A new analytic solution of complex Langevin differential equations

Purpose – In 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/approach – The methodology is based on the geometric function theory. Findings – The authors present a new analytic function for a class of complex LDEs. Originality/value – The authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.


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]). An arbitrary model of LDEs is studied in [6][7][8] including analytic solutions. The existence and stability of a class of LDEs with two Hilfer-Katugampola fractional derivatives is investigated in [9]. Moreover, the existence of LDE is illustrated suggesting different types of geometry [10,11].
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. 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]).
Here, we discuss the upper bound solution of LDEs of a complex variable in feature of geometric function theory. We illustrated a list of conditions that implies a univalent result in ∪ (the unit disk). The mechanism of our proof is considered utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.
Example 2.1. Let data given by We denote by PðA; BÞ, the class of functions where w satisfies w(0) 5 0 and jw(z)j < 1; and À1 ≤ B < A ≤ 1, then PðA; BÞ ⊂ P is the Janowski class. Next, we define a class of analytic functions.

and a convex function
(see [18]-P415). We note here that the coefficients are approximating to the Bernoulli numbers such that R e ξz À 1 ξz ≥ 1 2 ; 0 < ξ ≤ 1:793 . . . : Hence, R e ξz − 1 ξz ≥ 1=ρ e ð−1Þ ¼ 1 2 : Our design is generated by the Caratheodory functions, which are operated in [19]. In this situation, we establish the necessary conditions of the joining bounds of Ψ(z) consuming a Caratheodory function. Note that, when ς(z) is a constant, the class M ς (ρ) reduces to the well-known class in [20]. Corresponding to the above conclusion, we indicate that χ(z) is starlike.
Example 2.7. Let χ(z) 5 z/(1 À z) and ς(z) 5 1, then we have p(z) 5 1/(1 À z) and qðzÞ ¼ 1 þ zχ 00 ðzÞ Thus, we obtain Then by letting δ 5 1, we conclude that the integral existence result satisfies which achieves all indicated facts in Proposition 2.5. Since the coefficient bounds of W(z) are motorized by the coefficient bounds of χ(z), then we conclude that W(z) is convex. Moreover, the iteration of the integral existence theorem of a convex function (χ(z)) remains convex in the open unit disk (W(W. . .(W(z)))). As a conclusion, this example provides a chain of analytic convex solutions of LDEs in ∪. Next remark shows the important relation of W(z) with the function of nephroid plane curve. This leads to use W(z) as an official formula in a nephroid plane curve instead of using parametric functions. Moreover, Proposition 2.5 implies a positive real solution of LDEs; for example, by assuming β 1 5 1, β 2 5 0, we get R zW 0 ðzÞ W ðzÞ þ zq 0 ðzÞ qðzÞ > 0 because W(z) and q(z) are starlike in ∪ satisfying R zW 0 ðzÞ W ðzÞ > 0 and R zq 0 ðzÞ qðzÞ > 0. As a comparison with recent methods, our method provides in spite of an analytic solution, the strategy of the existing integral formula involving the analytic solution is still analytic in ∪. Note that this solution is univalent in ∪. All recent techniques provided an analytic solution without geometric presentations. Our method describes the analytic solution and its integration geometrically.

Computations
This section deals with some computational outcomes utilizing a sigmoid function. Note that a sigmoid function is bounded analytic in convex complex domain (see Figure 3).