This paper discuss the longstanding problems of fractional calculus such as too many definitions while lacking physical or geometrical meanings, and try to extend fractional calculus to any dimension. First, some different definitions of fractional derivatives, such as the Riemann-Liouville derivative, the Caputo derivative, Kolwankar’s local derivative and Jumarie’s modified Riemann-Liouville ...

As technology advances and the Internet makes our world a global village, it is important to understand prospective career of freelancing. A novel symmetric fractional mathematical model introduced in this study describe competitive market freelancing significance information its acceptance. In study, fixed point theory applied analyze uniqueness existence freelance model. Its numerical solutio...

We present conditions under which all solutions of the fractional differential equation with the Caputo derivative D ax(t) = f(t, x(t)), a > 1, α ∈ (1, 2), (1) are asymptotic to at+ b as t → ∞ for some real numbers a, b. AMS Classification: 34E10, 34A34

In this article, we consider a fractional differential equation (FDE) with Caputo derivative and study the existence and continuation of its solution. Firstly, we prove a theorem on the existence of local solutions. Then we extend the continuation theorems for ODEs to those FDEs. Also several global existence results for FDE are obtained.

In this paper, we introduce a family of fractional-order Chebyshev functions based on the classical Chebyshev polynomials. We calculate and derive the operational matrix of derivative of fractional order γ in the Caputo sense using the fractional-order Chebyshev functions. This matrix yields to low computational cost of numerical solution of fractional order differential equations to the soluti...

ABSTRACT. In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of order α are studied. We present four types of Ulam stability results for the fractional differential equation in the case of 0 < α < 1 and b = +∞ by virtue of the Henry-Gronwall inequality. Meanwhile, we give an interesting data dependence results for the fracti...

The numerical solution of a variable-order fractional financial system is calculated by using the Adams-Bashforth-Moulton method. The derivative is defined in the Caputo variable-order fractional sense. Numerical examples show that the Adams-Bashforth-Moulton method can be applied to solve such variable-order fractional differential equations simply and effectively. The convergent order of the ...

In this paper, we exhibit two methods to numerically solve the fractional integro differential equations and then proceed to compare the results of their applications on different problems. For this purpose, at first shifted Jacobi polynomials are introduced and then operational matrices of the shifted Jacobi polynomials are stated. Then these equations are solved by two methods: Caputo fractio...

The subdiffusion equation with a Caputo fractional derivative of order α ∈ (0,1) in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for discretizing the Caputo fractional derivative in time. The scheme was analyzed earlier i...