{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T02:08:57Z","timestamp":1771034937230,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,30]],"date-time":"2022-07-30T00:00:00Z","timestamp":1659139200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and\/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure\/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on.<\/jats:p>","DOI":"10.3390\/sym14081575","type":"journal-article","created":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T23:49:27Z","timestamp":1659397767000},"page":"1575","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6304-4965","authenticated-orcid":false,"given":"Joshua","family":"Sunday","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences, University of Jos, Jos 930003, Nigeria"}],"role":[{"role":"author","vocab":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2699-1490","authenticated-orcid":false,"given":"Ali","family":"Shokri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, University of Maragheh, Maragheh 83111-55181, Iran"}],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Joshua Amawa","family":"Kwanamu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Adamawa State University, Mubi 650001, Nigeria"}],"role":[{"role":"author","vocab":"crossref"}]},{"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":[{"role":"author","vocab":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.aml.2016.08.012","article-title":"A note on variable step size formulation of a Simpson\u2019s-type second derivative blocks method for solving stiff systems","volume":"64","author":"Ramos","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"ref_2","first-page":"27","article-title":"Uniqueness of solution of ordinary differential equations","volume":"74","author":"Wend","year":"1967","journal-title":"Am. Mon."},{"key":"ref_3","unstructured":"Aiken, R. (1985). Stiff Computation, Oxford University Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2215","DOI":"10.1016\/S1352-2310(97)88636-0","article-title":"Computational accuracy and efficiency of the time-splitting method in solving atmospheric transport\/chemistry equations","volume":"31","author":"Kin","year":"1997","journal-title":"Atmos. Environ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1073\/pnas.38.3.235","article-title":"Integration of stiff equations","volume":"38","author":"Curtiss","year":"1952","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/0377-0427(96)00009-X","article-title":"Stiffness in numerical initial value problems","volume":"72","author":"Spijker","year":"1996","journal-title":"J. Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Hairer, E., and Wanner, G. (1996). Solving Ordinary Differential Equations II: Stiff Differential-Algebraic Problems, Springer. [2nd ed.].","DOI":"10.1007\/978-3-642-05221-7"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"287","DOI":"10.46481\/jnsps.2022.777","article-title":"Implicit four-point hybrid block integrator for the simulations of stiff models","volume":"4","author":"Sunday","year":"2022","journal-title":"J. Nig. Soc. Phys. Sci."},{"key":"ref_9","first-page":"63","article-title":"An explicit trigonometrically fitted ten-step method with phase-lag of order infinity for the numerical solution of radial Schrodinger equation","volume":"14","author":"Shokri","year":"2015","journal-title":"Appl. Comput. Math."},{"key":"ref_10","first-page":"687","article-title":"P-stability, TF and VSDPL technique in Obrechkoff methods for the numerical solution of the Schr\u00f6dinger equation","volume":"42","author":"Shokri","year":"2016","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1186\/s42787-020-00095-3","article-title":"Seventh order hybrid block method for solution of first order stiff systems of initial value problems","volume":"28","author":"Akinfenwa","year":"2020","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_12","first-page":"283","article-title":"Extended block integrator for first-order stiff and oscillatory differential equations","volume":"3","author":"Sunday","year":"2013","journal-title":"Am. J. Comput. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"055016","DOI":"10.1088\/2399-6528\/ac7706","article-title":"Optimized two-step second derivative methods for the solutions of stiff systems","volume":"6","author":"Sunday","year":"2022","journal-title":"J. Phys. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hashim, I., Chowdhury, M.S.H., and Hosen, A. (2015, January 28\u201329). Solving linear and nonlinear stiff system of ordinary differential equations by multistage Adomian decomposition method. Proceedings of the Third International Conference on Advances in Applied Science and Environmental Technology, Bangkok, Thailand.","DOI":"10.15224\/978-1-63248-084-2-46"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Amat, S., Legaz, M.J., and Ruiz-Alvarez, J. (2019). On a Variational method for stiff differential equations arising from chemical kinetics. Mathematics, 7.","DOI":"10.3390\/math7050459"},{"key":"ref_16","first-page":"28","article-title":"The Symmetric P-Stable Hybrid Obrenchkoff Methods for the Numerical Solution of Second Order IVPs","volume":"5","author":"Shokri","year":"2012","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_17","first-page":"243","article-title":"Asymptotic reduction of solution space dimension for dynamical systems","volume":"12","author":"Pankov","year":"2021","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_18","first-page":"226","article-title":"An inverse boundary value problem for the boussineq-love equation with nonlocal integral condition","volume":"11","author":"Iskenderov","year":"2020","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_19","first-page":"119","article-title":"Mathematical and numerical modeling of the coupled dynamic thermoelastic problems for isotropic bodies","volume":"11","author":"Qalandarov","year":"2020","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_20","first-page":"366","article-title":"Periodic solutions for certain non-smooth oscillators with high nonlinearities","volume":"20","author":"Faydaoglu","year":"2021","journal-title":"Appl. Comput. Math."},{"key":"ref_21","first-page":"313","article-title":"On the solutions of fractional differential equations via Geraghty type hybrid contractions","volume":"20","author":"Adiguzel","year":"2021","journal-title":"Appl. Comput. Math."},{"key":"ref_22","first-page":"175","article-title":"Stability estimates for delay parabolic differential and difference equations","volume":"19","author":"Ashyralyev","year":"2020","journal-title":"Appl. Comput. Math."},{"key":"ref_23","unstructured":"Ibrahim, Z.B., Othman, K.I., and Suleiman, M. (2007, January 2\u20134). Variable step block backward differentiation formula for solving first order stiff ordinary differential equations. Proceedings of the World Congress on Engineering, London, UK."},{"key":"ref_24","first-page":"37","article-title":"A modified 3-point Adams block method of the variable step size strategy for solving neural delay differential equations","volume":"3","author":"Yashkun","year":"2019","journal-title":"Sukkur IBA J. Comput. Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"012031","DOI":"10.1088\/1742-6596\/1988\/1\/012031","article-title":"Variable step block backward differentiation formula with independent parameter for solving stiff ordinary differential equations","volume":"1988","author":"Zawawi","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_26","first-page":"49","article-title":"A new formula of variable step 3-point block backward differentiation formula method for solving stiff ordinary differential equations","volume":"12","author":"Abasi","year":"2014","journal-title":"J. Pure Appl. Math. Adv. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"929","DOI":"10.28924\/2291-8639-19-2021-929","article-title":"A computational strategy of variable step, variable order for solving stiff systems of ODEs","volume":"19","author":"Oghonyon","year":"2021","journal-title":"Int. J. Anal. Appl."},{"key":"ref_28","first-page":"565137","article-title":"Solving non-stiff higher order ODEs using variable order step size backward difference directly","volume":"565137","author":"Rasedee","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"375","DOI":"10.2306\/scienceasia1513-1874.2014.40.375","article-title":"Variable step 2-point block backward differentiation formula for index-1 differential algebraic equations","volume":"40","author":"Abasi","year":"2014","journal-title":"Sci. Asia"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1016\/S0898-1221(02)00188-8","article-title":"Variable order Adams codes","volume":"44","author":"Shampine","year":"2002","journal-title":"Comput. Math. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"101376","DOI":"10.1016\/j.jksus.2021.101376","article-title":"Two-point block variable order step size multistep method for solving higher order ordinary differential equations directly","volume":"33","author":"Rasedee","year":"2021","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s11075-010-9413-x","article-title":"A direct variable step block multistep method for solving general third order ordinary differential equations","volume":"57","author":"Mehrkanoon","year":"2011","journal-title":"Numer. Algorithms"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Soomro, H., Zainuddin, N., Daud, H., Sunday, J., Jamaludin, N., Abdullah, A., Apriyanto, M., and Kadir, E.A. (2022). Variable step block hybrid method for stiff chemical kinetics problems. Appl. Sci., 12.","DOI":"10.3390\/app12094484"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"9799627","DOI":"10.1155\/2021\/9799627","article-title":"Variable step size Adams methods for BSDEs","volume":"2021","author":"Han","year":"2021","journal-title":"J. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.17576\/jsm-2019-4808-23","article-title":"Variable order block method for solving second order ordinary differential equations","volume":"48","author":"Ibrahim","year":"2019","journal-title":"Sains Malays."},{"key":"ref_36","first-page":"210","article-title":"Variable step size selection methods for implicit integration schemes for ordinary differential equations","volume":"4","author":"Holsapple","year":"2007","journal-title":"Int. J. Numer. Anal. Model."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"96","DOI":"10.3923\/ajaps.2017.96.101","article-title":"A variable step size block predictor-corrector method for ordinary differential equations","volume":"10","author":"Oghonyon","year":"2017","journal-title":"Asian J. Appl. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Sunday, J., Shokri, A., and Marian, D. (2022). Variable step hybrid block method for the approximation of Kepler problem. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6060343"},{"key":"ref_39","unstructured":"Iserles, A. (1996). A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1137\/0710081","article-title":"Algorithms for changing the step size","volume":"10","author":"Krogh","year":"1973","journal-title":"SIAM J. Num. Anal."},{"key":"ref_41","unstructured":"Bettis, D.G. (1972, January 20). Changing step size in the integration of differential equations using modified divided differences. Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations, Austin, TX, USA."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1090\/S0025-5718-1980-0559191-X","article-title":"Numerical integrators for stiff and highly oscillatory differential equations","volume":"34","author":"Fatunla","year":"1980","journal-title":"Math Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/BF01963532","article-title":"A special stability problem for linear multistep methods","volume":"3","author":"Dahlquist","year":"1963","journal-title":"BIT Numer. Math."},{"key":"ref_44","unstructured":"Lambert, J.D. (1973). Computational Methods in Ordinary Differential Equations, John Wiley & Sons, Inc."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1023\/A:1013858030641","article-title":"A note on the step size selection in Adams multistep methods","volume":"27","author":"Calvo","year":"2001","journal-title":"Numer. Algorithms"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s11075-020-00900-1","article-title":"Local error estimation and step size control in adaptative linear multistep methods","volume":"86","author":"Arevalo","year":"2021","journal-title":"Numer. Algorithms"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3805","DOI":"10.1016\/j.cam.2011.06.032","article-title":"Step size strategies for the numerical integration of systems of differential equations","volume":"236","author":"Kizilkan","year":"2012","journal-title":"J. Comput. Appl. Math."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s11075-005-0413-1","article-title":"Second derivative methods with Runge-Kutta stability","volume":"40","author":"Butcher","year":"2005","journal-title":"Numer. Algorithms"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1093\/imanum\/drl040","article-title":"A family of A-stable Runge-Kutta collocation methods of higher order for initial value problems","volume":"27","author":"Ramos","year":"2007","journal-title":"IMA J. Numer. Anal."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1007\/s13370-015-0389-5","article-title":"Second derivative of high-order accuracy methods for the numerical integration of stiff initial value problems","volume":"27","author":"Yakubu","year":"2016","journal-title":"Afr. Mat."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1575\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:59:45Z","timestamp":1760140785000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1575"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,30]]},"references-count":50,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081575"],"URL":"https:\/\/doi.org\/10.3390\/sym14081575","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,30]]}}}