On tempered fractional calculus with respect to functions and the associated fractional differential equations

Abstract

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the Ψ-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the Ψ-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann-Liouville-type and Caputo-type cases of these operators. Furthermore, we study some non-linear fractional differential equations involving Ψ-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle and proving stability results by using Grönwall type inequalities.

A Novel Study on Tempered $({\upkappa},\uppsi)$-Hilfer Fractional Operators

“…Definition 2.3 (The ψ-tempered fractional Integral [19]). Let ℵ ∈ X p ψ (ϱ 1 , ϱ 2 ) and [ϱ 1 , ϱ 2 ] be a finite or infinite interval on the real axis R, ψ(δ) > 0 be an increasing function on (ϱ 1 , ϱ 2 ] and ψ ′ (δ) > 0 be continuous on (ϱ 1 , ϱ 2 ), λ ∈ R and σ > 0.…”

Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

“…As is known that there are dozens of fractional calculus which has been presented in recent decades. One of them of particular interest is the so-called tempered fractional calculus [15][16][17][18]. The tempered fractional calculus is a generalization of the classical fractional calculus, and it contains both the weak singular kernel and the exponential kernel.…”

“…Thus, it can describes the transition between normal and anomalous diffusions or some anomalous diffusions in finite time or bounded space domain. Many scholars have studied the tempered derivatives such as Psi-Caputo [16], Psi-Hilfer [19], Strum-Liouville [20]. In modelling, the tempered fractional calculus and the tempered fractional differential equations (TFDEs) have many real applications, such as describing rare or extreme events of stock price dynamics in finance [21] and understanding turbulence in geophysical flows [22].…”

Upper and Lower Solution Method for a Singular Tempered Fractional Equation with a p-Laplacian Operator

“…In comparison, the tempered derivative is a nonlocal fractional derivative with an exponential tempering factor, which possesses stronger nonlinearity. So, the study for tempered-type fractional differential equations is relatively difficult; for more background of tempered fractional operators, we refer the reader to [5][6][7].…”