In this paper, we study four cases of the Toeplitz plus Hankel integro - differential equations and general integro - differential equation. These equations can be solved in closed form using new convolutions and well-known convolutions. The obtained results were the expansions of these equation types. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2014 Vol. 59 No. 7 pp. 21-33 This paper is available online at http TOEPLITZ - HANKEL INTEGRO - DIFFERENTIAL EQUATION Nguyen Xuan Thao1 Nguyen Anh Dai2 and Nguyen Minh Phuong1 1 School of Applied Mathematics and Informatics Hanoi University of Science and Technology 2 Faculty of Mathematics Hung Yen University of Technology and Education Abstract. There are many applications of the Toeplitz plus Hankel equation in mathematics physics and medicine. However this is still an open problem nowadays. In this paper we study four cases of the Toeplitz plus Hankel integro - differential equations and general integro - differential equation. These equations can be solved in closed form using new convolutions and well-known convolutions. The obtained results were the expansions of these equation types. A new approach to solved classes of equations of this kind is to use the new and known convolutions to obtain explicit solutions as well as solutions in closed form. Keywords Toeplitz and Hankel integral equation integro-differential equation convolution Hartley transform Fourier transform Laplace transform and Fourier cosine transform. 1. Introduction The integral equation with Toeplitz plus Hankel is of the form 5 . f x k1 x y k2 x y f y dy g x x gt 0 0 where g k1 k2 are given and f is an unknown function. This equation has many useful applications in mathematics physics and medicine 1 3 . This integral equation can be solved in closed form only in some particular cases of the Hankel kernel k1 and the Toeplitz kernel k2 . However the solution of equation in the general case is still open. In various particular cases of equation the Toeplitz kernel k2 is an even function and equation takes the form f x k1 x y k2 x y f y dy g x x gt 0. 0 Received June 24 2014. Accepted September 10 2014. Contact Nguyen Xuan Thao e-mail address 21 Nguyen Xuan Thao Nguyen .
