Explicit Iteration and Unique Positive Solution for a Caputo-Hadamard Fractional Turbulent Flow Model

Hadamard fractional calculus theory has made many scholars enthusiastic and excited because of its special logarithmic function integral kernel. In this paper, we focus on a class of Caputo-Hadamard-type fractional turbulent flow model involving $p(t)$ -Laplacian operator and Erdélyi-Kober fractional integral operator. The $p(t)$ -Laplacian operator involved in our model is the non-standard growth operator which arises in many fields such as elasticity theory, physics, nonlinear electrorheological fluids, ect. It is the first paper that studies a Caputo-Hadamard-type fractional turbulent flow model involving $p(t)$ -Laplacian operator and Erdélyi-Kober fractional integral operator. Different from the constant growth operator, The non-standard growth characteristics of $p(t)$ -Laplacian operator bring great difficulties and challenges. In order to achieve a good survey result, we take advantage of the popular mixed monotonic iterative technique. With the help of this approach, we obtain the uniqueness of positive solution for the new Caputo-Hadamard-type fractional turbulent flow model. In the end, an example is also given to illustrate the main results.