In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control.

This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: $\sqrt{p+{\alpha }^2}$-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.

In recent years, we have abstracted physical fractal space from biological structures and movements within living organisms, revealing the profound intrinsic connections between fractional order time and fractional-dimensional space, and providing partial explanations for the sources and orders of fractional order. We have confirmed that the topological invariants of fractal cells, the order of physical components, and the mismatch of spatiotemporal order are important factors determining the fractional order of operators. This paper is a continuation of the previous work. Inspired by bone fractal operators, this article attempts to identify other factors that affect the order of operators. Specifically, the following contents are included: (1) originating from the bone fractal operators, we present the construction process of the “apparent half-order” system; (2) using the Schiessel–Blumen model as the comparative object, we analyze the origin and characteristics of the “γ-order” system; (3) using the continued fraction theory and operatorization thought as the link, we establish the design and control method for general fractional-order systems, and discuss the factors affecting the order of fractional-order operators.

This paper focuses on the invariant properties of fractal operators and aims to achieve the axiomatization of the theory of fractal operators. Building upon the derivative and integral theorems of the Laplace transform, we redefine the time differential operator and demonstrate that the newly defined operator exhibits form invariance under the Laplace transform. This property is further generalized to encompass broader classes of operators, including non-rational and fractional fractal operators. Inspired by Klein’s concept of “invariance under transformation groups”, we propose a postulate asserting the “form invariance of operators under the Laplace transform group”. Based on this postulate, we clarified the algebraic operational rules of fractal operators and constructed a rigorously axiomatized theory of fractal operators.

In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions.

In this paper, we explore the self-convolution of the kernel function of the aortic fractal operator. Previous research has established a model named “physical fractal”, and confirmed that the hemodynamics of the aorta can be inscribed by a fractal operator and that the dominant component of the kernel function of the fractal operator is a weighted first-order Bessel function. These studies primarily focus on solving the fractal operator kernel function and examining the overall properties of the physical fractal. As we began to investigate the internal structure of physical fractals, we discovered that studying the powers of fractal operators is a necessary step. In this paper, we introduce the concept of kernel function self-convolution, establish its connection with the power of the fractal operator, and derive a series of invariant properties for the self-convolution of the aortic operator kernel function. These invariant properties, in turn, are deeply and intrinsically related to the invariant properties of the Bessel functions. The research findings of this paper enrich hemodynamics and biomechanics in physical fractal space and extend the scope of using fractal operators to characterize the dynamics of living organisms.