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With　the　development　of　mordern　technology,　the　theory　of　perturbation　plays　a　more　and　more　important　role,　therefore　aroses　extensive　attentions　of　many　mathematicians.　All　kinds　of　perturbation　methods　are　the　main　part　of　perturbation　theory.　Differential　equations　with　a　small　positive　parameter　　　multiplying　the　highest　order　derivative　terms　are　said　to　be　singular　perturbed　problems　of　boundary　layer　type,　they　are　ubiquitous　in　the　mathematical　problems　which　come　from　many　areas　of　science　and　engineering.　In　general,　as　　　tends　to　zero,　the　solution　of　singularly　perturbed　problem　exhibits　a　multi-scale　character,　that　is,　the　solution　varies　rapidly　in　boundary　or　internal　layers,　but　the　solution　behaves　regularly　and　varies　slowly　away　from　the　layers.Problems　which　exhibit　the　so-called　resonance　phenomenon　have　been　studied　extensively.These　problems　have　stimulated　such　great　interest,　because　the　method　of　matched　of　asymptotic　expansions　can　not　find　unambiguously　a　global　asymptotic　solution,　which　is　uniformly　valid　as　a　certain　positive　parameter　approaches　zero.　It　is　well　known　that　matching　the　inner　and　outer　asymptotic　expansions　yields　a　one-parameter　family　of　possible　asymptotic　solutions,　but　the　perturbation　theory　is　unable　to　select　the　correct　member　of　the　family.　So　the　method　to　deal　for　this　kind　of　problems　will　be　studied　in　the　thesis.　The　outline　of　this　thesis　is　as　follows.In　Chapter　1　and　2,　we　introduce　the　background　and　research　actuality　of　singular　perturbation,　related　concepts　and　theorems　of　perturbation　theory,　and　related　basic　perturbation　method　(regular　perturbation　method).　In　Chapter　3,　some　kinds　of　singular　perturbation　method　for　boundary　layer　problems　arestudied　concretely.　The　method　of　matched　asymptotic　expansions　and　the　method　of　boundary　layer　correction　are　used　in　second　order　differential　equation　with　variable　coefficient.　After　analysing　the　process,　we　obtain　the　simplified　method　of　composited　expansion.In　Chapter　4,　we　apply　the　standard　method　of　matched　asymptotic　expansions　to　find　the　leading　terms　in　the　inner　and　outer　asymptotic　expansions　on　the　boundary　layer　resonance　problem.　We　describe　how　to　solve　the　selection　problem　by　extending　the　method　of　matched　asymptotic　expansions　to　include　exponentially　small　terms　in　the　outer　expansion　these　additional　terms　then　play　an　essential　role　in　the　matching　process,　with　the　end　result　that　the　parameter　left　undetermined　is　evaluated.

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