Fractional Black–Scholes equation

Abstract

In this paper, it has been shown that the combined use of exponential operators and special functions provides a powerful tool to solve certain class of generalized space fractional Laguerre heat equation. It is shown that exponential operators are powerful and effective method for solving certain singular integral equations and space fractional Black–Scholes equation.

Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

“…Thus we obtain infinitely many invariant subspaces for Equation (19), which in turn yield infinitely many particular solutions. …”

“…It follows from Theorem 1 applied to Equation (19) that Equation (19) has the exact solution that follows:…”

“…When α and β are other numbers, we can similarly obtain the exact solution of Equation (19). Next, we find the closed-form solutions of FPDEs satisfying initial conditions using the invariant subspace method.…”

“…More recently, much attention has been paid to the solutions of FDEs using various methods, such as the Adomian decomposition method (2005) [8], the first integral method (2014) [9], the Lie group theory method (2012,2015) [10,11], the homotopy analysis method (2016) [12], the inverse differential operational method (2016) [13][14][15], the F-expansion method (2017) [16], M-Wright transforms (2017) [17], exponential differential operators (2017,2018) [18,19], and so on. In reality, the finding of exact solutions of the FDEs is hard work and remains a problem.…”

Analytic solutions for singular integral equations and non-homogeneous fractional PDE

“…Several methods have been introduced to solve fractional differential equations, the popular Laplace transform method [1][2][3]6], the Fourier transform method [14], the iteration method [13] and operational method [7,9,10]. However, most of these methods are suitable for special types of fractional differential equations, mainly the linear with constant coefficients.…”