{"id":3107,"date":"2026-09-25T19:35:31","date_gmt":"2026-09-25T19:35:31","guid":{"rendered":"https:\/\/wameedalfikr.com\/?p=3107"},"modified":"2026-09-25T19:54:33","modified_gmt":"2026-09-25T19:54:33","slug":"numerical-solution-of-volterra-integral-equations-of-the-second-kind-with-logarithmic-kernels-by-orthogonal-polynomials","status":"publish","type":"post","link":"https:\/\/wameedalfikr.com\/?p=3107","title":{"rendered":"Numerical solution of Volterra integral equations of the second kind with logarithmic kernels by orthogonal polynomials"},"content":{"rendered":"<p style=\"text-align: center;\"><strong>Numerical solution of Volterra integral equations of the second kind with logarithmic kernels by orthogonal polynomials<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>\u0627\u0644\u062d\u0644 \u0627\u0644\u0639\u062f\u062f\u064a \u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0641\u0648\u0644\u062a\u064a\u0631\u0627 \u0627\u0644\u062a\u0643\u0627\u0645\u0644\u064a\u0629 \u0645\u0646 \u0627\u0644\u0646\u0648\u0639 \u0627\u0644\u062b\u0627\u0646\u064a \u0630\u0627\u062a \u0627\u0644\u0646\u0648\u0649 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a\u0629 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0643\u062b\u064a\u0631\u0627\u062a \u0627\u0644\u062d\u062f\u0648\u062f \u0627\u0644\u0645\u062a\u0639\u0627\u0645\u062f\u0629<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>By:<\/strong> <strong>Qays Atshan Khalaf Almusawi<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>\u00a0<\/strong><strong>Assistant Lecturer at the Open Educational College, Iraq<\/strong><\/p>\n<p style=\"text-align: center;\"><a href=\"mailto:qaysqaysatshan@gmail.com\"><strong>qaysqaysatshan@gmail.com<\/strong><\/a><\/p>\n<p style=\"text-align: center;\">p.p : 230-251<\/p>\n<p><strong>\u0645\u0633\u062a\u062e\u0644\u0635<\/strong><\/p>\n<p>\u062a\u064f\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a 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\u0627\u0644\u0637\u0631\u064a\u0642\u062a\u064a\u0646 \u0648\u0645\u0642\u0627\u0631\u0646\u062a\u0647\u0627 \u0628\u0628\u0639\u0636 \u0627\u0644\u0637\u0631\u0642 \u0627\u0644\u0639\u062f\u062f\u064a\u0629 \u0627\u0644\u0623\u062e\u0631\u0649 \u0628\u0645\u0627 \u0641\u064a \u0630\u0644\u0643 \u0637\u0631\u064a\u0642\u0629 \u0627\u0644\u062a\u0645\u0648\u0636\u0639 \u0627\u0644\u0645\u0634\u062a\u0631\u0643 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0643\u062b\u064a\u0631\u0627\u062a \u062d\u062f\u0648\u062f \u0628\u0631\u0648\u0646\u0631 \u0627\u0644\u0645\u062a\u0635\u0644\u0629 \u0627\u0644\u0642\u0637\u0639\u064a\u0629 \u0644\u0647\u0630\u0627 \u0627\u0644\u0646\u0648\u0639 \u0645\u0646 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0648\u0623\u0638\u0647\u0631\u062a \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0627\u0644\u062a\u064a \u062a\u0645\u0651 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u064a\u0647\u0627 \u0645\u0646 \u0647\u0630\u0627 \u0627\u0644\u0628\u062d\u062b \u0623\u0646\u0647 \u062d\u0642\u0642 \u0643\u0641\u0627\u0621\u0629\u064b \u0648\u0646\u062a\u0627\u0626\u062c \u0623\u0641\u0636\u0644 \u0645\u0646 \u0627\u0644\u0623\u0628\u062d\u0627\u062b \u0627\u0644\u0645\u0645\u0627\u062b\u0644\u0629 \u0627\u0644\u0623\u062e\u0631\u0649.<\/p>\n<p dir=\"ltr\"><strong>Abstract<\/strong><\/p>\n<p dir=\"ltr\">\u00a0\u00a0\u00a0\u00a0\u00a0 Integral equations with weak singular kernels are used in many problems of mathematical physics, applied mechanics, quantum physics, study of chemical reactions, electrochemistry, etc. Determining numerical solutions of this class of equations, especially its nonlinear form, has many complexities. Logarithmic kernels are among the prominent characteristics of this class of equations, which have attracted the attention of many researchers in recent years. This paper was based on the numerical solution of weak singular Volterra integral equations of the second kind with logarithmic kernels using orthogonal polynomials. The purpose of this research is to investigate the numerical solution of this type of equations using the finite expansion method of Chebyshev and Legendre polynomials, as well as their error and convergence analysis. Finally, the numerical results and convergence of these two methods were investigated and compared with some numerical methods, including the co-localization method with continuous piecewise Brunner polynomials for this type of equations. The results obtained from this research showed that it showed more efficient and better results than other similar research.<\/p>\n<p dir=\"ltr\"><strong>\u00a0\u00a0\u00a0\u00a0\u00a0 Keywords<\/strong>: Integra-Volterra equations \u2013 Legendre polynomials \u2013 Chebyshev polynomials \u2013 logarithmic kernels.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Numerical solution of Volterra integral equations of the second kind with logarithmic kernels by orthogonal polynomials \u0627\u0644\u062d\u0644 \u0627\u0644\u0639\u062f\u062f\u064a \u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0641\u0648\u0644\u062a\u064a\u0631\u0627 \u0627\u0644\u062a\u0643\u0627\u0645\u0644\u064a\u0629 \u0645\u0646 \u0627\u0644\u0646\u0648\u0639 \u0627\u0644\u062b\u0627\u0646\u064a \u0630\u0627\u062a \u0627\u0644\u0646\u0648\u0649 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a\u0629 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0643\u062b\u064a\u0631\u0627\u062a \u0627\u0644\u062d\u062f\u0648\u062f \u0627\u0644\u0645\u062a\u0639\u0627\u0645\u062f\u0629 By: Qays Atshan Khalaf Almusawi \u00a0Assistant Lecturer at the Open Educational College, Iraq qaysqaysatshan@gmail.com p.p : 230-251 \u0645\u0633\u062a\u062e\u0644\u0635 \u062a\u064f\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0627\u0644\u062a\u0643\u0627\u0645\u0644\u064a\u0629 \u0630\u0627\u062a \u0627\u0644\u0646\u0648\u0649 \u0627\u0644\u0636\u0639\u064a\u0641\u0629 &hellip;<\/p>\n","protected":false},"author":1,"featured_media":3086,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1,38],"tags":[71],"class_list":["post-3107","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-sections","category-foreign","tag-71"],"acf":[],"_links":{"self":[{"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=\/wp\/v2\/posts\/3107","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3107"}],"version-history":[{"count":1,"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=\/wp\/v2\/posts\/3107\/revisions"}],"predecessor-version":[{"id":3109,"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=\/wp\/v2\/posts\/3107\/revisions\/3109"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=\/wp\/v2\/media\/3086"}],"wp:attachment":[{"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3107"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3107"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wameedalfikr.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}