Abstract:
The complex variable is one of the most interesting and successful branches of mathematics and its results help in proving important theories. For mathematics, it opens up horizons for several concepts in other fields and many of the effective methods that are used in engineering and other sciences applications depend on complex functions. The complex variables also give an excellent introduction to modern mathematics because of its wide applications, its combination between engineering and analytical concepts and the ease of much of its results. Modern developments in function theory and complex variable theory are considered useful in giving applications in many engineering fields. The scientific purpose of using integral transforms lies in their ultra ability in disengagement of complex mathematical relations that the function lies under their effect or obtaining another form from the original function that can easily be dealt with mathematically. Of these transforms are Fourier transforms that are divided into two types. The first type is called Fourier finite transforms and second is called Fourier transforms. The greatest importance of Fourier transforms is generally their enormous capacity of obtaining solutions for normal and partial differential equations and also facilitating to obtain solutions for many of the integral equations. Fourier transforms depend on Fourier series and these series have greater importance in approximating functions in the form of infinite series of the sine and cosine functions simultaneously, which is termed combinations, or representing each of them separately. This assists in creating an enormous scientific revolution in solving differential equations particularly partial differential equations and initial boundary problems and boundary value problems by Fourier series and it is not necessary to satisfy the communication conditions throughout the period of the function definition. Two questions of elastodynamic crack were solved in this research by a theory of linear homogenous and heterogeneous properties elasticity. The two questions have received great scientific interest since the cracked bodies have finite boundaries in form of long strips. The method or the procedures of solving the cracked bodies were not unique but they were based on the complex variable method or the method of integral transform. The solution of the first problem was more easily obtained by the technique of the complex variable. Whereas, in the second problem the Wiener- Hopf method was used. This has been obtained despite that the first problem was more difficult due to the inclusion of the double link region in a finite strip instead of a simply linked domain for the second problem. This fact clearly shows the superiority of the complex variable methods, in some cases, over the more difficult Wiener- Hopf method.
Description:
A Dissertation Submitted to the University of Gezira in partial Fulfillment
of the Requirements for the Award of the Degree of Master of Science in Mathematics
Department of Mathematics Faculty of Mathematical and Computer Sciences ,ـFebruary,2014