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

Abstract

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p(t)$ </tex-math></inline-formula> -Laplacian operator and Erdélyi-Kober fractional integral operator. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p(t)$ </tex-math></inline-formula> -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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p(t)$ </tex-math></inline-formula> -Laplacian operator and Erdélyi-Kober fractional integral operator. Different from the constant growth operator, The non-standard growth characteristics of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p(t)$ </tex-math></inline-formula> -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.