Approximate solutions for a fractional hybrid initial value problem via the Caputo conformable derivative

Abstract

Our main purpose in this work is to derive an existence criterion for a Caputo conformable hybrid multi-term integro-differential equation equipped with initial conditions. In this way, we consider a partially ordered Banach space, and, by applying the lower solution property, the existence and successive approximations of solutions for the mentioned hybrid initial problem are investigated. Eventually, we formulate an illustrative example for this hybrid IVP to support our findings from a numerical point of view. Moreover, we plot the sequence of the obtained approximate solutions for different values of noninteger orders.

The generalized U–H and U–H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving φ-Caputo fractional operators

“…By using a prior estimate method, Nazir et al [9] turned to studying a sequential hybrid fractional equation and Vivek et al [10] analyzed dynamical behaviors of Hilfer-Hadamard type fractional pantograph equations by utilizing successive approximations. The latest achievements in this field can be found in references such as [11][12][13][14].…”

On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

“…3 we introduce the generalized Riesz-Caputo's fractional operators and derived some useful results, while in Sect. 4 we establish some equivalence results for boundary value problem (1) and establish the results for the existence and uniqueness of solutions for BVP (1). The last section of this paper presents the stability of solutions for BVP (1) by means of continuous dependence on parameters.…”

“…As we know, the existence theory is meat-and-potatoes in every field of science, as it is very applicative to comprehend whether there is a solution to a given differential equation beforehand; otherwise, all the attempts to find a numerical or analytic solution will become valueless. The analysis of fractional differential equations has been carried out by various authors (see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]).…”

Two fractional hybrid and non‐hybrid boundary value problems

“…What is more important to researchers today than anything else is to understand some qualitative properties of solutions for these fractional dynamical systems modeled based on different complicated fractional boundary value problems (BVPs) of hybrid or non‐hybrid type. In this direction, a lot of works have been published about different types of fractional integro‐differential and differential equations, 1‐6 integro‐differential equations involving the Caputo–Fabrizio derivative, 7‐11 hybrid differential equations, 12‐19 approximate solutions of different fractional differential equations, 20‐23 fractional mathematical modelings, 24‐31 and numerical techniques in different fields of fractional calculus 32‐45 …”