Phương trình đạo hàm riêng. Phương trình đạo hàm riêng tuyến tính cấp1, cấp 2. Bài toán Dirichlet của phương trình loại elliptic. Bài toán Cauchy của phương trình truyền sóng. Bài toán biên của phương trình truyền nhiệt. | THE LAPLACE TRANSFORM OF A PERIODIC FUNCTION Periodic functions frequently occur in engineering problems and we shall now show how to calculate their transform. They possess the property that f t T f t for t 0 and equal zero for t Ũ where T is the period of the function. For convenience lot US define a function x f which equals zero except over the interval 0 T where it equals f t l I . 0 I t T I u I By definition F s r Jo rT fZT y t l T f t e 9tdi f t e atdi f t e tdt -- Jq Jt JkT Now let z t - k7 where k 0 1 2 . . ill the fcth integral and F s becomes F s r f z T e i 1 T dz --- Jo Jo z 4-fcT e- i T dz Jo However z z z f z T . f z kT . because the range of integration in each integral is from 0 to T. Thus F s becomes F s x z e t2dz e tT x z e dz Jo Jo fT e-ksT t z e S2dz - Jo or F s 1 e- T . . . X S . 80 The first term on the right side of is a geometric series with common ratio e-iT. if e-iT 1 then the series converges and INVERSION BY PARTIAL FRACTIONS HEAVISIDE S EXPANSION THEOREM Tn the previous sections we have devoted our efforts to calculating the Laplace transform of a given function. Obviously we must have a method for going the other way. Given a transform we must find the corresponding function. This is often a very formidable task. In the next few sections wo shall present some general techniques for the inversion of a Laplace transform. The first technique involves transforms that we can express as the ratio of two polynomials F s q s p s . We shall assume that the order of ợ s is less than p s and we have divided out any common factor between them. In principle we know that. p s has n zeros where n is the order of the p s polynomial. Some of the zeros may be complex some of them may be real and some of them may be duplicates of other zeros. In the case when p s has n simple zeros nonrepeating roots a simple method exists for inverting the transform. We want to rewrite F s in the form w y . _ hh .
