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1.Hilbert space method
¹ØÓÚ΢·ÖËã×ÓµÄÒ»ÖÖ×ÔÈ»µÄ±íÊöÓ¦¸ÃÊǹãÒ庯ÊýϵĽâÊÍ¡£Òò¶øPDEs±»Àí½âΪ¹ãÒ庯Êý¿Õ¼äÖеÄÒ»¸öÔª£¬ÆäʵÊǼìÑ麯Êý¿Õ¼äµÄ¹éÄɼ«ÏÞÍØÆËʹµÃ΢·ÖÔËË㳩ͨÎÞ×裬¶øÇÒ»¹±£³ÖTVSµÄÍ걸ÐÔ¡£ÓÉÓÚ´óÁ¿µÄº¯Êý¿Õ¼äµÄ¶Ôż¿Õ¼ä¿ÉÒÔ±»Ç¶ÈëÔÚ¹ãÒ庯Êý¿Õ¼äÀïÃæ£¬ËùÒÔPDEs¿ÉÒÔ±»Àí½âΪһÖÖ¶ÔżÐÎʽ¡£Hilbert space Ò»·½ÃæÊÇ×Ô¶ÔżµÄ£¬ÁíÒ»·½ÃæÓÐÒ»¸ö×ÔÈ»µÄÄÚ»ý½á¹¹ÓëÎïÀíµÄÄÜÁ¿¹ÛÄî¶ÔÓ¦¡£ËùÒÔ´óÁ¿µÄÁ¦Ñ§¼°ÎïÀíÎÊÌâ¿ÉÓɱä·Ö²»µÈ·½³ÌÃèÊö¡£ÈÈѧ£¬µ¯ÐÔ£¬Õ³µ¯ÐÔ£¬ËÜÐÔ£¬¸÷ÖÖĦ²ÁÎÊÌ⣬BinghamÁ÷Ì壬µç´Å£¬etc
µ±È»Ò»°ã²»ÄÜÔÚ¹ãÒ庯Êý¿Õ¼äÏÂÃæÌÖÂÛÕâÀàÎÊÌ⣬һ¸öÖØÒªµÄÀ§ÄÑÔÚÓڳ˷¨ÔËËã¡£»¹ºÃÎÒÃÇÓоֲ¿¿É»ýº¯Êý¶ÔÓ¦µÄ¹ãÒ庯Êý£¬¼ÓÈë·¶Êý½á¹¹£¬¾Í±äµÃÊ®·ÖÓÐÓá£Ò»¸öÖøÃûµÄÀý×Ó¾ÍÊÇSobolev space¡£È·¶¨ÔÚʲôÑùµÄ¿Õ¼äÏÂÃæÌÖÂÛPDEsµÄ½â£¬ÕâÊÇÊ×ÏÈÃæ¶ÔµÄ´óÎÊÌâ¡£¡£¡£
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2.Weak convergence method
ÍÑÀëHilbert spaceµÄÁ¼ºÃ½á¹¹£¬ÕâÊÇBanach spaceÖÐÇó½âPDEsµÄÁíÒ»ÖÖ·½·¨¡£¾ßÌå²»Öª£¬ÎÞÏÞάBanach spaceÖе¥Î»Çò½ôÐÔȱʧ£¬±ØÐëÒª¿¼ÂǸ÷ÖÖ¸÷ÑùµÄÈõÍØÆËÁË¡£Í¨³£ÊÇÑ¡ÔñʹµÃÁ¬ÐøÏßÐÔ·ºº¯×壨·¶ÍØÆËÏ£©±£³ÖÁ¬ÐøµÄ×îÈõÍØÆË¡£Ò»ÏµÁеÄÌÖÂÛ×ÔÈ»¾ÍÀ´ÁË£¬½ôÐÔ£¬¿É·ÖÐÔ£¬etc
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3.Fixed point methd
ΪÁË˵Ã÷ijЩ´æÔÚÐÔÎÊÌ⣬³ýÁ˽ôÐÔ£¬fixed point µÄ¹ÛµãÁí±Ùõè¾¶¡£Ò»°ãÓÐmetrix fixed point theorem£¬µ«×îÖøÃûÒ²ÊÇ×îÓÐÓ°ÏìµÄÊÇtopological fixed point theorem¡£ÕâÄêÍ·£¬ÍØÆË»¯ÊÓºõÊǸöÇ÷ÊÆ¡£ÓëÍØÆËÏà¹ØµÄ½á¹ûÒ»°ã¾ßÓпÉÒÅ´«ÐÔ£¬¾ÍÊÇÔÚͬÅßϱ£³ÖijÖÖ²»±äÐÔ£¬ÕâÀïÎÒÃÇ˵µÄÊDz»¶¯µã´æÔÚÐÔ¡£¸÷ÖÖÑÓÍØµÄ¶¨ÀíÔڴ˱íÏÖµÃÒ쳣ǿ´ó¡£PDEsÎÊÌâÒÔÁíÒ»ÖֽǶȱ»½âÊÍÁË£¬ÎÒÃÇ¿ÉÒÔÔÚnormed spaceÏ¿¼ÂÇÕâÀàÎÊÌ⣬ֱ½ÓÇó½âËã×Ó·½³ÌµÄÇ¿´ó¹¤¾ß£¬ÕâÊÇ´úÊýÍØÆËµÄ÷ÈÁ¦°¡¡£¿Éϧ£¬Ò»°ãÎÒÃǺÜÄÑ»ñµÃµü´ú¸ñʽÓÃÓÚ¼ÆËã»ú¼ÆËã¡£ÓãºÍÐÜÕÆ²»¿É¼æµÃ°É¡£¡£¡£
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4.Lie group method
ÀîȺ·ÖÎöÓëPDEs£¬ÓÐһЩÊé½²Õâ¸ö£¬±ÈÈç¡°ÈðµäÈËNail H. Ibragimov±àдµÄÀîȺ·ÖÎöÓëÆ«Î¢·Ö·½³Ì¹ØÏµÊ½Ö®Êֲᡱ

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