There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Differential Equations

Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Fixed Point Theory

The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject in Newtonian fluids. In this article, the homotopy perturbation method (HPM), the variational iteration method (VIM), and the Akbari–Ganji method (AGM) were used to solve linear fractional differential equations order in fluid mechanics and fractional optimal control problems in the sense of Caputo. Achieving an exact solution to these equations has been one of the researchers’ goals; according to the research results, the proposed methods provide an exact solution. Comparing the results obtained by the proposed methods with those obtained by the Shehu method reveals that the present methods are very effective and convenient in applied fields. Also, the VIM and AGM methods are more proper than HPM in solving these equations. The solutions obtained by the proposed methods agree with the solutions available in the literature. 

Rigid plate submerged in a Newtonian fluid and fractional differential equation problems via Caputo fractional derivative

On the impulsive implicit Ψ-Hilfer fractional differential equations with delay

In this paper, the existence, uniqueness and stability of solutions to a boundary value problem of nonlinear FDEs of variable order are established. To do this, we first investigate some aspects of variable order operators of Hadamard type. Then, with the help of the generalized intervals and piecewise constant functions, we convert the variable order Hadamard FBVP to an equivalent standard Hadamard BVP of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used and, finally, the Ulam–Hyers–Rassias stability of the given variable order Hadamard FBVP is examined. These results are supported with the aid of a comprehensive example.

Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam–Hyers–Rassias Stability

In the present research study, for a given multiterm boundary value problem (BVP) involving the nonlinear fractional differential equation (NnLFDEq) of variable order, the uniqueness-existence properties are analyzed. To arrive at such an aim, we first investigate some specifications of this kind of variable order operator and then derive required criteria confirming the existence of solution. All results in this study are established with the help of two fixed-point theorems and examined by a practical example.

/pdf/a-study-on-the-solutions-of-a-multiterm-fbvp-of-variable-qbap9uumgd.pdf

A Study on the Solutions of a Multiterm FBVP of Variable Order

This study establishes the existence and stability of solutions for a general class of Riemann–Liouville (RL) fractional differential equations (FDEs) with a variable order and finite delay. Our findings are confirmed by the fixed-point theorems (FPTs) from the available literature. We transform the RL FDE of variable order to alternate RL fractional integral structure, then with the use of classical FPTs, the existence results are studied and the Hyers–Ulam stability is established by the help of standard notions. The approach is more broad-based and the same methodology can be used for a number of additional issues.

/pdf/existence-and-uniqueness-theorems-for-a-variable-order-3guxpc62.pdf

Existence and Uniqueness Theorems for a Variable-Order Fractional Differential Equation with Delay

A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results.

https://www.mdpi.com/2073-8994/13/5/896/pdf

Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order

In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.

Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings

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