{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T04:33:16Z","timestamp":1766377996269,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,12,3]],"date-time":"2019-12-03T00:00:00Z","timestamp":1575331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin\u2013Bona\u2013Mahony (BBM) equations. We present a generalised version of the modified tanh\u2013coth method. Using the generalised, modified tanh\u2013coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions.<\/jats:p>","DOI":"10.3390\/axioms8040134","type":"journal-article","created":{"date-parts":[[2019,12,4]],"date-time":"2019-12-04T04:30:35Z","timestamp":1575433835000},"page":"134","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin\u2013Bona\u2013Mahony Equations"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7556-8942","authenticated-orcid":false,"given":"Praveen","family":"Agarwal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9273-9512","authenticated-orcid":false,"given":"Abd-Allah","family":"Hyder","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"},{"name":"Department of Engineering Mathematics and Physics, Faculty of Engineering Al-Azhar University, Cairo P.O. Box 11371, Egypt"}]},{"given":"M.","family":"Zakarya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut P.O. Box 71524, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"AlNemer","sequence":"additional","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9333-5034","authenticated-orcid":false,"given":"Dario","family":"Assante","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Holden, H., \u00d8sendal, B., Ub\u00f8e, J., and Zhang, T. (2010). Stochastic Partial Differential Equations, Springer Science+Business Media, LLC.","DOI":"10.1007\/978-0-387-89488-1"},{"key":"ref_2","first-page":"47","article-title":"Model equations for long waves in nonlinear dispersive systems","volume":"272","author":"Benjamin","year":"1972","journal-title":"Trans. R. Soc. Lond. Ser. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.matcom.2008.04.018","article-title":"A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions","volume":"79","author":"Dehghan","year":"2008","journal-title":"Comput. Math. Simul."},{"key":"ref_4","unstructured":"Hereman, W. (2000). Exact Solutions of Nonlinear Partial Differential Equations. The tanh\/sech Method, Wolfram Research Academic Intern Program Inc."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Meryers, R.A. (2009). Shallow water waves and solitary waves. Encyclopedia of Complexity and Systems Science, Springer.","DOI":"10.1007\/978-0-387-30440-3"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"957","DOI":"10.2298\/FIL1205957K","article-title":"Exact solutions and conservation laws of a coupled integrable dispersionless system","volume":"26","author":"Khalique","year":"2012","journal-title":"Filomat"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1515\/eng-2017-0023","article-title":"Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modified equations of wave propagation","volume":"7","author":"Wazwaz","year":"2017","journal-title":"Open Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7420","DOI":"10.1016\/j.apm.2015.03.019","article-title":"Exact solutions for a Wick-type stochastic reaction Duffing equation","volume":"39","author":"Dai","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.1007\/s11071-015-2089-y","article-title":"Some discussions about variable separation of nonlinear models using Riccati equation expansion method","volume":"81","author":"Kong","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1331","DOI":"10.1007\/s11071-015-2406-5","article-title":"Re-study on localized structures based on variable separation solutions from the modified tanh-function method","volume":"83","author":"Wang","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2642","DOI":"10.1143\/JPSJ.52.2642","article-title":"Stochastic Korteweg-de Vries equation","volume":"52","author":"Wadati","year":"1983","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_12","first-page":"153","article-title":"White noise functional solutions for the Wick-type two dimensional stochastic Zakharov-Kuznetsov equations","volume":"6","author":"Ghany","year":"2012","journal-title":"Int. Rev. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"0805011","DOI":"10.1088\/1674-1056\/22\/8\/080501","article-title":"The fractional coupled KdV equations: Exact solutions and white noise functional approach","volume":"22","author":"Ghany","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_14","first-page":"75","article-title":"Exact solutions for the Wick-type stochastic time-fractional KdV equations","volume":"41","author":"Ghany","year":"2014","journal-title":"Kuwait J. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"0605031","DOI":"10.1088\/1674-1056\/23\/6\/060503","article-title":"Abundant solutions of Wick-type stochastic fractional 2D KdV equations","volume":"23","author":"Ghany","year":"2014","journal-title":"Chin. Phys. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"915","DOI":"10.18576\/amis\/110332","article-title":"Non-gaussian white noise functional solutions of \u03c7-Wick-type stochastic KdV equations","volume":"11","author":"Ghany","year":"2017","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_17","first-page":"191092","article-title":"Exact Solutions of Stochastic Fractional Korteweg de-Vries Equation with Conformable Derivatives","volume":"28","author":"Ghany","year":"2019","journal-title":"Chin. Phys. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.chaos.2004.04.015","article-title":"Exact solutions for generalized stochastic Wick- type KdV-mKdV equations","volume":"23","author":"Chen","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/j.cam.2005.11.009","article-title":"White noise functional solutions of Wick-type stochastic generalized Hirota-Satsuma coupled KdV equations","volume":"157","author":"Chen","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.cam.2006.04.002","article-title":"Periodic-like solutions of variable coefficient and Wick-type stochastic NLS equations","volume":"203","author":"Chen","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"539","DOI":"10.12732\/ijpam.v109i3.5","article-title":"Non-Gaussian Wick calculus based on hypercomplex systems","volume":"109","author":"Hyder","year":"2016","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s42787-019-0006-0","article-title":"The Well-Posedness of Stochastic Kawahara Equation: Fixed Point Argument and Fourier Restriction Method","volume":"27","author":"Hyder","year":"2019","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18642\/jmsaa_7100121781","article-title":"Wick-type stochastic KdV equation based on L\u00e9vy white noise","volume":"45","author":"Hyder","year":"2017","journal-title":"J. Math. Sci. Adv. Appl."},{"key":"ref_24","first-page":"39","article-title":"White noise analysis combined with hypercomplex systems for solving stochastic modified KdV equations with non-Gaussian parameters","volume":"24","author":"Hyder","year":"2018","journal-title":"Pioneer J. Adv. Appl. Math."},{"key":"ref_25","first-page":"423","article-title":"Well-Posedness of Stochastic Modified Kawahara Equation","volume":"2019","author":"Agarwal","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/S1007-5704(02)00109-0","article-title":"Soliton solutions in linear magnetic field and time-dependent laser field","volume":"9","author":"Liu","year":"2004","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_27","first-page":"201","article-title":"Newton-Tau numerical solution of a system of nonlinear Fredholm integral equations of second kind","volume":"5","author":"Ivaz","year":"2006","journal-title":"Appl. Comput. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.1016\/j.cnsns.2005.11.007","article-title":"New solitons and kinks solutions for the Gardner equation","volume":"12","author":"Wazwaz","year":"2007","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_29","first-page":"257","article-title":"An application of Sinc-Galerkin method for solving the Tzou equation","volume":"16","author":"Pourgholi","year":"2017","journal-title":"Appl. Comput. Math."},{"key":"ref_30","first-page":"151","article-title":"A reliable algorithm for solving linear and nonlinear Schr\u00f6dinger equation","volume":"17","author":"Momani","year":"2018","journal-title":"Appl. Comput. Math."},{"key":"ref_31","first-page":"248","article-title":"Solitons and other solutions to long-short wave resonance equation","volume":"14","author":"Mirzazadeh","year":"2015","journal-title":"Appl. Comput. Math."},{"key":"ref_32","first-page":"331","article-title":"Analytical solutions of nonlinear equations with proportional delays","volume":"15","author":"Bhalekar","year":"2016","journal-title":"Appl. Comput. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1119\/1.17120","article-title":"Solitary wave solutions of nonlinear wave equations","volume":"60","author":"Malfeit","year":"1992","journal-title":"Am. J. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/S0375-9601(00)00725-8","article-title":"Extended tanh-function method and its applications to nonlinear equations","volume":"277","author":"Fan","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_35","first-page":"1467","article-title":"The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations","volume":"188","author":"Wazwaz","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_36","first-page":"403","article-title":"Modified extended tanh-function method and its applications to nonlinear equations","volume":"161","author":"Zahran","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/j.physa.2005.07.008","article-title":"The modified extended tanh-function method for solving Burgers-type equations","volume":"361","author":"Soliman","year":"2006","journal-title":"Phys. A"},{"key":"ref_38","first-page":"37","article-title":"White-noise functional solutions for wick-type stochastic time-fractional Benjamin\u2013Bona\u2013Mahony equation","volume":"13","author":"Ghany","year":"2014","journal-title":"Int. J. Diff. Equ. Appl."},{"key":"ref_39","first-page":"1","article-title":"Novel approach for stochastic solutions of wick-type stochastic time-fractional Benjamin\u2013Bona\u2013Mahony equation for modeling long surface gravity waves of small amplitude","volume":"2019","author":"Sahoo","year":"2019","journal-title":"Stoch. Anal. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1080\/14786449508620739","article-title":"On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves","volume":"39","author":"Korteweg","year":"1895","journal-title":"Phil. Mag. Ser."},{"key":"ref_41","first-page":"10","article-title":"High-accuracy algorithms for solution of discrete periodic Riccati equations","volume":"6","author":"Larin","year":"2007","journal-title":"Appl. Comput. Math."},{"key":"ref_42","first-page":"765","article-title":"New multiple soliton-like solutions to the generalized (2+1)-dimensional KP equation","volume":"157","author":"Chen","year":"2004","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/8\/4\/134\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:39:44Z","timestamp":1760189984000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/8\/4\/134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,3]]},"references-count":42,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["axioms8040134"],"URL":"https:\/\/doi.org\/10.3390\/axioms8040134","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2019,12,3]]}}}