A new analytic solution of complex Langevin differential equations

Rabha Ibrahim

doi:10.1108/ajms-04-2021-0085

Abstract

PurposeIn this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.Design/methodology/approachThe methodology is based on the geometric function theory.FindingsThe authors present a new analytic function for a class of complex LDEs.Originality/valueThe authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.

Highlights

Langevin differential equation (LDE) is one of the most important differential equation in mathematical sciences, including fluid, Brownian motion, thermal and wavelet studies
An arbitrary model of LDEs is studied in [6,7,8] including analytic solutions
LDEs of a complex variable are applied to simulate special types of polymer and nanomaterials, including the conduct of the polymers [12]. Based on this priority of LDEs of a complex variable, we aim to study this class analytically

Summary

Introduction

Langevin differential equation (LDE) is one of the most important differential equation in mathematical sciences, including fluid, Brownian motion, thermal and wavelet studies It investigated wildly in view of various types of geometric, stochastic and analysis studies (see for example references [1,2,3,4,5]). The technique of the geometric function theory is used recently by Ibrahim and Baleanu [13] to determine the fractal solution They utilized different notions such as the subordination and super-ordination, majorization, Caratheodory functions, convex functions and special functions (see [14,15,16]). We discuss the upper bound solution of LDEs of a complex variable in feature of geometric function theory. Note that, when ς(z) is a constant, the class Mς(ρ) reduces to the well-known class in [20]

Geometric properties Some geometric properties are illustrated as follows

Conclusion