{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,3]],"date-time":"2023-09-03T02:30:41Z","timestamp":1693708241992},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2015,2,11]],"date-time":"2015-02-11T00:00:00Z","timestamp":1423612800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2015,10]]},"DOI":"10.1007\/s11075-014-9944-7","type":"journal-article","created":{"date-parts":[[2015,2,9]],"date-time":"2015-02-09T20:38:39Z","timestamp":1423514319000},"page":"227-248","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Efficient low-error analytical-numerical approximations for radial solutions of nonlinear Laplace equations"],"prefix":"10.1007","volume":"70","author":[{"given":"John","family":"Haussermann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert A.","family":"Van Gorder","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,2,11]]},"reference":[{"key":"9944_CR1","unstructured":"Liao, S.J.: On the proposed homotopy analysis techniques for nonlinear problems and its application, Ph.D. dissertation. Shanghai Jiao Tong University (1992)"},{"key":"9944_CR2","volume-title":"Introduction to the Homotopy Analysis Method, Chapman &","author":"SJ Liao","year":"2003","unstructured":"Liao, S.J., Perturbation, B.: Introduction to the Homotopy Analysis Method, Chapman & Hall\/CRC Press, Boca Raton (2003)"},{"key":"9944_CR3","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/S0020-7462(98)00056-0","volume":"34","author":"SJ Liao","year":"1999","unstructured":"Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Non-Linear Mech. 34, 759\u2013778 (1999)","journal-title":"Int. J. Non-Linear Mech."},{"key":"9944_CR4","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/S0096-3003(02)00790-7","volume":"147","author":"SJ Liao","year":"2004","unstructured":"Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499\u2013513 (2004)","journal-title":"Appl. Math. Comput."},{"key":"9944_CR5","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1111\/j.1467-9590.2007.00387.x","volume":"119","author":"SJ Liao","year":"2007","unstructured":"Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297\u2013354 (2007)","journal-title":"Stud. Appl. Math."},{"key":"9944_CR6","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1016\/j.cnsns.2008.04.013","volume":"14","author":"SJ Liao","year":"2009","unstructured":"Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14, 983\u2013997 (2009)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR7","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-25132-0","volume-title":"Homotopy Analysis Method in Nonlinear Differential Equations. Springer &","author":"SJ Liao","year":"2012","unstructured":"Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press, Heidelberg (2012)"},{"key":"9944_CR8","doi-asserted-by":"crossref","first-page":"4078","DOI":"10.1016\/j.cnsns.2009.03.008","volume":"14","author":"RA Van Gorder","year":"2009","unstructured":"Van Gorder, R.A., Vajravelu, K.: On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach. Commun. Nonlinear Sci. Numer. Simul. 14, 4078\u20134089 (2009)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR9","first-page":"2315","volume":"15","author":"S Liao","year":"2010","unstructured":"Liao, S.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2315\u20132332 (2010)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR10","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.physleta.2006.07.065","volume":"360","author":"S Abbasbandy","year":"2006","unstructured":"Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109\u2013113 (2006)","journal-title":"Phys. Lett. A"},{"key":"9944_CR11","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/j.icheatmasstransfer.2006.12.001","volume":"34","author":"S Abbasbandy","year":"2007","unstructured":"Abbasbandy, S.: Homotopy analysis method for heat radiation equations. Int. commun heat mass tran. 34, 380\u2013387 (2007)","journal-title":"Int. commun heat mass tran."},{"key":"9944_CR12","doi-asserted-by":"crossref","first-page":"2437","DOI":"10.1016\/j.ijheatmasstransfer.2006.01.030","volume":"49","author":"SJ Liao","year":"2006","unstructured":"Liao, S.J., Su, J., Chwang, A.T.: Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body. Int. J. Heat and Mass Transfer 49, 2437\u20132445 (2006)","journal-title":"Int. J. Heat and Mass Transfer"},{"key":"9944_CR13","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1017\/S0022112001007169","volume":"453","author":"SJ Liao","year":"2002","unstructured":"Liao, S.J., Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 453, 411\u2013425 (2002)","journal-title":"J. Fluid Mech."},{"key":"9944_CR14","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/S0020-7462(98)00056-0","volume":"34","author":"SJ Liao","year":"1999","unstructured":"Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Non-Linear Mech. 34, 759\u2013778 (1999)","journal-title":"Int. J. Non-Linear Mech."},{"key":"9944_CR15","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1017\/S0022112099004292","volume":"385","author":"SJ Liao","year":"1999","unstructured":"Liao, S.J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate. J. Fluid Mech. 385, 101\u2013128 (1999)","journal-title":"J. Fluid Mech."},{"key":"9944_CR16","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1017\/S0022112003004865","volume":"488","author":"SJ Liao","year":"2003","unstructured":"Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189\u2013212 (2003)","journal-title":"J. Fluid Mech."},{"key":"9944_CR17","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.amc.2008.12.073","volume":"210","author":"FT Akyildiz","year":"2009","unstructured":"Akyildiz, F.T., Vajravelu, K., Mohapatra, R.N., Sweet, E., Van Gorder, R.A.: Implicit Differential Equation Arising in the Steady Flow of a Sisko Fluid. Appl. Math. Comput. 210, 189\u2013196 (2009)","journal-title":"Appl. Math. Comput."},{"key":"9944_CR18","doi-asserted-by":"crossref","first-page":"053601","DOI":"10.1063\/1.3392770","volume":"22","author":"X Hang","year":"2010","unstructured":"Hang, X., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls. Phys. Fluids 22, 053601 (2010)","journal-title":"Phys. Fluids"},{"key":"9944_CR19","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s11071-006-9140-y","volume":"50","author":"M Sajid","year":"2007","unstructured":"Sajid, M., Hayat, T., Asghar, S.: Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dynamics 50, 27\u201335 (2007)","journal-title":"Nonlinear Dynamics"},{"key":"9944_CR20","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.physleta.2006.09.060","volume":"361","author":"T Hayat","year":"2007","unstructured":"Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Physics Letters A 361, 316\u2013322 (2007)","journal-title":"Physics Letters A"},{"key":"9944_CR21","doi-asserted-by":"crossref","first-page":"106104","DOI":"10.1063\/1.3249752","volume":"21","author":"M Turkyilmazoglu","year":"2009","unstructured":"Turkyilmazoglu, M.: Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer. Phys. Fluids 21, 106104 (2009)","journal-title":"Phys. Fluids"},{"key":"9944_CR22","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s11071-006-9193-y","volume":"51","author":"S Abbasbandy","year":"2008","unstructured":"Abbasbandy, S., Zakaria, F.S.: Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. Nonlinear Dyn. 51, 83\u201387 (2008)","journal-title":"Nonlinear Dyn."},{"key":"9944_CR23","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.chaos.2004.12.016","volume":"26","author":"W Wu","year":"2005","unstructured":"Wu, W., Liao, S.J.: Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos. Solitons & Fractals 26, 177\u2013185 (2005)","journal-title":"Solitons & Fractals"},{"key":"9944_CR24","doi-asserted-by":"crossref","first-page":"2783","DOI":"10.1016\/j.amc.2010.03.128","volume":"216","author":"E Sweet","year":"2010","unstructured":"Sweet, E., Van Gorder, R.A.: Analytical solutions to a generalized Drinfel\u2019d - Sokolov equation related to DSSH and KdV6. Appl. Math. Comput. 216, 2783\u20132791 (2010)","journal-title":"Appl. Math. Comput."},{"key":"9944_CR25","doi-asserted-by":"crossref","first-page":"1733","DOI":"10.1016\/S0960-0779(04)00437-0","volume":"23","author":"Y Wu","year":"2005","unstructured":"Wu, Y., Wang, C., Liao, S.J.: Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos. Solitons & Fractals 23, 1733\u20131740 (2005)","journal-title":"Solitons & Fractals"},{"key":"9944_CR26","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.jmaa.2008.01.050","volume":"343","author":"J Cheng","year":"2008","unstructured":"Cheng, J., Liao, S.J., Mohapatra, R.N., Vajravelu, K.: Series solutions of Nano-boundary-layer flows by means of the homotopy analysis method. J. Math. Anal. Appl. 343, 233\u2013245 (2008)","journal-title":"J. Math. Anal. Appl."},{"key":"9944_CR27","doi-asserted-by":"crossref","first-page":"1494","DOI":"10.1016\/j.cnsns.2009.06.004","volume":"15","author":"RA Van Gorder","year":"2010","unstructured":"Van Gorder, R.A., Sweet, E., Vajravelu, K.: Nano boundary layers over stretching surfaces. Commun. Nonlinear Sci. Numer. Simul. 15, 1494\u20131500 (2010)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR28","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.physleta.2007.05.094","volume":"371","author":"AS Bataineh","year":"2007","unstructured":"Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method. Phys. Lett. A 371, 72\u201382 (2007)","journal-title":"Phys. Lett. A"},{"key":"9944_CR29","doi-asserted-by":"crossref","first-page":"1121","DOI":"10.1016\/j.cnsns.2008.02.004","volume":"14","author":"AS Bataineh","year":"2009","unstructured":"Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Homotopy analysis method for singular IVPs of Emden-Fowler type. Commun. Nonlinear Sci. Numer. Simul. 14, 1121\u20131131 (2009)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR30","doi-asserted-by":"crossref","first-page":"6060","DOI":"10.1016\/j.physleta.2008.08.002","volume":"372","author":"RA Van Gorder","year":"2008","unstructured":"Van Gorder, R.A., Vajravelu, K.: Analytic and numerical solutions to the Lane-Emden equation. Phys. Lett. A 372, 6060\u20136065 (2008)","journal-title":"Phys. Lett. A"},{"key":"9944_CR31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0096-3003(02)00943-8","volume":"142","author":"S Liao","year":"2003","unstructured":"Liao, S.: A new analytic algorithm of Lane-Emden type equations. Appl. Math. Comput. 142, 1\u201316 (2003)","journal-title":"Appl. Math. Comput."},{"key":"9944_CR32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijnonlinmec.2012.01.004","volume":"47","author":"RA Van Gorder","year":"2012","unstructured":"Van Gorder, R.A.: Analytical method for the construction of solutions to the F\u00f6ppl - von K\u00e1rm\u00e1n equations governing deflections of a thin flat plate. Int. J. Non-Linear Mech. 47, 1\u20136 (2012)","journal-title":"Int. J. Non-Linear Mech."},{"key":"9944_CR33","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1016\/j.cnsns.2011.07.036","volume":"17","author":"RA Van Gorder","year":"2012","unstructured":"Van Gorder, R.A.: Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function H(x) in the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 17, 1233\u20131240 (2012)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR34","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1016\/j.cnsns.2011.08.003","volume":"17","author":"M Ghoreishi","year":"2012","unstructured":"Ghoreishi, M., Ismail, A.I.B., Alomari, A.K., Sami Bataineh, A.: The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models. Commun. Nonlinear Sci. Numer. Simul. 17, 1163\u20131177 (2012)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR35","doi-asserted-by":"crossref","first-page":"4268","DOI":"10.1016\/j.cnsns.2011.03.031","volume":"16","author":"S Abbasbandy","year":"2011","unstructured":"Abbasbandy, S., Shivanian, E., Vajravelu, K.: Mathematical properties of h-curve in the frame work of the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 16, 4268\u20134275 (2011)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR36","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1007\/s11075-012-9554-1","volume":"61","author":"RA Van Gorder","year":"2012","unstructured":"Van Gorder, R.A.: Control of error in the homotopy analysis of semi-linear elliptic boundary value problems. Numer. Algorithms 61, 613\u2013629 (2012)","journal-title":"Numer. Algorithms"},{"key":"9944_CR37","doi-asserted-by":"crossref","first-page":"4057","DOI":"10.1016\/j.cnsns.2009.02.016","volume":"14","author":"S Liang","year":"2009","unstructured":"Liang, S., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation. Commun. Nonlinear Sci. Numer. Simul. 14, 4057\u20134064 (2009)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9944_CR38","doi-asserted-by":"crossref","first-page":"2296","DOI":"10.1016\/j.nonrwa.2007.08.007","volume":"9","author":"M Sajid","year":"2008","unstructured":"Sajid, M., Hayat, T.: Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations. Nonlinear Anal.: Real World Appl. 9, 2296\u20132301 (2008)","journal-title":"Nonlinear Anal.: Real World Appl."},{"key":"9944_CR39","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.amc.2006.12.074","volume":"190","author":"FM Allan","year":"2007","unstructured":"Allan, F.M.: Derivation of the Adomian decomposition method using the homotopy analysis method. Appl. Math. Comput. 190, 6\u201314 (2007)","journal-title":"Appl. Math. Comput."},{"key":"9944_CR40","doi-asserted-by":"crossref","first-page":"1901","DOI":"10.1088\/0305-4470\/26\/8\/017","volume":"26","author":"J Miller","year":"1993","unstructured":"Miller, J., Rubel, L.A.: Functional separation of variables for Laplace equations in two dimensions. J. Phys. A 26, 1901\u20131913 (1993)","journal-title":"J. Phys. A"},{"key":"9944_CR41","doi-asserted-by":"crossref","first-page":"2801","DOI":"10.1103\/PhysRevD.54.2801","volume":"54","author":"AL Larsen","year":"1996","unstructured":"Larsen, A.L., Sanchez, N.: sinh-Gordon, cosh-Gordon, and Liouville equations for strings and multistrings in constant curvature spacetimes. Phys. Rev. D 54, 2801\u20132807 (1996)","journal-title":"Phys. Rev. D"},{"key":"9944_CR42","doi-asserted-by":"crossref","first-page":"1763","DOI":"10.1134\/S0012266111120068","volume":"47","author":"SS Kharibegashvili","year":"2011","unstructured":"Kharibegashvili, S.S., Dzhokhadze, O.M.: Cauchy problem for a generalized nonlinear liouville equation. Differential Equations 47, 1763\u20131775 (2011)","journal-title":"Differential Equations"},{"key":"9944_CR43","doi-asserted-by":"crossref","first-page":"2237","DOI":"10.1142\/S0217979211100771","volume":"25","author":"BI Lev","year":"2011","unstructured":"Lev, B.I.: Nonequilibirum self-gravitating system. Int. J. Mod. Phys. B 25, 2237 (2011)","journal-title":"Int. J. Mod. Phys. B"},{"key":"9944_CR44","doi-asserted-by":"crossref","first-page":"6494","DOI":"10.1016\/j.amc.2012.12.049","volume":"219","author":"M Russo","year":"2013","unstructured":"Russo, M., Van Gorder, R.A.: Control of error in the homotopy analysis of nonlinear Klein-Gordon initial value problems. Appl. Math. Comput. 219, 6494\u20136509 (2013)","journal-title":"Appl. Math. Comput."},{"key":"9944_CR45","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.jde.2005.05.005","volume":"216","author":"SY Alice Chang","year":"2005","unstructured":"Alice Chang, S.Y., Han, Z.C., Yang, P.: Classification of singular radial solutions to the [sigma] k Yamabe equation on annular domains. J. Differential Equations 216, 482\u2013501 (2005)","journal-title":"J. Differential Equations"},{"key":"9944_CR46","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/BF02096944","volume":"149","author":"L Andersson","year":"1992","unstructured":"Andersson, L., Chrusciel, P.T., Friedrich, H.: On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein\u2019s field equations. Commun. Math. Phys. 149, 587\u2013612 (1992)","journal-title":"Commun. Math. Phys."},{"key":"9944_CR47","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1090\/S0894-0347-07-00575-9","volume":"21","author":"S Brendle","year":"2008","unstructured":"Brendle, S.: Blow-up phenomena for the Yamabe equation. J. American Math. Soc. 21, 951\u2013980 (2008)","journal-title":"J. American Math. Soc."},{"key":"9944_CR48","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.newast.2010.08.005","volume":"16","author":"RA Van Gorder","year":"2011","unstructured":"Van Gorder, R.A.: An elegant perturbation solution for the Lane-Emden equation of the second kind. New Astron. 16, 65\u201367 (2011)","journal-title":"New Astron."},{"key":"9944_CR49","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s10569-010-9318-y","volume":"109","author":"RA Van Gorder","year":"2011","unstructured":"Van Gorder, R.A.: Analytical solutions to a quasilinear differential equation related to the Lane-Emden equation of the second kind. Celest. Mech. Dyn. Astron. 109, 137\u2013145 (2011)","journal-title":"Celest. Mech. Dyn. Astron."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-014-9944-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-014-9944-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-014-9944-7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,20]],"date-time":"2019-08-20T16:04:18Z","timestamp":1566317058000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-014-9944-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,11]]},"references-count":49,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,10]]}},"alternative-id":["9944"],"URL":"https:\/\/doi.org\/10.1007\/s11075-014-9944-7","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2,11]]}}}