{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T07:06:25Z","timestamp":1782975985195,"version":"3.54.5"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T00:00:00Z","timestamp":1776384000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T00:00:00Z","timestamp":1776384000000},"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":[[2026,6]]},"DOI":"10.1007\/s00500-025-10999-w","type":"journal-article","created":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T10:35:05Z","timestamp":1776422105000},"page":"3799-3827","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["F-expansion method analysis of Navier-Bernoulli hypothesis and Love\u2019s hypothesis driven nonlinear dispersive waves in Murnaghan\u2019s cylindrical elastic rod"],"prefix":"10.1007","volume":"30","author":[{"given":"Rathinavel","family":"Silambarasan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adem","family":"Kilicman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zakia","family":"Hammouch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,17]]},"reference":[{"issue":"12","key":"10999_CR1","doi-asserted-by":"publisher","first-page":"4839","DOI":"10.1007\/s00542-018-3895-1","volume":"24","author":"MY Abou-zeid","year":"2018","unstructured":"Abou-zeid MY (2018) Homotopy perturbation method for couple stresses effect on MHD peristaltic flow of a non-Newtonian nanofluid. Microsyst Technol 24(12):4839\u20134846. https:\/\/doi.org\/10.1007\/s00542-018-3895-1","journal-title":"Microsyst Technol"},{"issue":"2","key":"10999_CR2","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/j.cnsns.2006.04.002","volume":"13","author":"M Amkadni","year":"2008","unstructured":"Amkadni M, Azzouzi A, Hammouch Z (2008) On the exact solutions of laminar MHD flow over a stretching flat plate. Commun Nonlinear Sci Numer Simul 13(2):359\u2013368","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"10999_CR3","doi-asserted-by":"publisher","unstructured":"Arshed S (2019) Sub-pico second chirped optical pulses with Triki-Biswas equation by $$\\exp \\left(\\Phi \\left(\\xi \\right)\\right)-$$ expansion method and the first integral method. Optik 179:518\u2013525. https:\/\/doi.org\/10.1016\/j.ijleo.2018.10.220","DOI":"10.1016\/j.ijleo.2018.10.220"},{"issue":"4","key":"10999_CR4","doi-asserted-by":"publisher","first-page":"720","DOI":"10.1080\/17455030.2015.1080392","volume":"25","author":"HM Baskonus","year":"2015","unstructured":"Baskonus HM, Bulut H (2015) On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method. Waves in Random and Complex Media 25(4):720\u2013728. https:\/\/doi.org\/10.1080\/17455030.2015.1080392","journal-title":"Waves Random Complex Media"},{"issue":"1","key":"10999_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jaubas.2013.12.002","volume":"17","author":"A Bekir","year":"2018","unstructured":"Bekir A, Ta\u015fcan F (2018) Exact solutions of the zoomeron and Klein-Gordon-Zakharov equations. J Assoc Arab Univ Basic Appl Sci 17(1):1\u20135. https:\/\/doi.org\/10.1016\/j.jaubas.2013.12.002","journal-title":"J Assoc Arab Univ Basic Appl Sci"},{"key":"10999_CR6","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1016\/j.ijleo.2017.11.054","volume":"156","author":"A Biswas","year":"2018","unstructured":"Biswas A, Mirzazadeh M, Triki H, Zhou Q, Ullah MZ, Moshokoa SP, Belic M (2018) Perturbed resonant 1-soliton solution with anti-cubic nonlinearity by Riccati-Bernoulli sub-ODE method. Optik 156:346\u2013350. https:\/\/doi.org\/10.1016\/j.ijleo.2017.11.054","journal-title":"Optik"},{"key":"10999_CR7","doi-asserted-by":"publisher","first-page":"930","DOI":"10.1016\/j.ijleo.2019.01.041","volume":"182","author":"A Biswas","year":"2019","unstructured":"Biswas A, Ekici M, Sonmezoglu A, Belic MR (2019) Highly dispersive optical solitons with quadratic-cubic law by F-expansion. Optik 182:930\u2013943. https:\/\/doi.org\/10.1016\/j.ijleo.2019.01.041","journal-title":"Optik"},{"issue":"8","key":"10999_CR8","doi-asserted-by":"publisher","first-page":"2978","DOI":"10.1016\/j.camwa.2018.01.025","volume":"75","author":"C Chen","year":"2018","unstructured":"Chen C, Jiang Y-L (2018) Simplest equation method for some time-fractional partial differential equations with conformable derivative. Comput Math Appl 75(8):2978\u20132988. https:\/\/doi.org\/10.1016\/j.camwa.2018.01.025","journal-title":"Comput Math Appl"},{"key":"10999_CR9","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1177\/1081286503035199","volume":"9","author":"H-H Dai","year":"2014","unstructured":"Dai H-H, Fan X (2014) Asmptoticaliy approximate model equations for weakly nonlinear long Waves in compressible elastic rods and their comparisons with other simplified model equations. Math Mech Solids 9:61\u201379. https:\/\/doi.org\/10.1177\/1081286503035199","journal-title":"Math Mech Solids"},{"issue":"2","key":"10999_CR10","doi-asserted-by":"publisher","first-page":"9","DOI":"10.3390\/mca21020011","volume":"21","author":"ST Demiray","year":"2016","unstructured":"Demiray ST, Bulut H (2016) New exact solutions of the system of equations for the ion sound and Langmuir waves by ETEM. Math Comput Appl 21(2):9. https:\/\/doi.org\/10.3390\/mca21020011","journal-title":"Math Comput Appl"},{"issue":"1","key":"10999_CR11","first-page":"1","volume":"44","author":"ST Demiray","year":"2017","unstructured":"Demiray ST, Bulut H (2017) New exact solutions for generalized Gardner equation. Kuwait J Sci 44(1):1\u20138","journal-title":"Kuwait J Sci"},{"issue":"20","key":"10999_CR12","doi-asserted-by":"publisher","first-page":"9621","DOI":"10.1016\/j.ijleo.2016.07.045","volume":"127","author":"W Djoudi","year":"2016","unstructured":"Djoudi W, Zerarka A (2016) Exact structures for the KdV-mKdV equation with variable coefficients via the functional variable method. Optik 127(20):9621\u20139626. https:\/\/doi.org\/10.1016\/j.ijleo.2016.07.045","journal-title":"Optik"},{"issue":"2","key":"10999_CR13","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.wavemoti.2011.11.003","volume":"49","author":"A Ebaid","year":"2012","unstructured":"Ebaid A, Aly EH (2012) Exact solutions for the transformed reduced Ostrovsky equation via the $$F-$$ expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions. Wave Motion 49(2):296\u2013308. https:\/\/doi.org\/10.1016\/j.wavemoti.2011.11.003","journal-title":"Wave Motion"},{"issue":"9","key":"10999_CR14","doi-asserted-by":"publisher","first-page":"3655","DOI":"10.1016\/j.cnsns.2010.12.040","volume":"16","author":"A Elsaid","year":"2011","unstructured":"Elsaid A (2011) Homotopy analysis method for solving a class of fractional partial differential equations. Commun Nonlinear Sci Numer Simul 16(9):3655\u20133664. https:\/\/doi.org\/10.1016\/j.cnsns.2010.12.040","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"3","key":"10999_CR15","doi-asserted-by":"publisher","first-page":"451","DOI":"10.2478\/s13531-013-0104-y","volume":"3","author":"M Eslami","year":"2013","unstructured":"Eslami M, Mirzazadeh M (2013) Functional variable method to study nonlinear evolution equations. Central Eur J Eng 3(3):451\u2013458. https:\/\/doi.org\/10.2478\/s13531-013-0104-y","journal-title":"Central Eur J Eng"},{"issue":"11","key":"10999_CR16","doi-asserted-by":"publisher","first-page":"1425","DOI":"10.1007\/s12648-018-1203-7","volume":"92","author":"A Fatahi-Vajari","year":"2018","unstructured":"Fatahi-Vajari A, Azimzadeh Z (2018) Analysis of nonlinear axial vibration of single-walled carbon nanotubes using Homotopy perturbation method. Indian J Phys 92(11):1425\u20131438. https:\/\/doi.org\/10.1007\/s12648-018-1203-7","journal-title":"Indian J Phys"},{"key":"10999_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2019.123772","volume":"545","author":"W Gao","year":"2020","unstructured":"Gao W, Silambarasan R, Baskonus HM, Vijay Anand R, Rezazadeh H (2020) Periodic waves of the non dissipative double dispersive micro strain wave in the micro structured solids. Phys A 545:123772. https:\/\/doi.org\/10.1016\/j.physa.2019.123772","journal-title":"Phys A"},{"issue":"1","key":"10999_CR18","doi-asserted-by":"publisher","first-page":"79","DOI":"10.2478\/ijmce-2023-0006","volume":"1","author":"B Gasmi","year":"2023","unstructured":"Gasmi B, Ciancio A, Moussa A, Alhakim L, Mati Y (2023) New analytical solutions and modulation instability analysis for the nonlinear (1+1)-dimensional Phi-four model. Int J Math Comput Eng 1(1):79\u201390","journal-title":"Int J Math Comput Eng"},{"key":"10999_CR19","doi-asserted-by":"publisher","DOI":"10.1080\/00207160.2018.1487555","author":"KA Gepreel","year":"2018","unstructured":"Gepreel KA, Nofal TA, Al-Asmari AA (2018) Abundant traveling wave solutions for nonlinear kawahara partial differential equation using extended trial equation method. Int J Comput Math. doi: 10.1080\/00207160.2018.1487555.","journal-title":"Int J Comput Math"},{"issue":"4","key":"10999_CR20","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1515\/zna-2014-0345","volume":"70","author":"KA Gepreel","year":"2015","unstructured":"Gepreel KA, Nofal TA (2015) Extended trial equation method for nonlinear partial differential equations. Zeitschrift f\u00fcr Naturforschung A 70(4):269\u2013279. https:\/\/doi.org\/10.1515\/zna-2014-0345","journal-title":"Zeitschrift f\u00fcr Naturforschung A"},{"key":"10999_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/s11082-017-1311-1","volume":"50","author":"O Guner","year":"2018","unstructured":"Guner O (2018) New exact solutions to the space-time fractional nonlinear wave equation obtained by the ansatz and functional variable methods. Opt Quant Electron 50:38. https:\/\/doi.org\/10.1007\/s11082-017-1311-1","journal-title":"Opt Quant Electron"},{"issue":"18","key":"10999_CR22","first-page":"879","volume":"6","author":"Z Hammouch","year":"2012","unstructured":"Hammouch Z, Mekkaoui T (2012) A laplace-variational iteration method for solving the homogeneous Smoluchowski coagulation equation. Appl Math Sci 6(18):879\u2013886","journal-title":"Appl Math Sci"},{"key":"10999_CR23","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1016\/j.cam.2018.05.011","volume":"343","author":"B Hazarika","year":"2018","unstructured":"Hazarika B, Srivastava HM, Arab R, Rabbani M (2018) Existence of solution for infinite system of nonlinear integral equations via measure of noncompactness and homotopy perturbation method to solve it. J Comput Appl Math 343:341\u2013352. https:\/\/doi.org\/10.1016\/j.cam.2018.05.011","journal-title":"J Comput Appl Math"},{"key":"10999_CR24","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1016\/j.ijleo.2016.10.136","volume":"130","author":"K Hosseini","year":"2017","unstructured":"Hosseini K, Mayeli P, Ansari R (2017) Modified Kudryashov method for solving the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities. Optik 130:737\u2013742. https:\/\/doi.org\/10.1016\/j.ijleo.2016.10.136","journal-title":"Optik"},{"key":"10999_CR25","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/j.ijleo.2016.12.032","volume":"132","author":"K Hosseini","year":"2017","unstructured":"Hosseini K, Bekir A, Ansari R (2017) New exact solutions of the conformable time-fractional Cahn-Allen and Cahn-Hilliard equations using the modified Kudryashov method. Optik 132:203\u2013209. https:\/\/doi.org\/10.1016\/j.ijleo.2016.12.032","journal-title":"Optik"},{"issue":"4","key":"10999_CR26","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1134\/S156035472004005X","volume":"25","author":"K Hosseini","year":"2020","unstructured":"Hosseini K, Samavat M, Mirzazadeh M, Ma WX, Hammouch Z (2020) A new-dimensional Hirota bilinear equation: its B\u00e4cklund transformation and rational-type solutions. Regular and Chaotic Dynamics 25(4):383\u2013391","journal-title":"Regular Chaotic Dyn"},{"issue":"4","key":"10999_CR27","doi-asserted-by":"publisher","DOI":"10.1088\/1402-4896\/ab5055","volume":"95","author":"A Houwe","year":"2020","unstructured":"Houwe A, Sabi\u2019u J, Hammouch Z, Doka SY (2020) Solitary pulses of a conformable nonlinear differential equation governing wave propagation in low-pass electrical transmission line. Phys Scr 95(4):045203","journal-title":"Phys Scr"},{"issue":"55","key":"10999_CR28","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/s11082-017-1307-x","volume":"50","author":"M Ilie","year":"2018","unstructured":"Ilie M, Biazar J, Ayati Z (2018) The first integral method for solving some conformable fractional differential equations. Optical and Quantum Electron 50(55):11. https:\/\/doi.org\/10.1007\/s11082-017-1307-x","journal-title":"Optic Quantum Electron"},{"key":"10999_CR29","doi-asserted-by":"publisher","first-page":"374","DOI":"10.1016\/j.ijleo.2017.02.058","volume":"136","author":"M Inc","year":"2017","unstructured":"Inc M, Inan IE, Ugurlu Y (2017) New applications of the functional variable method. Optik 136:374\u2013381. https:\/\/doi.org\/10.1016\/j.ijleo.2017.02.058","journal-title":"Optik"},{"issue":"2","key":"10999_CR30","doi-asserted-by":"publisher","first-page":"837","DOI":"10.1007\/s11071-019-04825-w","volume":"96","author":"B Kaur","year":"2019","unstructured":"Kaur B, Gupta RK (2019) Dispersion analysis and improved F-expansion method for space-time fractional differential equations. Nonlinear Dyn 96(2):837\u2013852. https:\/\/doi.org\/10.1007\/s11071-019-04825-w","journal-title":"Nonlinear Dyn"},{"issue":"10","key":"10999_CR31","doi-asserted-by":"publisher","first-page":"527","DOI":"10.3390\/sym10100527","volume":"10","author":"A Kilicman","year":"2018","unstructured":"Kilicman A, Silambarasan R (2018) Modified Kudryashov method to solve generalized Kuramoto-Sivashinsky equation. Symmetry 10(10):527. https:\/\/doi.org\/10.3390\/sym10100527","journal-title":"Symmetry"},{"key":"10999_CR32","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/j.ijleo.2017.09.066","volume":"149","author":"D Kumar","year":"2017","unstructured":"Kumar D, Hosseini K, Samadani F (2017) The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitz\u00e9ica type equations in nonlinear optics. Optik - Int J Light and Electron Optics 149:439\u2013446. https:\/\/doi.org\/10.1016\/j.ijleo.2017.09.066","journal-title":"Optik Int J Light and Electron Optics"},{"issue":"1","key":"10999_CR33","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.cjph.2017.11.020","volume":"56","author":"D Kumar","year":"2018","unstructured":"Kumar D, Seadawy AR, Joardar AK (2018) Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chinese J Physi 56(1):75\u201385. https:\/\/doi.org\/10.1016\/j.cjph.2017.11.020","journal-title":"Chinese J Phys"},{"issue":"3","key":"10999_CR34","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1016\/j.amc.2012.07.021","volume":"219","author":"W-W Li","year":"2012","unstructured":"Li W-W, Tian Y, Zhang Z (2012) $$F-$$ expansion method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations. Appl Math Comput 219(3):1135\u20131143. https:\/\/doi.org\/10.1016\/j.amc.2012.07.021","journal-title":"Appl Math Comput"},{"issue":"1","key":"10999_CR35","doi-asserted-by":"publisher","first-page":"11","DOI":"10.2478\/ijmce-2023-0002","volume":"1","author":"AA Mahmud","year":"2023","unstructured":"Mahmud AA, Tanriverdi T, Muhamad KA (2023) Exact traveling wave solutions for (2+1)-dimensional Konopelchenko-Dubrovsky equation by using the hyperbolic trigonometric functions methods. Int J Math Comput Eng 1(1):11\u201324","journal-title":"Int J Math Comput Eng"},{"issue":"4","key":"10999_CR36","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1007\/s12043-014-0912-5","volume":"85","author":"M Matinfar","year":"2015","unstructured":"Matinfar M, Eslami M, Kordy M (2015) The functional variable method for solving the fractional Korteweg-de Vries equations and the coupled Korteweg-de Vries equations. Pramana 85(4):583\u2013592. https:\/\/doi.org\/10.1007\/s12043-014-0912-5","journal-title":"Pramana"},{"issue":"2","key":"10999_CR37","first-page":"251","volume":"39","author":"T Mekkaoui","year":"2012","unstructured":"Mekkaoui T, Hammouch Z (2012) Approximate analytical solutions to the Bagley-Torvik equation by the fractional iteration method. Annals of the University of Craiova-Math Comput Sci Series 39(2):251\u2013256","journal-title":"Ann Univ Craiova-Math Comput Sci Ser"},{"issue":"2","key":"10999_CR38","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1007\/s00009-013-0309-6","volume":"11","author":"IE Mhlanga","year":"2014","unstructured":"Mhlanga IE, Khalique CM (2014) Solutions of two nonlinear evolution equations using Lie symmetry and simplest equation methods. Mediterr J Math 11(2):487\u2013496. https:\/\/doi.org\/10.1007\/s00009-013-0309-6","journal-title":"Mediterr J Math"},{"key":"10999_CR39","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/j.ijleo.2017.07.011","volume":"145","author":"M Mirzazadeh","year":"2017","unstructured":"Mirzazadeh M, Alqahtani RT, Biswas A (2017) Optical soliton perturbation with quadratic-cubic nonlinearity by Riccati-Bernoulli sub-ode method and Kudryashov\u2019s scheme. Optik 145:74\u201378. https:\/\/doi.org\/10.1016\/j.ijleo.2017.07.011","journal-title":"Optik"},{"key":"10999_CR40","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1016\/j.enconman.2014.01.041","volume":"80","author":"S Panda","year":"2014","unstructured":"Panda S, Bhowmik A, Das R, Repaka R, Martha SC (2014) Application of homotopy analysis method and inverse solution of a rectangular wet fin. Energy Convers Manage 80:305\u2013318. https:\/\/doi.org\/10.1016\/j.enconman.2014.01.041","journal-title":"Energy Convers Manage"},{"key":"10999_CR41","doi-asserted-by":"publisher","unstructured":"Pandir Y, Duzgun HH (2018) New exact solutions of the space-time fractional cubic Schr$$\\ddot{o}$$dinger equation using the new type F-expansion method. Waves in Random and Complex Media 29(3):10. https:\/\/doi.org\/10.1080\/17455030.2018.1449987","DOI":"10.1080\/17455030.2018.1449987"},{"issue":"6","key":"10999_CR42","doi-asserted-by":"publisher","first-page":"962","DOI":"10.1108\/MMMS-06-2022-0119","volume":"18","author":"Z Pinar Izgi","year":"2022","unstructured":"Pinar Izgi Z (2022) The solutions of dissipation-dispersive models arising in material science. Multidiscip Modeling in Mater Struct 18(6):962\u2013967","journal-title":"Multidiscip Model Mater Struct"},{"issue":"1","key":"10999_CR43","doi-asserted-by":"publisher","first-page":"45","DOI":"10.36753\/mathenot.799596","volume":"10","author":"Z Pinar","year":"2023","unstructured":"Pinar Z (2023) An analytical approach to an elastic circular rod equation. Math Sci Appl E-Notes 10(1):45\u201349","journal-title":"Math Sci Appl E-Notes"},{"key":"10999_CR44","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/j.cnsns.2018.06.012","volume":"66","author":"P Rana","year":"2019","unstructured":"Rana P, Shukla N, Gupta Y, Pop I (2019) Homotopy analysis method for predicting multiple solutions in the channel flow with stability analysis. Commun Nonlinear Sci Numer Simul 66:183\u2013193. https:\/\/doi.org\/10.1016\/j.cnsns.2018.06.012","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"3","key":"10999_CR45","doi-asserted-by":"publisher","DOI":"10.3390\/fractalfract5030094","volume":"5","author":"S Rashid","year":"2021","unstructured":"Rashid S, Hammouch Z, Aydi H, Ahmad AG, Alsharif AM (2021) Novel computations of the time-fractional Fisher\u2019s model via generalized fractional integral operators by means of the Elzaki transform. Fractal and Fractional 5(3):94","journal-title":"Fractal and Fractional"},{"key":"10999_CR46","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/i2019-12541-2","volume":"134","author":"R Silambarasan","year":"2019","unstructured":"Silambarasan R, Baskonus HM, Bulut H (2019) Jacobi elliptic function solutions of the double dispersive equation in the Murnaghan\u2019s rod. The European Physical Journal Plus 134:125. https:\/\/doi.org\/10.1140\/epjp\/i2019-12541-2","journal-title":"The European Physical Journal Plus"},{"key":"10999_CR47","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2020.11.011","author":"R Silambarasan","year":"2020","unstructured":"Silambarasan R, Baskonus HM, Anand RV, Dinakaran M, Balusamy B, Gao W (2020) Longitudinal strain waves propagating in an infinitely long cylindrical rod composed of generally incompressible materials and its Jacobi elliptic function solutions,. Math Comput Simul In press. https:\/\/doi.org\/10.1016\/j.matcom.2020.11.011","journal-title":"Math Comput Simul In press"},{"issue":"1","key":"10999_CR48","doi-asserted-by":"publisher","first-page":"71","DOI":"10.2478\/ijmce-2024-0006","volume":"2","author":"S Sivasundaram","year":"2024","unstructured":"Sivasundaram S, Kumar A, Singh RK (2024) On the complex properties to the first equation of the Kadomtsev-Petviashvili hierarchy. Int J Math Comput Eng 2(1):71\u201384","journal-title":"Int J Math Comput Eng"},{"issue":"8","key":"10999_CR49","doi-asserted-by":"publisher","DOI":"10.1088\/1674-1056\/19\/8\/080201","volume":"19","author":"F Ta\u015fcan","year":"2010","unstructured":"Ta\u015fcan F, Bekir A (2010) Applications of the first integral method to nonlinear evolution equations. Chin Phys B 19(8):4","journal-title":"Chin Phys B"},{"issue":"1","key":"10999_CR50","doi-asserted-by":"publisher","first-page":"1055","DOI":"10.1016\/j.aej.2020.10.030","volume":"60","author":"MF Uddin","year":"2021","unstructured":"Uddin MF, Hafez MG, Hammouch Z, Rezazadeh H, Baleanu D (2021) Traveling wave with beta derivative spatial-temporal evolution for describing the nonlinear directional couplers with metamaterials via two distinct methods. Alex Eng J 60(1):1055\u20131065","journal-title":"Alex Eng J"},{"issue":"5","key":"10999_CR51","doi-asserted-by":"publisher","first-page":"1047","DOI":"10.1088\/0253-6102\/50\/5\/06","volume":"50","author":"Y Xian-Lin","year":"2008","unstructured":"Xian-Lin Y, Jia-Shi T (2008) Travelling wave solutions for Konopelchenko-Dubrovsky equation using an extended sinh-Gordon equation expansion method. Commun Theor Phys 50(5):1047\u20131051","journal-title":"Commun Theor Phys"},{"issue":"2","key":"10999_CR52","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1016\/S0960-0779(02)00321-1","volume":"16","author":"Z Yan","year":"2003","unstructured":"Yan Z (2003) A sinh-Gordon equation expansion method to construct doubly periodic solutions for nonlinear differential equations. Chaos, Solitons Fractals 16(2):291\u2013297. https:\/\/doi.org\/10.1016\/S0960-0779(02)00321-1","journal-title":"Chaos Solitons Fractals"},{"issue":"3","key":"10999_CR53","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1016\/j.chaos.2004.05.003","volume":"23","author":"Z Yan","year":"2005","unstructured":"Yan Z (2005) A new sine-gordon equation expansion algorithm to investigate some special nonlinear differential equations. Chaos, Solitons Fractals 23(3):767\u2013775. https:\/\/doi.org\/10.1016\/j.chaos.2004.05.003","journal-title":"Chaos Solitons Fractals"},{"key":"10999_CR54","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-015-0452-4","author":"X-F Yang","year":"2015","unstructured":"Yang X-F, Deng Z-C, Wei Y (2015) A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv Differ Equ. doi: 10.1186\/s13662-015-0452-4.","journal-title":"Adv Differ Equ"},{"issue":"15","key":"10999_CR55","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.cam.2018.07.023","volume":"346","author":"H Y\u00e9pez-Mart\u00ednez","year":"2019","unstructured":"Y\u00e9pez-Mart\u00ednez H, G\u00f3mez-Aguilar JF (2019) A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM). J Comput Appl Math 346(15):247\u2013260. https:\/\/doi.org\/10.1016\/j.cam.2018.07.023","journal-title":"J Comput Appl Math"},{"issue":"4","key":"10999_CR56","doi-asserted-by":"publisher","first-page":"2449","DOI":"10.1007\/s11071-016-2837-7","volume":"85","author":"J Yu","year":"2016","unstructured":"Yu J, Wang D-S, Sun Y, Wu S (2016) Modified method of simplest equation for obtaining exact solutions of the Zakharov-Kuznetsov equation, the modified Zakharov-Kuznetsov equation, and their generalized forms. Nonlinear Dyn 85(4):2449\u20132465. https:\/\/doi.org\/10.1007\/s11071-016-2837-7","journal-title":"Nonlinear Dyn"},{"key":"10999_CR57","doi-asserted-by":"publisher","first-page":"626","DOI":"10.1016\/j.ijleo.2019.03.112","volume":"185","author":"EM Zayed","year":"2019","unstructured":"Zayed EM, Shohib RMA (2019) Optical solitons and other solutions to Biswas-Arshed equation using the extended simplest equation method. Optik 185:626\u2013635. https:\/\/doi.org\/10.1016\/j.ijleo.2019.03.112","journal-title":"Optik"},{"issue":"10","key":"10999_CR58","doi-asserted-by":"publisher","first-page":"3390","DOI":"10.1016\/j.camwa.2019.05.007","volume":"78","author":"EM Zayed","year":"2019","unstructured":"Zayed EM, Shohib RMA, Al-Nowehy A-G (2019) On solving the $$(3+1)-$$ dimensional NLEQZK equation and the $$(3+1)-$$ dimensional NLmZK equation using the extended simplest equation method. Comput Math Appl 78(10):3390\u20133407. https:\/\/doi.org\/10.1016\/j.camwa.2019.05.007","journal-title":"Comput Math Appl"},{"issue":"8","key":"10999_CR59","doi-asserted-by":"publisher","first-page":"1382","DOI":"10.1134\/S0965542509080089","volume":"49","author":"HA Zedan","year":"2009","unstructured":"Zedan HA, Tantawy SS (2009) Solution of Davey-Stewartson equations by homotopy perturbation method. Comput Math Math Phys 49(8):1382\u20131388. https:\/\/doi.org\/10.1134\/S0965542509080089","journal-title":"Comput Math Math Phys"},{"issue":"8","key":"10999_CR60","first-page":"249","volume":"10","author":"B Zheng","year":"2011","unstructured":"Zheng B (2011) Traveling wave solutions for some nonlinear evolution equations by the first integral method. Wseas Trans Math 10(8):249\u2013258","journal-title":"Wseas Trans Math"},{"issue":"4","key":"10999_CR61","doi-asserted-by":"publisher","first-page":"1883","DOI":"10.1007\/s11071-016-2613-8","volume":"84","author":"Q Zhou","year":"2016","unstructured":"Zhou Q, Ekici M, Sonmezoglu A, Mirzazadeh M, Eslami M (2016) Optical solitons with Biswas-Milovic equation by extended trial equation method. Nonlinear Dyn 84(4):1883\u20131900. https:\/\/doi.org\/10.1007\/s11071-016-2613-8","journal-title":"Nonlinear Dyn"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-025-10999-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-025-10999-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-025-10999-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T06:32:42Z","timestamp":1782973962000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-025-10999-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,17]]},"references-count":61,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["10999"],"URL":"https:\/\/doi.org\/10.1007\/s00500-025-10999-w","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,17]]},"assertion":[{"value":"23 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 April 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}