CINXE.COM

<!DOCTYPE html><html lang="en-US"><head><title>Just a moment...</title><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta http-equiv="X-UA-Compatible" content="IE=Edge"><meta name="robots" content="noindex,nofollow"><meta name="viewport" content="width=device-width,initial-scale=1"><style>*{box-sizing:border-box;margin:0;padding:0}html{line-height:1.15;-webkit-text-size-adjust:100%;color:#313131;font-family:system-ui,-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Helvetica Neue,Arial,Noto Sans,sans-serif,Apple Color Emoji,Segoe UI Emoji,Segoe UI Symbol,Noto Color Emoji}body{display:flex;flex-direction:column;height:100vh;min-height:100vh}.main-content{margin:8rem auto;max-width:60rem;padding-left:1.5rem}@media (width <= 720px){.main-content{margin-top:4rem}}.h2{font-size:1.5rem;font-weight:500;line-height:2.25rem}@media (width <= 720px){.h2{font-size:1.25rem;line-height:1.5rem}}#challenge-error-text{background-image:url(data:image/svg+xml;base64,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);background-repeat:no-repeat;background-size:contain;padding-left:34px}@media (prefers-color-scheme:dark){body{background-color:#222;color:#d9d9d9}}</style><meta http-equiv="refresh" content="120"></head><body class="no-js"><div class="main-wrapper" role="main"><div class="main-content"><noscript><div class="h2"><span id="challenge-error-text">Enable JavaScript and cookies to continue</span></div></noscript></div></div><script>(function(){window._cf_chl_opt={cvId: '3',cZone: "publications.waset.org",cType: 'non-interactive',cRay: '9135335ddf7a48e8',cH: 'gi9eoEpzjdhtjXZiZrBmY3Lqznch0z5loD2N4u2ZPuw-1739789260-1.2.1.1-xZgTmhCSwohtT7Kpr3TfHg0mh__r_usL_jEEvJ0B0Hn2HOHc9BtYeXgglSZArgWs',cUPMDTk: "\/search?q=Harmonic%20equations&__cf_chl_tk=du10RFmJ.1YjWC7GlFCS0Qnh0UgUFEyWcJa4h0n4kI0-1739789260-1.0.1.1-dyb3eX.0769HkDjiLqqUPYMFmm.1HEa1MWgMXc7Bq9s",cFPWv: 'b',cITimeS: '1739789260',cTTimeMs: '1000',cMTimeMs: '120000',cTplC: 0,cTplV: 5,cTplB: 'cf',cK: "",fa: "\/search?q=Harmonic%20equations&__cf_chl_f_tk=du10RFmJ.1YjWC7GlFCS0Qnh0UgUFEyWcJa4h0n4kI0-1739789260-1.0.1.1-dyb3eX.0769HkDjiLqqUPYMFmm.1HEa1MWgMXc7Bq9s",md: "cTj4E0A7S.SxlMRZq4wkuyePnserP.NAqMUr8XkZ7NU-1739789260-1.2.1.1-bqpzEcU9XfajX6iydkJe8ZyJKjWsE.SjiYWJeVWoysbwZFb1OMG3qlgqzjl5S.TBnW0vHTdWBxbv3VAbIjoJxovVlf9v0VCtZxuBayY9B81E6yyqgvP28fB2k0sbkp.EIzv_5kc7SYG2LXG.ln9X3lZwvCcHt9N.KX.QAMlLAfh0REN7hJhtJzhyMsFgQQzZOAr2wBAgeenbdSl9SGzDxLiYmzntb3kUHkdazPp8NhJ4jdyzEyW2HTfapEF9IB59pLDa3piKK9NPniTQSVPldRPBzyhOvwm5WIEU7Mmmg8SHQMpU6nBJqq8kbzcA7VUqG2suDJyNNvSIhrDW_3lgOK2aLhUkYwxbUGY7MpepU_uL8kFWYVdEOMlMFOS6wX.tTkgv.abt9xjthSLzExJnShwVybXdNPBC5jCC_yYhfS8gyPGmPln.XnRmHaisJA5StRltzJxTWwRSKjbsVkN5I8A3MEyzOMlNh.UU352WzTdSD.edJzm1ou70C83Rn3Tqtox9yf9QvQ7YPD5Xhf.TQh4PpltwJqaBF.EREIYhGHGDbXHVtffbhcNBpqoflAl0T4P7xHLzwXu6WP8PIiiHoiAtn.BRcWMhHhMI6K9pZrakRib7Jzyx6SImEGS6v8g_75jiHHLNQZ8vBQ_auTE5LX0wGYU_fyP3DaReEkdeqhFBjOiRJaSLk68NnjhLMgP7tt_INlODadEn3_J3EDJ_zWE1YhFuzxGHcENJFJHWAW9IoL0mRwBKyaoGEyiUADWjOt63st_C19mKZCpTAS6blSBKnvHuZc6peNuRtOC1ZvlwTv8DOH6leVKc5Wx2EFx4b1_qPQ22M7vwyVFG1nGG0o0NoNQTlYLyToOlz2NlKMER1UcN_.dgAG5woLii7YVIHK8cr5JpLT7DSYCcVdBJBFcc9tkEjBeNahxmwKDjGvpt3jgzZdJYdfI_Oc.qD0.Q0UbN5cRPOzEVimLm2L9dX67YhkbVeIprjawhtbyoNXnFx79HnljZVeWFEAfuPURgnbX843LXXuNJFVhCY1Um9u4QUCWv2I1RDZDNKF1uZiFGG1Fha6CkHQqQ8cNygtZCRnvmh2o_kTHxyV.nDsxuo1d1kG9Rz.yBgmQIcVOdDxlCDBH5eJXXYQCCxU.3nTFotzvtnq0BToyWiktZJW8EKC8bweSfD3mMCyNZAmAEQrAKn.cKYh_4.7h9xaP5.K.VeZIy6NUwx8z25VniQW0jXrruOFgdkDM9ml690tUQUt.BZrglGOcS7tq2HJ.HV9sXKpvGvkdg14YFxzmwjjev0XoSgMKz5PU6QBc3xxe9CZg4nzQhp33q1vsYT0t4aD5olas0JzmvyI0nt83_kF9ny1ZEUxKGI5i47cITEtzBKDqhKWT_u2xztPRdScwLEu0W7RdIJAqTAe5pfvAQqLUbEBPE9RCRLORapXqO.sGyKIr5lqHFgfGqVrh8iTRrY_r0V9YrALAt0jJXq_cJu2PczYDNdGH6rgYwVyx4bWa4n_6EKNLim2Ta6sptcTVMq9Z4ISKW6c6na5H2Ybec75OaXbQ5SEwF_MOaSSMpu85DECLWpYQOAn.T8qHowU1OwNQhcf_8fybWibWbS.weqjm7KUT_6kzIaUdA9k0flUw8Oi6Wub3hr.IqLuHt6e7rEjbJlMUj3AAeY5lPaDZ_g8yBeh3m_Jvofsl6SRLjU8N3BO43Uq8i2o2R.pl2k2qX_7Xzjl2JV9Wp0xKzrhGoc9GTnAGzqBXYCB51g4zmh_pF4Lvk2BC.6sKfuDPWINpwQIGzOAtWu4..Dfs54lHtbl.N6ZIXkODMNdOj2UDZqjSso.z8GT4q.hzqhqFvQ61gU9p_rv4Oro8ZoSB35WPrGrIDy1sA0A2eGcY9v01qCuvt8XF2y2WZylxvwOX8YyuOT8sWta3Nk8_asfUOqx1T498w_A",mdrd: "AmKgPzKBssRUBwejV.dEsZtBq4AH9hqsynOll62H9DE-1739789260-1.2.1.1-QY7hbOnn.bF53SXfD2ZB04zWbwPXcutosM_0DmmKOKBAT69Sw1NGyI4zxyA21z5nAM61e23P50YQp5BNhDLMkjcsKaMAQeg3CzkDS.nvJzBa8BZxnLLG3av49jeuMTDAyhHa8BmRLgM2RirV25IXIhMlajXXo792AhZqz7MpDwItQYObYL3jHNjImxCgpoWXgND83iqPZ_ubm9MoNCcvP_Z32FTd_lBb4AJUJbQ76iXvXqE3mRfCirWR0YzWGQ0KSdyHJjQmmbLPkd7TTpjOhL2AFFfexyTZ38fzRpDOqh4v7S4RZtYmfh2rnsInYAOVkQ.3MxO3zqgLrqPPo2fm0MxfXbCDLzecIJN3TnMcLR4gl8URF_1EQvO6rlHNUVzSYkP3qJTKtWpI3rS_DC3x_eLx5GdgrtlRHbg.sOvwlzbBqA.dKBnfT3.XdHW9gXUbE5B.ojNlaO8KajxFz7fvLEXXPDEYAwRYJUJPydb8OcFwk8lZH46h9IL5qilpDi.V31l_4S4NBQto4L8eR7JnPh7nDZpfOFwtgisBeM5ZIz4lQuG9a9amyTxWI0nKMq3c5QkAuYit7zdb4iTQ.gYVpuOc29ENA7raAuLn1COmuYEsCz5PdKhpuvGlr3kzWykgKfk.6KSDUkalkqjhcf8kS1Nf3SUqC6kMM7py4sL2.DFyhbfWh99JR.ClMVDyd_sN_dgABkJgld6NeH05fHVcdAWXT_Fw9K2VDmu5mJcjip1JBr5zJVLmTZ9p7WJYFyZFnY0_BYPwizop6TfO0ejggv6gI5l.Ws77SvgJ9OJfq0ZsCzmgKCFbFOcrfsLrFv.DP_.11dYASRr1Ql8mjn7lrQ.MsoYH9mOwv6.ZW_WehO1LGEllihOfh5hJHQzURrsqfj9h6Uo70OcjQP0D09nX4Ij4uzRjCuFTfEGZpZkvnYeAaDlgLFOHtMoF6GK0dNMb43KJxT9GP9wQNDMVLQqRNdHK7fXu9b.xbg60oL_QlUWA.P4UAU_ciZe5YYlE55E6IUY_Nelj7WF.WsdYN09Ndhv5B_xZ2L5MQ3kvCduU5FTzW.U5ZXJDETr71X3F.iN9Xr_0EafyiT1pSfuMRbIIKvJH6GyR4HRVuC35u1wb6EofN93_cFVEb6ounV3A7JUMgi.eoxjs7mb_34PiPezZhoHMKWeZr8.G.6WoVbJXDbF46RJtzJJr7owg3gJs6OZP0uuSGip28bO46SS0HUVrO9S_2Z0AB14BgmU6xByHeeXXCfj0hJNbWZWs0RDPHD_B4jyxV7Yv2fEZJ4m5uV193hSpY0sFq0r26qN13DjzRt1n._30XD6k_af9hBrCkDvxKcclGxaHa_3lQ.Tt8w7vVZnyehlMYE4.QHRjwRlN9xTGjNjUOYAcSGFRqSp2JHI2ZITiWVBcsoJuqdEQ9b2PtqbNvkcAeYysnwJM_hFWFUw5SWAH7mddLm_.ASRNYmDkzAjHoHHXHup5YjtT1YA9dTb8rub26PwTFBgENVAskITfIOdmeLnZaxMXf7Q09jRYb0YsOW3KnTeNl6n58YQoFZm6ysjBx2t32NJUxWMya44mt8rCFIyN2ek6M.i2QY6A6rtoO0zgAKXQc9QRUcKOxdKBfN6g0vPtamZuTXb8hxIo0zeEoW0lYTOqOtNLB0tpLN70VKXLOi2NelJvcHUg5D6wmDjTyhOrJNeTXST2M95y9.RwO5atlrwSCwEH0NFtoCM9G4vRb47tI6l4qELWAbDJ..wr2FmF_WCS60DTY1sw4YSrYy47uHK7wZ.xaNEBfc2sYMYjMvTlP5T0zN1CqLMUmPVVyAIyWN1fvHbQWDZGhB4OjrNF_GUY5S6ZmK557e24RasiGPKk8z4bX3ReJhEO9R5J0yTfopTSLDttm1N_FYoABahoqZDZjMCeK30IfH7bmGrfwLWuGRgyoxr92xTylT2p29dd2semXfNEbeKIVHjcHQGjYblOEYcJoUWpVwJmACeMGqy38pMLzX2cbQ"};var cpo = document.createElement('script');cpo.src = '/cdn-cgi/challenge-platform/h/b/orchestrate/chl_page/v1?ray=9135335ddf7a48e8';window._cf_chl_opt.cOgUHash = location.hash === '' && location.href.indexOf('#') !== -1 ? '#' : location.hash;window._cf_chl_opt.cOgUQuery = location.search === '' && location.href.slice(0, location.href.length - window._cf_chl_opt.cOgUHash.length).indexOf('?') !== -1 ? '?' : location.search;if (window.history && window.history.replaceState) {var ogU = location.pathname + window._cf_chl_opt.cOgUQuery + window._cf_chl_opt.cOgUHash;history.replaceState(null, null, "\/search?q=Harmonic%20equations&__cf_chl_rt_tk=du10RFmJ.1YjWC7GlFCS0Qnh0UgUFEyWcJa4h0n4kI0-1739789260-1.0.1.1-dyb3eX.0769HkDjiLqqUPYMFmm.1HEa1MWgMXc7Bq9s" + window._cf_chl_opt.cOgUHash);cpo.onload = function() {history.replaceState(null, null, ogU);}}document.getElementsByTagName('head')[0].appendChild(cpo);}());</script></body></html>