M. A. Abdou Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 8, Issue 10 (Part -I) Oct 2018, pp 23-27 www.ijera.com DOI: 10.9790/9622-081001232723|Page On A New Technique For Studying The Resolving Kernel of Volterra Integral Equation M. A. Abdou 1 , A. A. Abbas 2 Mathematics Department, Faculty of Education, Alexandria University Mathematics Department, Faculty of Science, Benha University Corresponding Author:M. A. Abdou ABSTRACT:Here, we use resolving kernel method as a successive approximation method to solve the solution of Volterra integral equation. Some numerical examples are considered and the error of the method is computed Key Wards:Volterra integral equation, resolving kernel, approximate method, the error of the method. --------------------------------------------------------------------------------------------------------------------------------------- Date Of Submission:01-10-2018 Date Of Acceptance: 12-10-2018 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION: The theory of integral equations has close contact with many different areas of different sciences. These different problems have led researches to establish different methods for solving integral equations of different kinds with continuous kernel. There are many well–written texts on the theory and applications of integral equation in different sciences. Among such, we noteGreen,1969);(Hochstadt,1971); (Golberg.ed,1979), (Tricomi, 1985); (Burten1983); (Kanwal, 1996); ( Schiavone at.al.,2002) and (Muskhelishvili, 1953). The reader must know that the importance of the singular integral equations came from the work of(Muskhelishvili1953); , who has established the theory of singular integral equation ( Cauchy method ), that gives the solution of the singular integral equation, analytically. At the same time, approximately from 1960, many new numerical methods have been developed for the solution of many types of integral equation. We note especially (Linz, 1985); ( Atkinson, 1976. 1997);(Baker , 1082), (Delves and Mohamed, 1985) and (Golberg ,ed. 1990). Consider the linear Volterra integral equation of the second kind,      0 , t t f t kts s ds       (1) Here, k(t, s) and f(t) are known continuous functions called the kernel and free term, respectively, while φ(s) is the unknown function. Theorem 1.(without proof):If k(t, s) and f(t) are continuous in 0≤t ≤ T, then the integral equation (1) possesses a unique continuous solution in 0 ≤ t ≤ T< 1. Here, in this paper the existence and uniqueness solution of Volterra integral equation of the second kind is considered. In addition, the solution of the linear Volterra equation with continuous kernel is obtained using a new technique for studying the resolvent kernel. Some examples are considered and the estimate error, with respect to the kernel,is computed. II. THE RESOLVENT KERNEL METHOD We pick up continuous function φ 0 (x) = f(t) then, from (1) we define the sequences      1 0 , , 1 , 2 ,... (2) t n n t f t kts s ds n         and      1 2 0 , (3) t n n t f t kts s ds         By subtracting, we have       1 1 2 0 , [ ] t n n n n t t kts s s ds             (4) For easy of manipulation it is convenient to introduce      1 0 ; , 1 , 2 ,... (5) n n n n t t t t f t n           By using equation (5), the formula (4) becomes     1 0 , , 1 , 2 ,... (6). t n n t kts s ds n       In addition, from equation (5), we get   0 (7) n i n i i t t       Using the recurrence relations and mathematical and the factthat: If the kernel   , kts and the RESEARCH ARTICLE OPEN ACCESS