Editorial for Special Issue “New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus”

Fractional calculus has reshaped science and technology since its first appearance in a letter received to Gottfried Wilhelm Leibniz from Guil-laume de l’Hôpital in the year 1695. The existence of fractional behavior in nature cannot be denied. Any phenomenon with a pulse, rhythm, or pattern has the potential to be a fractal. The goal of this Special Issue is to explore new developments in both pure and applied mathematics as a result of fractional behavior. This assertion is supported by the papers in this Special Issue. The variety of topics covered here demonstrates the importance of fractional calculus in various fields and provides adequate coverage to appeal to the interests of each reader. This Special Issue of Fractal and Fractional was posted in early 2021 with the goal of exploring the various connections between fractional calculus and its applications in pure and applied mathematics. Initially, a deadline was set and has been extended to 5 April 2022, in consideration of the author’s interest. In total, we received 74 submissions. Following a thorough peer-review process, seventeen of them were eventually published and, keeping with the original concept of this Special Issue, have now been compiled into this book. The following are details of the papers published in our Special Issue:

Ali et al. [1] developed a new version of generalized fractional Hadamard and Fejér–Hadamard-type integral inequalities that can be used to investigate the stability and control of corresponding fractional dynamic equations.

Fisher’s equation is a precise mathematical result derived from population dynamics and genetics, specifically chemistry. Rashid et al. [2] used a hybrid technique in conjunction with a new iterative transform method to solve the nonlinear fractional Fisher model. Furthermore, while the proposed procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, it has the potential to be consistently applied to other multifaceted physical processes.

It is worth noting that the proposed fuzziness approach is to validate the superiority and dependability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures. As a result, in [3], the authors evaluate a semi-analytical method in conjunction with a new hybrid fuzzy integral transform and the Adomian decomposition method using the fuzziness concept known as the Elzaki Adomian decomposition method (EADM).

In [4], the authors analyzed the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To that end, the fractional-order vector operators of divergence, curl, and gradient were identified as components of the quaternionic fractional Dirac operator. General solutions to some non-homogeneous div-curl systems were derived that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types as one of the most important results of this manuscript.

An integro-differential kinetic equation was derived in [5] by using novel fractional operators and its solution using weighted generalized Laplace transforms. The weighted (k,s)-Riemann–Liouville fractional integral and differential operators are defined by the authors. The paper includes some specific properties of the operators as well as the weighted generalized Laplace transform of the new operators.

The models that include vaccination as a control measure are very important. In light of this, the authors developed and mathematically investigated integer and fractional models of typhoid fever transmission dynamics in [6]. Several numerical simulations were run, allowing us to conclude that such diseases may be combated through vaccination combined with environmental sanitation.

Chemical, electrical, biochemical, geometrical, and meteorological models are examples of nonlinear models used in science and engineering. The authors of [7] investigated the global fractal behavior of a new nonlinear three-step method with tenth-order convergence. Basins of attraction consider various types of complex functions. When compared to other well-known methods, the proposed method achieves the specified tolerance in the smallest number of iterations while assuming different initial guesses.

The authors investigate the existence results for the hybrid Caputo–Hadamard fractional boundary value problem in [8]. The proposed BVP’s inclusion version with three-point hybrid Caputo–Hadamard terminal conditions is also considered, and the related existence results are provided. To accomplish these objectives, Dhage’s well-known fixed-point theorems for both BVPs are applied. Furthermore, two numerical examples are presented to validate the analytical findings.

The authors of [9] developed a feedback-control strategy to control the chaos caused by bifurcation. The proposed model’s fractal dimensions were computed. To further confirm the complexity and chaotic behavior, the maximum Lyapunov exponents and phase portraits were depicted. Finally, numerical simulations were presented to validate the theoretical and analytical results.

Numerical analysis is always necessary to demonstrate the efficacy of proposed schemes. Keeping this in mind, the authors in [10] concentrated on numerically addressing the time fractional Cattaneo equation involving the Caputo–Fabrizio derivative using spline-based numerical techniques. The main advantage of the schemes is that the approximation solution is generated as a smooth piecewise continuous function, which allows to approximate a solution at any point in the domain of interest.

Certain convex and s-convex functions have applications in optimization theory. As a result, in [11], the authors investigated a variety of mean-type integral inequalities for a well-known Hilfer fractional derivative. Some identities were also established in order to infer more interesting mean inequalities. The Caputo fractional derivative consequences were presented as special cases to their general conclusions.

The authors of [12] proposed a numerical method for solving Caputo fractional-order differential equations based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) (FDEs). A new operational matrix of fractional-order derivatives in the Caputo sense was derived, which was then used in conjunction with the spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Numerical examples were provided to demonstrate the accuracy of this method, which demonstrated that the obtained results agree well with the analytical solutions for both linear and nonlinear FDEs.

A semi-analytical analysis of the fractional-order non-linear coupled system of Whitham–Broer–Kaup equations was presented in [13]. The fractional derivative was considered in the Caputo–Fabrizio sense. When the analytical and actual solutions are compared, it is clear that the proposed approaches effectively solve complex nonlinear problems. Furthermore, the proposed methodologies control and manipulate the obtained numerical solutions in an extreme manner in a large acceptable region.

The authors of [14] derived some suitable results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) with nonlinear boundary conditions (NBCs). The aforementioned outcomes were obtained by employing the monotone iterative method, which employs the procedure of upper and lower solutions. There are two sequences of extremal solutions generated, one of which converges to the upper solution and the other to the corresponding lower solution. The method does not require any prior discretization or collocation to generate the aforementioned upper and lower solution sequences.

Q-calculus is a non-trivial and useful generalization of calculus. The authors of [15] presented two new identities involving q-Riemann–Liouville fractional integrals. New q-fractional estimates of trapezoidal-like inequalities were derived using these identities as auxiliary results, in essence of the class of generalized exponential convex functions.

The definition and applicability of new families of polynomials generating function and operational representations are always of great interest. The authors of [16] used operational techniques to investigate a new type of polynomial, specifically the Gould–Hopper–Laguerre–Sheffer matrix polynomials. Furthermore, these particular matrix polynomials were interpreted in terms of quasi-monomiality. The integral transform was used to investigate the properties of the extended versions of the Gould–Hopper–Laguerre–Sheffer matrix polynomials. There were also examples of how these results apply to specific members of the matrix polynomial family.

Laplace transform of the Riemann zeta function using its distributional representation was computed, which played a critical role in applying the operators of generalized fractional calculus to this well-studied function [17]. As a result, as special cases, similar new images can be obtained using various other popular fractional transforms. The Riemann zeta function was used to formulate and solve a new fractional kinetic equation. Following that, a new relationship involving the Laplace transform of the Riemann zeta function and the Fox–Wright function was investigated, which significantly simplified the results.

To summarize, this special selection covers the scope of ongoing activities in the context of fractional calculus by presenting alternative perspectives, viable methods, new derivatives, and strategies to solve practical issues. As editors, we presume that this will be followed by a set of Special Issues and texts to further investigate this theme.

As the guest editors of this Special Issue, we would like to take this opportunity to thank all of the reviewers, editorial board members, and editors who assisted us in perfecting the content of this volume. We would also like to thank Ms. Cecile Zheng from the journal office for her prompt assistance across the Special Issue management process.

All author contributions to this Special Issue are greatly acknowledged with thanks.