{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:53Z","timestamp":1760147033882,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T00:00:00Z","timestamp":1672704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Umm Al-Qura University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.<\/jats:p>","DOI":"10.3390\/sym15010138","type":"journal-article","created":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T01:42:44Z","timestamp":1672796564000},"page":"138","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6102-671X","authenticated-orcid":false,"given":"Waleed Mohamed","family":"Abd-Elhameed","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Seraj Omar","family":"Alkhamisi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Jeddah, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0670-6712","authenticated-orcid":false,"given":"Amr Kamel","family":"Amin","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Adham University College, Umm AL-Qura University, Makkah 21955, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0403-8797","authenticated-orcid":false,"given":"Youssri Hassan","family":"Youssri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,3]]},"reference":[{"key":"ref_1","unstructured":"Thomas, J.W. (2013). Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media."},{"key":"ref_2","unstructured":"Smith, G. (1985). Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hesthaven, J.S., Gottlieb, S., and Gottlieb, D. (2007). Spectral Methods for Time-Dependent Problems, Cambridge University Press.","DOI":"10.1017\/CBO9780511618352"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. (1988). Spectral Methods in Fluid Dynamics, Springer.","DOI":"10.1007\/978-3-642-84108-8"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Abd-Elhameed, W.M. (2021). Novel expressions for the derivatives of sixth-kind Chebyshev polynomials: Spectral solution of the non-linear one-dimensional Burgers\u2019 equation. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5020053"},{"key":"ref_6","first-page":"636","article-title":"Jacobi collocation method for the fractional advection-dispersion equation arising in porous media","volume":"38","author":"Singh","year":"2022","journal-title":"Numer. Methods Part. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.apnum.2020.10.024","article-title":"A spectral collocation method for solving fractional KdV and KdV-Burgers equations with non-singular kernel derivatives","volume":"161","author":"Khader","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1007\/s40314-022-02096-7","article-title":"Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel","volume":"41","author":"Atta","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3376","DOI":"10.1016\/j.physb.2010.05.008","article-title":"A Taylor\u2013Galerkin finite element method for the KdV equation using cubic B-splines","volume":"405","author":"Sari","year":"2010","journal-title":"Phys. B Cond. Matter"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1137\/S0036142902410271","article-title":"A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: Application to the KDV equation","volume":"41","author":"Shen","year":"2003","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_11","first-page":"1148","article-title":"Application of Petrov-Galerkin finite element method to shallow water waves model: Modi ed Korteweg-de Vries equation","volume":"24","author":"Turgut","year":"2017","journal-title":"Sci. Iran. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2126","DOI":"10.1016\/j.camwa.2011.06.060","article-title":"Some new solitonary solutions of the modified Benjamin\u2013Bona\u2013Mahony equation","volume":"62","author":"Noor","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s11082-017-1303-1","article-title":"On the analytical and numerical solutions of the Benjamin\u2013Bona\u2013Mahony equation","volume":"50","author":"Yokus","year":"2018","journal-title":"Opt. Quantum Electron."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5617","DOI":"10.1002\/mma.7135","article-title":"Solitary and periodic wave solutions of the generalized fourth-order Boussinesq equation via He\u2019s variational methods","volume":"44","author":"Wang","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1143\/JPSJ.33.260","article-title":"Oscillatory solitary waves in dispersive media","volume":"33","author":"Kawahara","year":"1972","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.chaos.2008.10.033","article-title":"Crank-Nicolson\u2013differential quadrature algorithms for the Kawahara equation","volume":"42","author":"Korkmaz","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1016\/j.enganabound.2010.07.009","article-title":"RBFs approximation method for Kawahara equation","volume":"35","author":"Haq","year":"2011","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/s10915-007-9158-4","article-title":"A dual-Petrov-Galerkin method for the Kawahara-type equations","volume":"34","author":"Yuan","year":"2008","journal-title":"J. Sci. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/S0096-3003(02)00412-5","article-title":"An explicit and numerical solutions of some fifth-order KdV equation by decomposition method","volume":"144","author":"Kaya","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1016\/j.camwa.2007.06.018","article-title":"Symbolic computation and new families of exact travelling solutions for the Kawahara and modified Kawahara equations","volume":"55","author":"Bekir","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.mcm.2008.06.017","article-title":"Application of variational iteration method and homotopy perturbation method to the modified Kawahara equation","volume":"49","author":"Jin","year":"2009","journal-title":"Math. Comput. Model."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.1016\/j.chaos.2006.10.012","article-title":"Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine\u2013Cosine method","volume":"37","author":"Bekir","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Mason, J.C., and Handscomb, D.C. (2003). Chebyshev Polynomials, CRC.","DOI":"10.1201\/9781420036114"},{"key":"ref_24","unstructured":"Boyd, J.P. (2000). Chebyshev and Fourier Spectral Methods, Dover. [2nd ed.]."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1080\/16583655.2018.1451063","article-title":"Numerical solution of systems of differential equations using operational matrix method with Chebyshev polynomials","volume":"12","year":"2018","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1377","DOI":"10.1007\/s00366-019-00889-9","article-title":"Chebyshev polynomials for the numerical solution of fractal\u2013fractional model of nonlinear Ginzburg\u2013Landau equation","volume":"37","author":"Heydari","year":"2021","journal-title":"Eng. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s40096-021-00401-9","article-title":"An operational matrix method to solve linear Fredholm\u2013Volterra integro-differential equations with piecewise intervals","volume":"15","year":"2021","journal-title":"Math. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1515\/nleng-2018-0062","article-title":"An efficient numerical algorithm for solving system of Lane\u2013Emden type equations arising in engineering","volume":"8","year":"2019","journal-title":"Nonlinear Eng."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Abdelhakem, M., Alaa-Eldeen, T., Baleanu, D., Alshehri, M.G., and El-Kady, M. (2021). Approximating real-life BVPs via Chebyshev polynomials\u2019 first derivative pseudo-Galerkin method. Fractal Frac., 5.","DOI":"10.3390\/fractalfract5040165"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2250061","DOI":"10.1142\/S0129183122500619","article-title":"Tau and Galerkin operational matrices of derivatives for treating singular and Emden\u2013Fowler third-order-type equations","volume":"33","author":"Ahmed","year":"2022","journal-title":"Int. J. Mod. Phys."},{"key":"ref_31","first-page":"3966135","article-title":"New fractional derivative expression of the shifted third-kind Chebyshev polynomials: Application to a type of nonlinear fractional pantograph differential equations","volume":"2022","author":"Youssri","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Duangpan, A., Boonklurb, R., and Juytai, M. (2021). Numerical solutions for systems of fractional and classical integro-differential equations via finite integration method based on shifted Chebyshev polynomials. Fractal Frac., 5.","DOI":"10.3390\/fractalfract5030103"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3569","DOI":"10.1007\/s00366-020-01018-7","article-title":"Application of shifted Chebyshev polynomial-based Rayleigh\u2013Ritz method and Navier\u2019s technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation","volume":"37","author":"Jena","year":"2021","journal-title":"Eng. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2227","DOI":"10.1002\/mma.2969","article-title":"An operational matrix method for solving Lane\u2013Emden equations arising in astrophysics","volume":"37","year":"2014","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1007\/s40314-021-01610-7","article-title":"A spectral collocation matrix method for solving linear Fredholm integro-differential\u2013difference equations","volume":"40","author":"Demir","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1016\/j.apnum.2011.09.003","article-title":"Markov\u2013Bernstein inequalities for generalized Gegenbauer weight","volume":"61","author":"Draux","year":"2011","journal-title":"Appl. Numer. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1016\/S0196-8858(02)00017-9","article-title":"An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials","volume":"29","author":"Xu","year":"2002","journal-title":"Adv. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Abd-Elhameed, W.M., and Alkhamisi, S.O. (2021). New results of the fifth-kind orthogonal Chebyshev polynomials. Symmetry, 13.","DOI":"10.3390\/sym13122407"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., and Youssri, Y.H. (2022). Modal shifted fifth-kind Chebyshev tau integral approach for solving heat conduction equation. Fractal Frac., 6.","DOI":"10.3390\/fractalfract6110619"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2897","DOI":"10.1007\/s40314-017-0488-z","article-title":"Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations","volume":"37","author":"Youssri","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Masjed-Jamei, M. (2006). Some New Classes of Orthogonal Polynomials and Special Functions: A Symmetric Generalization of Sturm-Liouville Problems and Its Consequences. [Ph.D. Thesis, Department of Mathematics, University of Kassel].","DOI":"10.1080\/10652460701510949"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Abd-Elhameed, W.M., and Youssri, Y.H. (2021). New formulas of the high-order derivatives of fifth-kind Chebyshev polynomials: Spectral solution of the convection\u2013diffusion equation. Numerical Methods for Partial Differential Equations, Wiley.","DOI":"10.1002\/num.22756"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Koepf, W. (2014). Hypergeometric Summation, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4471-6464-7"},{"key":"ref_44","first-page":"188","article-title":"B-spline collocation method for numerical solution of nonlinear Kawahara and modified Kawahara equations","volume":"7","author":"Bagherzadeh","year":"2017","journal-title":"TWMS J. App. Eng. Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1016\/j.asej.2017.03.004","article-title":"Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves","volume":"9","author":"Goswami","year":"2018","journal-title":"Ain Shams Eng. J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/138\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:57:18Z","timestamp":1760119038000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/138"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,3]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010138"],"URL":"https:\/\/doi.org\/10.3390\/sym15010138","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,3]]}}}