Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral

Abstract

We study the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo type, which increases the interest of this study. In order to study this problem we have introduced a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo, therefore, the proofs are based upon the reduction of the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and we have built the minimum conditions to obtain the existence and uniqueness of this solution. The analytical study is followed by a complete numerical study.

Multiple and Singular Soliton Solutions for Space–Time Fractional Coupled Modified Korteweg–De Vries Equations

“…We engage in an analytical exploration of the core principles of CFD, as discussed in [15][16][17][18][19][20][21][22][23].…”

“…In order to calculate 𝛽 and 𝛽 1 in this scenario, we first insert Equation (21) into Equation (17), and then substitute the resulting equation into system (3). This gives 𝛽 = 𝑐 and 𝛽 1 = − 1 a𝑐 , where c is constant.…”

“…The singular two-soliton solutions are obtained by substituting Equation (25) into Equation (17) to get…”

“…CFD represents a significant advancement in fractional calculus, exhibiting fundamental characteristics that are instrumental across various fields, including mathematics, engineering, and physics [14][15][16][17][18][19][20][21][22][23]. Within the above systems, the CFD related to time (𝑡) and space (𝑥) is denoted as 𝐷 𝑡 𝛼 and 𝐷 𝑥 𝛼 , respectively.…”

Numerical solution of a fractional coupled system with the Caputo-Fabrizio fractional derivative

“…In 2015, Caputo and Fabrizio [18] suggested a new fractional derivative with non-singular kernel. On the other hand, the Caputo-Fabrizio fractional derivative has many significant properties, such as its ability to describe matter hetrogeneities and configuration with different scales, many works studied the existence and uniqueness solution of boundary value problems involving such operator [19][20][21]. In the articles [22][23][24] several methods and issues for solving and modeling solutions to problems in applied mathematics and proofs for important theories were presented.…”