Theories of tempered fractional calculus applied to tempered fractional Langevin and Vasicek equations

Abstract

Our main goal in the current research work is to explore proofs of newly discovered theorems related to tempered fractional calculus. We use a new mechanism, namely, the natural tempered fractional transformation method, which can be used to solve important tempered fractional differential equations that are important in science, such as the linear and nonlinear tempered fractional differential equations. Indeed, we found new exact solutions to both tempered fractional Langevin and Vasicek differential equations and an exact solution for the famous tempered fractional diffusion equation. The new method makes it easier to do the calculations than with the traditional methods, and all you need is a few simple manipulations. Our new research technique is straightforward to use and highly accurate.

Using the decomposition method to solve the fractional order temperature distribution equation: A new approach

“…Here are some useful N‐transforms properties, and we shall use them throughout this paper; see previous works [27–32]: $$$ {N}^{+}\left[K\right]=\frac{K}{r} $$$. $$$ {N}^{+}\left[{\eta}^{\beta}\right]=\frac{\Gamma \left(\beta +1\right)\kern0.1em {v}^{\beta }}{r^{\beta +1}},\beta >-1 $$$. $$$ {N}^{+}\left[{e}^{b\eta}\right]=\frac{1}{\left(r- bv\right)}. $$$ Suppose that $$$ k>0 $$$, where $$$ k-1<\beta \le k $$$ and $$$ M\left(r,v\right) $$$ is the N‐transformation of $$$ \zeta \left(\eta \right) $$$, then the N‐transformation of the fractional derivative in the Caputo sense of the function $$$ \zeta \left(\eta \right) $$$ of order $$$ \beta $$$ denoted by …”

“…Here are some useful N-transforms properties, and we shall use them throughout this paper; see previous works [27][28][29][30][31][32]:…”

“…Furthermore, finding exact solutions to fractional linear and nonlinear differential equations in applied mathematics and physics remains a challenging task that calls for novel approaches and strategies. The fractional complex transform method [13, 14], the fractional subequation method [15, 16], the $$$ \left(G/{G}^{\prime}\right) $$$‐expansion method [17], the exp‐function method [18], the first integral method [19], the functional variable method [20], the trial equation method [21], and the spline function method are just a few of the influential and effective techniques to generate approximate analytical solutions to FDEs have recently been developed; see earlier studies [22–32]. To the best of our knowledge, no approach in the literature can, however, handle every type of BVP.…”

Enhancing the Mathematical Theory of Nabla Tempered Fractional Calculus: Several Useful Equations

“…With special selection of weight function, this has led to new insights, inspirations, and developments in the field of fractional calculus, as it provides a way to accurately model complex systems with adjustable memory. These advantages have motivated many researchers to explore the subject of tempered fractional calculus, leading to significant progress in the field in recent years [6][7][8]. Tempered fractional calculus has become a hot topic in fractional order fields.…”