2 Preface recently emerged and fascinating subjects for equations of mathematical physics.Chapter 9 covers the inverse scattering perturbation theory,the method of formal parameter expansion and Gel'fand-Levitan-Marchenko method on the inverse problem,and Hirota transformations,Backlund transformations and inverse scattering transformations on nonlinear problems. Both differential and integral methods can be used to derive mathematical models,but here only the differential method is adopted. Equations of mathematical physics have very wide subjects.As far as the linear system is concerned,what we have focused here on are second-order linear scalar partial differential equations of hyperbolic,parabolic and elliptic types.Little or no attention is given to vector differential equations,higher than second-order,mixed-type,integral and integro-differential equations.Vector partial differential equations play an increasingly important role in the theories of electro-magnetic waves and elastic waves with the deepening of scientific research,but they are too advanced for the undergraduate.Also no attention is given to the Wiener-Hopf method and other important methods related to the function theory of one complex variable,and many important approximate methods. The third part has six chapters.Wavelet transforms are intermediate ones between Fourier and Impulse transforms.Wavelets are a class of representations of very special importance.Considering their special applications,many restraints must be imposed on the function series of expansion.But the issues follow:Do there exist such function series of expansion that meet these restraints?If so,how can they (i.e.,wavelets)be constructed?And do they also have a fast algorithm,which is analogous to FFT? In the first chapter of the third part,wavelets'definitions are given and whys for putting so many restraints on wavelets are expounded.The subsequent four chapters are devoted to the above-mentioned three issues and the last chapter discuses wavelets applications to approximate solution for equations of mathematical physics besides mentioning briefly some other applications. Wavelets'rapid development just occurred after 1988.Now their use has almost become ubiquitous in every domain in natural sciences,including every branch of physics. The knowledge of wavelet transforms is urgent to be spread among undergraduates, graduates and researchers of natural sciences.But up to now,wavelets are still mysterious to many physical workers and engineers.In the author's point of view,there are three reasons for such a mystery.Two of them are objective:First,functional analysis is involved,and second,there are many classes of wavelets,almost each of which has also many members,and those wavelets meeting required constraints have only numerical and/ or geometrical representations,and no analytical representation.The third is subjective: Many problems-formulation are very long and it is sure to result in confusion of idea and pproach for someone to seek a solution to a problem in haste without knowing it clearly and completely.The author tries to address these problems in simple terms.ｒｅｃｅｎｔｌｙｅｍｅｒｇｅｄａｎｄｆａｓｃｉｎａｔｉｎｇｓｕｂｊｅｃｔｓｆｏｒｅｑｕａｔｉｏｎｓｏｆｍａｔｈｅｍａｔｉｃａｌｐｈｙｓｉｃｓ．Ｃｈａｐｔｅｒ９ ｃｏｖｅｒｓｔｈｅｉｎｖｅｒｓｅ ｓｃａｔｔｅｒｉｎｇ ｐｅｒｔｕｒｂａｔｉｏｎ ｔｈｅｏｒｙ，ｔｈｅ ｍｅｔｈｏｄ ｏｆｆｏｒｍａｌ ｐａｒａｍｅｔｅｒ ｅｘｐａｎｓｉｏｎａｎｄＧｅｌｆａｎｄＬｅｖｉｔａｎＭａｒｃｈｅｎｋｏ ｍｅｔｈｏｄｏｎｔｈｅｉｎｖｅｒｓｅｐｒｏｂｌｅｍ，ａｎｄ Ｈｉｒｏｔａ ｔｒａｎｓｆｏｒｍａｔｉｏｎｓ，Ｂｃｋｌｕｎｄ ｔｒａｎｓｆｏｒｍａｔｉｏｎｓ ａｎｄｉｎｖｅｒｓｅ ｓｃａｔｔｅｒｉｎｇ ｔｒａｎｓｆｏｒｍａｔｉｏｎｓ ｏｎ ｎｏｎｌｉｎｅａｒｐｒｏｂｌｅｍｓ． Ｂｏｔｈｄｉｆｆｅｒｅｎｔｉａｌａｎｄｉｎｔｅｇｒａｌｍｅｔｈｏｄｓｃａｎｂｅｕｓｅｄｔｏｄｅｒｉｖｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｓ，ｂｕｔ ｈｅｒｅｏｎｌｙｔｈｅｄｉｆｆｅｒｅｎｔｉａｌｍｅｔｈｏｄｉｓａｄｏｐｔｅｄ． Ｅｑｕａｔｉｏｎｓｏｆ ｍａｔｈｅｍａｔｉｃａｌｐｈｙｓｉｃｓｈａｖｅｖｅｒｙ ｗｉｄｅｓｕｂｊｅｃｔｓ．Ａｓｆａｒａｓｔｈｅｌｉｎｅａｒ ｓｙｓｔｅｍｉｓｃｏｎｃｅｒｎｅｄ，ｗｈａｔｗｅｈａｖｅｆｏｃｕｓｅｄｈｅｒｅｏｎａｒｅｓｅｃｏｎｄｏｒｄｅｒｌｉｎｅａｒｓｃａｌａｒｐａｒｔｉａｌ ｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｏｆｈｙｐｅｒｂｏｌｉｃ，ｐａｒａｂｏｌｉｃａｎｄｅｌｌｉｐｔｉｃｔｙｐｅｓ．Ｌｉｔｔｌｅｏｒｎｏａｔｔｅｎｔｉｏｎｉｓ ｇｉｖｅｎｔｏｖｅｃｔｏｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ，ｈｉｇｈｅｒｔｈａｎｓｅｃｏｎｄｏｒｄｅｒ，ｍｉｘｅｄｔｙｐｅ，ｉｎｔｅｇｒａｌａｎｄ ｉｎｔｅｇｒｏｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ．Ｖｅｃｔｏｒｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｐｌａｙａｎｉｎｃｒｅａｓｉｎｇｌｙ ｉｍｐｏｒｔａｎｔｒｏｌｅｉｎｔｈｅｔｈｅｏｒｉｅｓｏｆｅｌｅｃｔｒｏｍａｇｎｅｔｉｃ ｗａｖｅｓａｎｄ ｅｌａｓｔｉｃ ｗａｖｅｓ ｗｉｔｈｔｈｅ ｄｅｅｐｅｎｉｎｇｏｆｓｃｉｅｎｔｉｆｉｃｒｅｓｅａｒｃｈ，ｂｕｔｔｈｅｙａｒｅｔｏｏａｄｖａｎｃｅｄｆｏｒｔｈｅｕｎｄｅｒｇｒａｄｕａｔｅ．Ａｌｓｏｎｏ ａｔｔｅｎｔｉｏｎｉｓｇｉｖｅｎｔｏｔｈｅＷｉｅｎｅｒＨｏｐｆｍｅｔｈｏｄａｎｄｏｔｈｅｒｉｍｐｏｒｔａｎｔｍｅｔｈｏｄｓｒｅｌａｔｅｄｔｏｔｈｅ ｆｕｎｃｔｉｏｎｔｈｅｏｒｙｏｆｏｎｅｃｏｍｐｌｅｘｖａｒｉａｂｌｅ，ａｎｄｍａｎｙｉｍｐｏｒｔａｎｔａｐｐｒｏｘｉｍａｔｅｍｅｔｈｏｄｓ． Ｔｈｅｔｈｉｒｄｐａｒｔｈａｓｓｉｘｃｈａｐｔｅｒｓ．Ｗａｖｅｌｅｔｔｒａｎｓｆｏｒｍｓａｒｅｉｎｔｅｒｍｅｄｉａｔｅｏｎｅｓｂｅｔｗｅｅｎ ＦｏｕｒｉｅｒａｎｄＩｍｐｕｌｓｅｔｒａｎｓｆｏｒｍｓ．Ｗａｖｅｌｅｔｓａｒｅａｃｌａｓｓｏｆｒｅｐｒｅｓｅｎｔａｔｉｏｎｓｏｆｖｅｒｙｓｐｅｃｉａｌ ｉｍｐｏｒｔａｎｃｅ．Ｃｏｎｓｉｄｅｒｉｎｇｔｈｅｉｒｓｐｅｃｉａｌａｐｐｌｉｃａｔｉｏｎｓ，ｍａｎｙｒｅｓｔｒａｉｎｔｓｍｕｓｔｂｅｉｍｐｏｓｅｄｏｎ ｔｈｅｆｕｎｃｔｉｏｎｓｅｒｉｅｓｏｆｅｘｐａｎｓｉｏｎ．Ｂｕｔｔｈｅｉｓｓｕｅｓｆｏｌｌｏｗ：Ｄｏｔｈｅｒｅｅｘｉｓｔｓｕｃｈｆｕｎｃｔｉｏｎｓｅｒｉｅｓ ｏｆｅｘｐａｎｓｉｏｎｔｈａｔ ｍｅｅｔｔｈｅｓｅｒｅｓｔｒａｉｎｔｓ？Ｉｆｓｏ，ｈｏｗ ｃａｎｔｈｅｙ（ｉ．ｅ．，ｗａｖｅｌｅｔｓ）ｂｅ ｃｏｎｓｔｒｕｃｔｅｄ？Ａｎｄｄｏｔｈｅｙａｌｓｏｈａｖｅａｆａｓｔａｌｇｏｒｉｔｈｍ，ｗｈｉｃｈｉｓａｎａｌｏｇｏｕｓｔｏＦＦＴ？ Ｉｎｔｈｅｆｉｒｓｔｃｈａｐｔｅｒｏｆｔｈｅｔｈｉｒｄｐａｒｔ，ｗａｖｅｌｅｔｓｄｅｆｉｎｉｔｉｏｎｓａｒｅｇｉｖｅｎａｎｄ ｗｈｙｓｆｏｒ ｐｕｔｔｉｎｇｓｏｍａｎｙｒｅｓｔｒａｉｎｔｓｏｎｗａｖｅｌｅｔｓａｒｅｅｘｐｏｕｎｄｅｄ．Ｔｈｅｓｕｂｓｅｑｕｅｎｔｆｏｕｒｃｈａｐｔｅｒｓａｒｅ ｄｅｖｏｔｅｄｔｏｔｈｅａｂｏｖｅｍｅｎｔｉｏｎｅｄｔｈｒｅｅｉｓｓｕｅｓａｎｄｔｈｅｌａｓｔｃｈａｐｔｅｒｄｉｓｃｕｓｅｓ ｗａｖｅｌｅｔｓ ａｐｐｌｉｃａｔｉｏｎｓｔｏ ａｐｐｒｏｘｉｍａｔｅ ｓｏｌｕｔｉｏｎ ｆｏｒ ｅｑｕａｔｉｏｎｓ ｏｆ ｍａｔｈｅｍａｔｉｃａｌ ｐｈｙｓｉｃｓ ｂｅｓｉｄｅｓ ｍｅｎｔｉｏｎｉｎｇｂｒｉｅｆｌｙｓｏｍｅｏｔｈｅｒａｐｐｌｉｃａｔｉｏｎｓ． Ｗａｖｅｌｅｔｓｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｊｕｓｔｏｃｃｕｒｒｅｄａｆｔｅｒ１９８８．Ｎｏｗ ｔｈｅｉｒｕｓｅｈａｓａｌｍｏｓｔ ｂｅｃｏｍｅｕｂｉｑｕｉｔｏｕｓｉｎｅｖｅｒｙｄｏｍａｉｎｉｎｎａｔｕｒａｌｓｃｉｅｎｃｅｓ，ｉｎｃｌｕｄｉｎｇｅｖｅｒｙｂｒａｎｃｈｏｆｐｈｙｓｉｃｓ． Ｔｈｅｋｎｏｗｌｅｄｇｅｏｆｗａｖｅｌｅｔｔｒａｎｓｆｏｒｍｓｉｓｕｒｇｅｎｔｔｏｂｅｓｐｒｅａｄａｍｏｎｇｕｎｄｅｒｇｒａｄｕａｔｅｓ， ｇｒａｄｕａｔｅｓａｎｄｒｅｓｅａｒｃｈｅｒｓｏｆｎａｔｕｒａｌｓｃｉｅｎｃｅｓ．Ｂｕｔｕｐｔｏｎｏｗ，ｗａｖｅｌｅｔｓａｒｅｓｔｉｌｌｍｙｓｔｅｒｉｏｕｓ ｔｏｍａｎｙｐｈｙｓｉｃａｌｗｏｒｋｅｒｓａｎｄｅｎｇｉｎｅｅｒｓ．Ｉｎｔｈｅａｕｔｈｏｒｓｐｏｉｎｔｏｆｖｉｅｗ，ｔｈｅｒｅａｒｅｔｈｒｅｅ ｒｅａｓｏｎｓｆｏｒｓｕｃｈａ ｍｙｓｔｅｒｙ．Ｔｗｏｏｆｔｈｅｍ ａｒｅｏｂｊｅｃｔｉｖｅ：Ｆｉｒｓｔ，ｆｕｎｃｔｉｏｎａｌａｎａｌｙｓｉｓｉｓ ｉｎｖｏｌｖｅｄ，ａｎｄｓｅｃｏｎｄ，ｔｈｅｒｅａｒｅｍａｎｙｃｌａｓｓｅｓｏｆｗａｖｅｌｅｔｓ，ａｌｍｏｓｔｅａｃｈｏｆｗｈｉｃｈｈａｓａｌｓｏ ｍａｎｙｍｅｍｂｅｒｓ，ａｎｄｔｈｏｓｅｗａｖｅｌｅｔｓｍｅｅｔｉｎｇｒｅｑｕｉｒｅｄｃｏｎｓｔｒａｉｎｔｓｈａｖｅｏｎｌｙｎｕｍｅｒｉｃａｌａｎｄ／ ｏｒｇｅｏｍｅｔｒｉｃａｌｒｅｐｒｅｓｅｎｔａｔｉｏｎｓ，ａｎｄｎｏａｎａｌｙｔｉｃａｌｒｅｐｒｅｓｅｎｔａｔｉｏｎ．Ｔｈｅｔｈｉｒｄｉｓｓｕｂｊｅｃｔｉｖｅ： Ｍａｎｙｐｒｏｂｌｅｍｓｆｏｒｍｕｌａｔｉｏｎａｒｅｖｅｒｙｌｏｎｇａｎｄｉｔｉｓｓｕｒｅｔｏｒｅｓｕｌｔｉｎｃｏｎｆｕｓｉｏｎｏｆｉｄｅａａｎｄ ａｐｐｒｏａｃｈｆｏｒｓｏｍｅｏｎｅｔｏｓｅｅｋａｓｏｌｕｔｉｏｎｔｏａｐｒｏｂｌｅｍｉｎｈａｓｔｅｗｉｔｈｏｕｔｋｎｏｗｉｎｇｉｔｃｌｅａｒｌｙ ａｎｄｃｏｍｐｌｅｔｅｌｙ．Ｔｈｅａｕｔｈｏｒｔｒｉｅｓｔｏａｄｄｒｅｓｓｔｈｅｓｅｐｒｏｂｌｅｍｓｉｎｓｉｍｐｌｅｔｅｒｍｓ． ２ 犘狉犲犳犪犮犲