Page 42 - programme book

AN-002
Nonlinear Fredholm Functional-Integral Equation of First Kind with
Degenerate Kernel and Maxima

T.K. Yuldashev 1,b) , Z.K. Eshkuvatov 2,3,a) , N.M.A. Nik Long 4,c)
1 Uzbek-Israel Joint Faculty of High Technology and Engineering Mathematics National University of
Uzbekistan (NUUz), Tashkent, Uzbekistan
2 Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu (UMT), Kuala
Nerus, Terengganu
3 Independent researcher, Faculty of Applied Mathematics and Intellectual Technologies, National University of
Uzbekistan (NUUz), Tashkent, Uzbekistan
4 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM),
Serdang, Selangor Malaysia.
a) Corresponding author: zainidin@umt.edu.my
b)tursun.k.yuldashev@gmail.com ,
c) nmasri@upm.edu.my

Abstract. In this note, the solvability and solution construction of a nonlinear Fredholm functional-
integral equation of the first kind with degenerate kernel and maxima are considered. Using the
regularization method combined with the method of the degenerate kernel, we obtained an implicit
functional equation with maxima. Since the Fredholm functional-integral equation of the first kind is
ill-posed (non-correct), we used boundary conditions to ensure the uniqueness of the solution. Using
the method of successive approximations, we transform the implicit functional equation to the
nonlinear Volterra type functional-integral equation of the second kind. The solvability and uniqueness
of the solution of the latter integral equations are proved. Two examples are analyzed with an exact
and approximate solution, which is in line with the theoretical findings.

Keywords: Fredholm functional-integral equation, first kind, nonlinear equation, degenerate kernel,
maxima, boundary conditions, regularization, one value solvability.

40