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Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations and Their Numerical Solutions Hamed Bazgir1 · Bahman Ghazanfari1 © Springer Nature India Private Limited 2020

Abstract In this paper, we investigate the existence and uniqueness of solution for a multi-term fractional integro-differential problem with nonlocal four-point fractional boundary conditions via the Caputo differentiation. We obtain operational matrix of Riemann–Liouville fractional integral operator of Bernstein polynomials and investigate the numerical solutions of the problem by using the collocation method. By appling these matrices fractional integrodifferential equations convert to a linear system of equations. In this way, we give some examples to illustrate our results. The numerical method is computer oriented and produces very accurate and stable numerical results. Keywords Fractional integro-differential equation · α-ψ-Contraction theorem · Bernstein polynomial · Collocation method

Introduction In the past decades, fractional differential equations have excited a considerable interest both in mathematics and in applications. They were used in the mathematical modeling of systems and processes occurring in many engineering and scientific disciplines; for instance, see [1–5]. A significant feature of a fractional order differential operator, in contrast to its counterpart in classical calculus, is its nonlocal behavior. Nonlocal boundary value problems of nonlinear fractional order differential equations have recently been investigated by several researchers. For some recent development of the subject for instance see [6–8]. Ahmad and Nieto [1] studied the existence of solution for the nonlinear fractional equation  c D q x(t) = f (t, x(t)), x(0) = −x(T ), c D q x(0) = − c D q x(T ),

H. Bazgir, B. Ghazanfari: Equal contributor.

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Bahman Ghazanfari [email protected] Hamed Bazgir [email protected]

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Department of Mathematics, Lorestan University, Khorramabad 68137-17133, Iran 0123456789().: V,-vol

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Int. J. Appl. Comput. Math

(2020) 6:122

where t ∈ [0, T ], T > 0, 1 < q < 2, 0 < p < 1 and f is a given continuous function. Agarwal et al. [2] discussed the existence of solutions for a boundary value problem of integro-differential equation −D α x(t) = A f (t, x(t)) + B I β g(t, x(t)), t ∈ [0, 1] of fractional order 2 < α ≤ 3 with nonlocal three-point boundary conditions D δ x(t) = 0, D δ+1 x(t) = 0, D δ x(1) − D δ x(η) = a, where 0 < δ ≤ 1, α − δ > 3, 0 < β < 1, 0 < η < 1. Motivated by the mentioned works, in this paper, we discuss the existence, uniqueness and numerical solutions for a boundary value problem integro-differential equation of fractional order 1 < p ≤ 2 with nonlocal four-point boundary conditions ⎧ c p q c r ⎪ ⎨ D u(t) = f (t, λu(t), I u(t), D u(t)), (1) au(0) + bu  (1) = c, ⎪ 1 ⎩c r c r D u(α) + D u(β) = d 0 u(s)ds, where q > 0, 0 < r ≤ 1, λ > 0, 0 < α, β < 1, c, d ∈ R, p > r + 1, a, b > 0, b ≤ a, t ∈ I := [0, 1] and f ∈ C(I × R3 , R) is considered. The giv