volume":"201","author":"Caglar","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_12","first-page":"219","article-title":"Analytic treatment for variable coefficient fourth-order parabolic partial differential equations","volume":"123","author":"Wazwaz","year":"2001","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3443","DOI":"10.1080\/00207160903085820","article-title":"Sextic spline solution of variable coefficient fourth-order parabolic equations","volume":"87","author":"Rashidinia","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_14","first-page":"153","article-title":"Spline methods for the solution of fourth-order parabolic partial differential equations","volume":"167","author":"Aziz","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1016\/j.camwa.2006.12.049","article-title":"Hes variational iteration method for fourth-order parabolic equations","volume":"54","author":"Biazar","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e630","DOI":"10.1016\/j.na.2008.11.011","article-title":"Modified variational iteration technique for solving singular fourth-order parabolic partial differential equations","volume":"71","author":"Noor","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1515\/zna-2009-7-803","article-title":"The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method","volume":"64","author":"Dehghan","year":"2009","journal-title":"Z. Naturforschung A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2052","DOI":"10.1016\/j.camwa.2019.03.053","article-title":"Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients","volume":"78","author":"Nadeem","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1002\/mma.3984","article-title":"Finding exact solutions of nonlinear PDEs using the natural decomposition method","volume":"40","author":"Rawashdeh","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_20","first-page":"111","article-title":"The natural transform decomposition method for linear and nonlinear partial differential equations","volume":"5","author":"Baskonus","year":"2014","journal-title":"Math. Eng. Sci. Aerosp. (MESA)"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1016\/0022-247X(84)90182-3","article-title":"A new approach to nonlinear partial differential equations","volume":"102","author":"Adomian","year":"1984","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","first-page":"99","article-title":"Theory of natural transform","volume":"3","author":"Belgacem","year":"2012","journal-title":"J. MESA"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Khan, H., Shah, R., Kumam, P., and Arif, M. (2019). Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method. Entropy, 21.","DOI":"10.3390\/e21060597"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Shah, R., Khan, H., Kumam, P., Arif, M., and Baleanu, D. (2019). Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay. Mathematics, 7.","DOI":"10.3390\/math7060532"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Shah, R., Khan, H., Mustafa, S., Kumam, P., and Arif, M. (2019). Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method. Entropy, 21.","DOI":"10.3390\/e21060557"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1186\/s13662-016-0960-x","article-title":"New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM","volume":"1","author":"Rawashdeh","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Eltayeb, H., Abdalla, Y.T., Bachar, I., and Khabir, M.H. (2019). Fractional telegraph equation and its solution by natural transform decomposition method. Symmetry, 11.","DOI":"10.3390\/sym11030334"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3409","DOI":"10.1002\/mma.5533","article-title":"Numerical solution for (2+1)-dimensional time-fractional coupled Burger equations using fractional natural decomposition method","volume":"42","author":"Prakasha","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1063\/1.4765477","article-title":"Advances in the natural transform","volume":"1493","author":"Belgacem","year":"2012","journal-title":"AIP Conf. Proc."},{"key":"ref_30","first-page":"127","article-title":"N-transform properties and applications","volume":"1","author":"Khan","year":"2008","journal-title":"NUST J. Eng. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Sci. Publishing.","DOI":"10.1142\/9789812817747"},{"key":"ref_32","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier."},{"key":"ref_33","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley. [1st ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1086\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:48:31Z","timestamp":1760165311000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1086"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,21]]},"references-count":33,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23081086"],"URL":"https:\/\/doi.org\/10.3390\/e23081086","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,21]]}}}