Abstract
In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].
Highlights
Let’s consider a problem under boundary condition containing non-local and global terms for a fractional order integro-differential equation x ba q m 1, m, x a,b, (1)m ij y j 1 a ij y j 1 b bHi t y t dt di, j =1 a i 1, m, (2)ij, ij, i and di, i = 1, m, j = 1, m are real constants, and boundary conditions (2) are linearly independent.2
We try to get some basic relations. The first of these relations is Lagrange’s formula. We multiply both ides of Equation (3) by fundamental solution (5) and integrate the obtained expression on a,b
If we take into account that (5) is the fundamental solution, the last relation (m-th) will be as follows:
Summary
Ij , ij , i and di , i = 1, m , j = 1, m are real constants, and boundary conditions (2) are linearly independent. The boundary value problem (1)-(2) has unique solution. We multiply both ides of Equation (3) by fundamental solution (5) and integrate the obtained expression on a,b (see [4,5])
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