The science of strategy (game theory) is known as the optimal decision-making of autonomous and challenging players in a strategic background. There are different strategies to complete the optimal decision. One of these strategies is the similarity technique. Similarity technique is a generalization of the symmetric strategy, which depends only on the other approaches employed, which can be formulated by altering diversities. One of these methods is the fractal theory. In this investigation, we present a new method studying the similarity analytic solution (SAS) of a 3D-fractal nanofluid system (FNFS). The dynamic evolution is completely given by the concept of differential subordination and majorization. Subordination and majorization relationships are the sets of observable individualities. Game theory can simplify the conditions under which particular sets combine. We offer an explicit construction for the complex possible velocity, energy and thermal functions of two-dimensional fluid flow (the complex variable is suggested in the open unit disk, where the disk is selected at a constant temperature and concentration with uniform velocity). We establish that whenever the 3D-fractal nanofluid system is approximated by a fractal function, the solution has the same property, so a class of fractal tangent function gives SAS. Finally, we demonstrate some simulations and examples that give the consequences of this methodology.

Similarity solution is an extension of the symmetric solution of dynamic processing systems, where its stability introduces the stability of the dynamic processing of any system. Maschler et al. [

The theory of fractal (local fractional calculus), which was first offered by Kolwankar et al. [

SAS is a type of solution which is similar to itself whenever the independent and dependent variables are accordingly sized. SAS approaches have been used in numerous engineering categories, particularly in the boundary layer flows. Because of its suitable computational performance and accuracy, there are restricted search in the field of fluid flow and heat transfer in absorbent media such as thermal sinks but the usage of this method is common in numerous other problems [

The viewpoint of SAS including univalent solution (one-one) in the open unit disk deals with the preparation of functions, agreeing with the control and the uniqueness of the geometrical and analytic possessions. To dominate the dynamic process, we employ the subordination and majorization concepts, where the game theory provides the main conditions to get a stability. Recently, this concept is applied in different real life examples (see [

The geometry in this study is about the analysis of 2D-flow, which is called 2D-flow when the velocity sets on a fixed plane and controls not by the coordinate plane; and thermal transfer in a unit disk (or even in a cylinder).

For

The 2D-fractal derivative is considered for a complex function

For example

Therefore, in general for an analytic function

where

In order to study SAS of any nanofluid system, we introduce the similarity variables, which are formulated in the following chaotic system for the fractal function

where

System becomes

where

We selected the tangent fractal function because, in general, the tangent family provides a complete classification of their stable behavior (see

In addition, it can be described by the hyperbolic mechanisms and gave an explanation their placement in the parameter plane. Precisely, the symmetry of the maps with respect to 0 implies that the stable and unstable sets are symmetric with respect to the origin, satisfying

We proceed to generalize System by using the fractal difference operator. For

where

In view of

We aim to find SAS at

We request the following definition [

Note that, there is a deep relation between these two concepts. It is well known, under some conditions on

We aim to find the conditions on System

In addition

In this section, we construct the analytic method to investigate the SAS of System

We have the following result:

Proposition 1. Consider System

then

Proof. By using the series method of a fractal constructions of the functions

A comparison implies

But in view of the first condition of the theorem, we have

which means that

Now, we proceed to determine the upper bound of

Consequently, we obtain

By the second condition, we have

which leads to

Finally, a computation implies that

The last condition of the theorem implies that

which confirms that

As a conclusion of the result, we obtain the desired inequalities

Corollary 2. Let the conditions of Proposition 1 hold. Then the solution of

Proof. In view of Proposition 1, we have the majorization inequalities (

The essential objective of the similar solutions is to exploit them from the game theory perspective. Moreover, due to the confidence, these solutions may correct their approaches in changing the variables and the parameters of the system. Therefore, the relationship with these variables is not infinitely considerable. On the other hand, throughout each obtaining cycle, these variables may regulate their schemes (or occupation performance) to maximize the solution in order to make dominated information periodically. Thus, in order to describe the phenomenon of similarity solutions among the variables of the system, an evolutionary game model is established in the green attaining relationship [

Proposition 3. The replicator dynamic system of

Then the origin is the stable fixed and equilibrium point in the open unit disk.

Proof. Since the only zero of the function

In this section, we illustrate some special cases of System

Let

Let

Let

Let

where

In this study, we introduced a new approach of similarity analytic solution (SAS) for a class of nanofluid systems dominated by the surface of the open unit disk. The outcomes presented that the solution of the system can be bounded by a fractal tangent function. These functions are solutions of the Lagrange equation and are computed through required performed flow conditions. We further demarcated the fluid flow formulated by a unique source and hypothesis a univalent function (one-one conformal function), so that the image of a source is also a source for a specific complex potential. The outcomes were selected

Remark

• The optimal solution of System

which means, that (

The authors would like to thanks the editor office for the deep advice to improve our work.