{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:34:52Z","timestamp":1760243692333,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T00:00:00Z","timestamp":1663891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper introduces a direct method derived from the global radial basis function (RBF) interpolation over arbitrary collocation nodes occurring in variational problems involving functionals that depend on functions of a number of independent variables. This technique parameterizes solutions with an arbitrary RBF and transforms the two-dimensional variational problem (2DVP) into a constrained optimization problem via arbitrary collocation nodes. The advantage of this method lies in its flexibility in selecting between different RBFs for the interpolation and parameterizing a wide range of arbitrary nodal points. Arbitrary collocation points for the center of the RBFs are applied in order to reduce the constrained variation problem into one of a constrained optimization. The Lagrange multiplier technique is used to transform the optimization problem into an algebraic equation system. Three numerical examples indicate the high efficiency and accuracy of the proposed technique.<\/jats:p>","DOI":"10.3390\/e24101345","type":"journal-article","created":{"date-parts":[[2022,9,25]],"date-time":"2022-09-25T23:13:27Z","timestamp":1664147607000},"page":"1345","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2047-5081","authenticated-orcid":false,"given":"Ahmad","family":"Golbabai","sequence":"first","affiliation":[{"name":"School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2461-3891","authenticated-orcid":false,"given":"Nima","family":"Safaei","sequence":"additional","affiliation":[{"name":"School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1242-4989","authenticated-orcid":false,"given":"Mahboubeh","family":"Molavi-Arabshahi","sequence":"additional","affiliation":[{"name":"School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,23]]},"reference":[{"key":"ref_1","unstructured":"Schechter, R.S. (1967). The Variational Method in Engineering, McGraw-Hill."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/0016-0032(75)90199-4","article-title":"A Walsh series direct method for solving variational problems","volume":"300","author":"Chen","year":"1975","journal-title":"J. Frankl. Inst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/S0378-4754(00)00170-1","article-title":"Legendre wavelets direct method for variational problems","volume":"53","author":"Razzaghi","year":"2000","journal-title":"Math. Comput. Simul."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/BF00934535","article-title":"Shifted Legendre direct method for variational problems","volume":"39","author":"Chang","year":"1983","journal-title":"J. Optim. Theory Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/BF00934611","article-title":"Laguerre series direct method for variational problems","volume":"39","author":"Hwang","year":"1983","journal-title":"J. Optim. Theory Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1080\/00207728508926718","article-title":"Shifted Chebyshev direct method for solving variational problems","volume":"16","author":"Horng","year":"1985","journal-title":"Int. J. Syst. Sci."},{"key":"ref_7","first-page":"140","article-title":"An analytic study on the Euler-Lagrange equation arising in calculus of variations","volume":"2","author":"Saadatmandi","year":"2014","journal-title":"Comput. Methods Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1080\/00207160802283047","article-title":"The use of He\u2019s variational iteration method for solving variational problems","volume":"87","author":"Yousefi","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_9","first-page":"221","article-title":"An investigation of radial basis function approximation methods with application in dynamic investment model","volume":"39","author":"Golbabai","year":"2015","journal-title":"Iran. J. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1029\/JB076i008p01905","article-title":"Multiquadric equations of topography and other irregular surfaces","volume":"76","author":"Hardy","year":"1971","journal-title":"J. Geophys. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/0898-1221(90)90270-T","article-title":"Multiquadrics\u2014A scattered data approximation scheme with applications to computational fluid-dynamics\u2014I surface approximations and partial derivative estimates","volume":"19","author":"Kansa","year":"1990","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0898-1221(90)90271-K","article-title":"Multiquadrics\u2014A scattered data approximation scheme with applications to computational fluid-dynamics\u2014II solutions to parabolic, hyperbolic and elliptic partial differential equations","volume":"19","author":"Kansa","year":"1990","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/i2019-12748-1","article-title":"Solitary wave solution of the nonlinear KdV-Benjamin-Bona-Mahony-Burgers model via two meshless methods","volume":"134","author":"Nikan","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_14","first-page":"22","article-title":"Collocation methods based on radial basis functions for the coupled Klein\u2013Gordon\u2013Schr\u00f6dinger equations","volume":"39","author":"Golbabai","year":"2012","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref_15","first-page":"3685","article-title":"Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves","volume":"14","author":"Nikan","year":"2021","journal-title":"Discret. Contin. Dyn. Syst. S"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2298\/TSCI20S1049C","article-title":"Numerical computation of the time non-linear fractional generalized equal width model arising in shallow water channel","volume":"24","author":"Can","year":"2020","journal-title":"Therm. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Avazzadeh, Z., Nikan, O., and Machado, J.A.T. (2020). Solitary wave solutions of the generalized Rosenau-KdV-RLW equation. Mathematics, 8.","DOI":"10.3390\/math8091601"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.jare.2021.03.002","article-title":"A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer","volume":"32","author":"Nikan","year":"2021","journal-title":"J. Adv. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1016\/j.enganabound.2021.07.001","article-title":"An efficient local meshless method for the equal width equation in fluid mechanics","volume":"131","author":"Rasoulizadeh","year":"2021","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"101243","DOI":"10.1016\/j.jksus.2020.101243","article-title":"An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model","volume":"33","author":"Nikan","year":"2021","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"101394","DOI":"10.1016\/j.jocs.2021.101394","article-title":"Numerical study of the nonlinear anomalous reaction-subdiffusion process arising in the electroanalytical chemistry","volume":"53","author":"Nikan","year":"2021","journal-title":"J. Comput. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"104048","DOI":"10.1016\/j.rinp.2021.104048","article-title":"An improved localized radial basis-pseudospectral method for solving fractional reaction\u2013subdiffusion problem","volume":"23","author":"Nikan","year":"2021","journal-title":"Results Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s40096-021-00375-8","article-title":"The impact of LRBF-FD on the solutions of the nonlinear regularized long wave equation","volume":"15","author":"Rasoulizadeh","year":"2021","journal-title":"Math. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105755","DOI":"10.1016\/j.cnsns.2021.105755","article-title":"Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport","volume":"99","author":"Nikan","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"111220","DOI":"10.1016\/j.chaos.2021.111220","article-title":"Numerical simulation of a degenerate parabolic problem occurring in the spatial diffusion of biological population","volume":"151","author":"Nikan","year":"2021","journal-title":"Chaos Solit. Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.enganabound.2021.05.019","article-title":"An efficient localized meshless technique for approximating nonlinear sinh-Gordon equation arising in surface theory","volume":"130","author":"Nikan","year":"2021","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1007\/s11071-021-06822-4","article-title":"Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory","volume":"106","author":"Nikan","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_28","first-page":"126063","article-title":"A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics","volume":"401","author":"Nikan","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.enganabound.2022.05.026","article-title":"Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaces","volume":"143","author":"Nikan","year":"2022","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"105892","DOI":"10.1016\/j.icheatmasstransfer.2022.105892","article-title":"Numerical treatment of microscale heat transfer processes arising in thin films of metals","volume":"132","author":"Nikan","year":"2022","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"113695","DOI":"10.1016\/j.cam.2021.113695","article-title":"Coupling of the Crank\u2013Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow","volume":"398","author":"Nikan","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/j.matcom.2022.04.006","article-title":"A locally stabilized radial basis function partition of unity technique for the sine\u2013Gordon system in nonlinear optics","volume":"199","author":"Nikan","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Nikan, O., Avazzadeh, Z., Machado, J., and Rasoulizadeh, M. (2022). An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals. Eng. Comput., 1\u201318.","DOI":"10.1007\/s00366-022-01630-9"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.apm.2021.07.025","article-title":"Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model","volume":"100","author":"Nikan","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.apnum.2021.07.008","article-title":"Numerical simulation of fractional evolution model arising in viscoelastic mechanics","volume":"169","author":"Nikan","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Buhmann, M.D. (2003). Radial Basis Functions: Theory and Implementations, Cambridge University Press.","DOI":"10.1017\/CBO9780511543241"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Wendland, H. (2005). Scattered Data Approximation, Cambridge University Press.","DOI":"10.1017\/CBO9780511617539"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Micchelli, C.A. (1984). Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Approximation Theory and Spline Functions, Springer.","DOI":"10.1007\/978-94-009-6466-2_7"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Franke, R. (1979). A Critical Comparison of Some Methods for Interpolation of Scattered Data, Technical Report; Naval Postgraduate School.","DOI":"10.21236\/ADA081688"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.1016\/j.enganabound.2012.04.001","article-title":"A meshfree method based on radial basis functions for the eigenvalues of transient Stokes equations","volume":"36","author":"Golbabai","year":"2012","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1016\/j.enganabound.2009.07.003","article-title":"A random variable shape parameter strategy for radial basis function approximation methods","volume":"33","author":"Sarra","year":"2009","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.robot.2016.10.015","article-title":"An RBF collocation method for solving optimal control problems","volume":"87","author":"Mirinejad","year":"2017","journal-title":"Robot. Auton. Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1394","DOI":"10.1177\/1077546312472919","article-title":"Radial basis functions approach on optimal control problems: A numerical investigation","volume":"20","author":"Rad","year":"2014","journal-title":"J. Vib. Control"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. (2007). Spectral Methods: Fundamentals in Single Domains, Springer Science & Business Media.","DOI":"10.1007\/978-3-540-30728-0"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1345\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:38:22Z","timestamp":1760143102000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1345"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,23]]},"references-count":44,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["e24101345"],"URL":"https:\/\/doi.org\/10.3390\/e24101345","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,9,23]]}}}