{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:58:30Z","timestamp":1760057910908,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,2]],"date-time":"2025-03-02T00:00:00Z","timestamp":1740873600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Agencia Estatal de Investigaci\u00f3n (AEI) of Spain","award":["PID2020-113275GB-I00"],"award-info":[{"award-number":["PID2020-113275GB-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We propose a wavelet collocation method for solving the fractional Riccati equation, using the M\u00fcntz\u2013Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton\u2013Raphson\u2019s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance.<\/jats:p>","DOI":"10.3390\/axioms14030185","type":"journal-article","created":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T03:22:44Z","timestamp":1740972164000},"page":"185","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["M\u00fcntz\u2013Legendre Wavelet Collocation Method for Solving Fractional Riccati Equation"],"prefix":"10.3390","volume":"14","author":[{"given":"Fatemeh","family":"Soleyman","sequence":"first","affiliation":[{"name":"K.N. Toosi University of Technology, Tehran P.O. Box 16315-1618, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0872-5017","authenticated-orcid":false,"given":"Iv\u00e1n","family":"Area","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada II, IFCAE, Universidade de Vigo, E.E. Aeron\u00e1utica e do Espazo, Campus As Lagoas-Ourense, 32004 Ourense, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear models of dissipation whose Q is almost frequency independent\u2014II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. Int."},{"key":"ref_2","unstructured":"Tarasov, V.E. (2019). Handbook of Fractional Calculus with Applications, de Gruyter."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","article-title":"A new collection of real-world applications of fractional calculus in science and engineering","volume":"64","author":"Sun","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1515\/fca-2017-0040","article-title":"Life and Science of Alexey Gerasimov, One of the Pioneers of Fractional Calculus in Soviet Union","volume":"20","author":"Novozhenova","year":"2017","journal-title":"FCAA"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3043","DOI":"10.1007\/s11012-017-0652-y","article-title":"On the notion of fractional derivative and applications to the hysteresis phenomena","volume":"52","author":"Caputo","year":"2017","journal-title":"Meccanica"},{"key":"ref_6","first-page":"85","article-title":"Using Riccati Equation to Construct New Solitary Solutions of Nonlinear Evolution Equations","volume":"31","author":"Tariq","year":"2024","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_7","first-page":"255","article-title":"Optical Solitons in Nonlinear Optical Media: A Riccati Equation Approach","volume":"56","author":"Singh","year":"2024","journal-title":"Opt. Quantum Electron."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"101001","DOI":"10.1115\/1.4054884","article-title":"A robust time-varying Riccati-based control for uncertain nonlinear dynamical systems","volume":"144","author":"Azimi","year":"2022","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_9","first-page":"179","article-title":"M\u00fcntz orthogonal polynomials and their numerical evaluation","volume":"Volume 131","author":"Gautschi","year":"1999","journal-title":"Applications and Computation of Orthogonal Polynomials"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1137\/040621533","article-title":"Gaussian type quadrature rules for M\u00fcntz systems","volume":"27","year":"2005","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.cam.2014.10.009","article-title":"Generalized quadrature rules of Gaussian type for numerical evaluation of singular integrals","volume":"278","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.apnum.2021.05.017","article-title":"Fractional-order Boubaker wavelets method for solving fractional Riccati differential equations","volume":"168","author":"Rabiei","year":"2021","journal-title":"Appl. Num. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.cam.2017.09.031","article-title":"A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations","volume":"339","author":"Mohammadi","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"109124","DOI":"10.1016\/j.cpc.2024.109124","article-title":"Quadrature of functions with endpoint singular and generalised polynomial behaviour in computational physics","volume":"299","author":"Lombardi","year":"2024","journal-title":"Comput. Phys. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1130","DOI":"10.1016\/j.physa.2019.04.120","article-title":"Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients","volume":"523","author":"Singh","year":"2019","journal-title":"Physica A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"115265","DOI":"10.1088\/1402-4896\/ad85a7","article-title":"The multi-resolution Haar wavelets collocation procedure for fractional Riccati equations","volume":"99","author":"Ahsan","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s40819-024-01696-7","article-title":"Spectral collocation approach with shifted Chebyshev third-kind series approximation for nonlinear generalized fractional Riccati equation","volume":"10","author":"Atta","year":"2024","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_18","first-page":"3075","article-title":"An efficient computational intelligence approach for solving fractional-order Riccati equations using ANN and SQP","volume":"39","author":"Raja","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"372","DOI":"10.29020\/nybg.ejpam.v17i1.5013","article-title":"Solving Fractional Riccati Differential equation with Caputo-Fabrizio fractional derivative","volume":"17","author":"Abuteen","year":"2024","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_20","first-page":"1027","article-title":"A new numerical technique for solving \u03c8-fractional Riccati differential equations","volume":"13","author":"Ali","year":"2023","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1515\/ijnsns-2018-0146","article-title":"An algorithm for the approximate solution of the fractional Riccati differential equation","volume":"20","author":"Machado","year":"2019","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"A2182","DOI":"10.1137\/19M1264217","article-title":"Order Reduction Methods for Solving Large-Scale Differential Matrix Riccati Equations","volume":"42","author":"Kirsten","year":"2020","journal-title":"Siam J. Sci. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"126401","DOI":"10.1016\/j.amc.2021.126401","article-title":"Galerkin trial spaces and Davison-Maki methods for the numerical solution of differential Riccati equations","volume":"410","author":"Behr","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ray, S.S., and Gupta, A.K. (2018). Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations, Chapman and Hall\/CRC.","DOI":"10.1201\/9781315167183"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s40314-022-01772-y","article-title":"A wavelet-based novel technique for linear and nonlinear fractional Volterra\u2013Fredholm integro-differential equations","volume":"41","author":"Behera","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_26","first-page":"24048","article-title":"Multiwavelet-based operator learning for differential equations","volume":"34","author":"Gupta","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.cam.2018.09.016","article-title":"An efficient algorithm for solving Volterra integro-differential equations based on Alpert\u2019s multi-wavelets Galerkin method","volume":"348","author":"Saray","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1002\/mma.6068","article-title":"Sparse multiscale representation of Galerkin method for solving linear-mixed Volterra-Fredholm integral equations","volume":"43","author":"Saray","year":"2020","journal-title":"Math. Method Appl. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/s10543-020-00832-1","article-title":"Sparse representation based on multiwavelets for Abel\u2019s integral operator","volume":"61","author":"Saray","year":"2021","journal-title":"BIT Numer. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"9915551","DOI":"10.1155\/2021\/9915551","article-title":"A new scheme for solving multiorder fractional differential equations based on M\u00fcntz\u2013Legendre wavelets","volume":"2021","author":"Jebreen","year":"2021","journal-title":"Complexity"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1916","DOI":"10.1002\/oca.2456","article-title":"Numerical solution of a class of 2D fractional optimal control problems using 2D M\u00fcntz-Legendre wavelets","volume":"39","author":"Rahimkhani","year":"2018","journal-title":"Optim. Control Appl. Meth."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1283","DOI":"10.1007\/s11075-017-0363-4","article-title":"M\u00fcntz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations","volume":"77","author":"Rahimkhani","year":"2018","journal-title":"Numer. Algor."},{"key":"ref_33","first-page":"152","article-title":"M\u00fcntz type theorems","volume":"3","author":"Almira","year":"2007","journal-title":"Surv. Approx. Theory"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2357","DOI":"10.1137\/15M1052391","article-title":"M\u00fcntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems","volume":"38","author":"Shen","year":"2016","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Torres-Hernandez, A., Brambila-Paz, F., Iturrar\u00e1n-Viveros, U., and Caballero-Cruz, R. (2021). Fractional Newton\u2013Raphson Method Accelerated with Aitken\u2019s Method. Axioms, 10.","DOI":"10.3390\/axioms10020047"},{"key":"ref_36","unstructured":"Kilbas, A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier B. V."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"8087","DOI":"10.1016\/j.apm.2016.04.026","article-title":"Fractional-order Bernoulli wavelets and their applications","volume":"40","author":"Rahimkhani","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Tverdyi, D., and Parovik, R. (2022). Investigation of finite-difference schemes for the numerical solution of a fractional nonlinear equation. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6010023"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Chang, C.-W., Qureshi, S., Argyros, I.K., Saraz, K.M., and Hincal, E. (2024). A modified fractional Newton\u2019s solver. Axioms, 13.","DOI":"10.3390\/axioms13100689"},{"key":"ref_40","first-page":"141","article-title":"A new approach using fractional Bernoulli wavelets for solving fractional differential equations","volume":"300","author":"Rahimkhani","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_41","first-page":"219","article-title":"Fractional Legendre wavelet method for solving fractional differential equations","volume":"115","author":"Babolian","year":"2017","journal-title":"Appl. Numer. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/185\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:45:53Z","timestamp":1760028353000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/185"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,2]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030185"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030185","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,3,2]]}}}