Fractal and Fractional, Volume 3, Issue 2

In the present research paper, our aim is to introduce a new subfamily of p-valent (multivalent) functions of reciprocal order. We investigate sufficiency criterion for such defined family.

We look at estimates for the Green’s function of time-fractional evolution equations of the form $D_{0+\ast }^{\nu }u=Lu$ , where $D_{0+\ast }^{\nu }$ is a Caputo-type time-fractional derivative, depending on a Lév...

In this paper, we investigate a new class of boundary value problems involving fractional differential equations with mixed nonlinearities, and nonlocal multi-point and Riemann–Stieltjes integral-multi-strip boundary conditions. Based on the st...

In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag–Leffler function of two variables. We establish several new refinements of Hermite–Hadamard-like inequalities via...

In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this ki...

In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we c...

In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fr...

In this paper, firstly we have established a new generalization of Hermite–Hadamard inequality via p-convex function and fractional integral operators which generalize the Riemann–Liouville fractional integral operators introduced by Rain...

The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time bet...

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and he...

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equatio...

In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, wh...

The main objective of this paper is to obtain the Hermite–Hadamard-type inequalities for exponentially s-convex functions via the Katugampola fractional integral. The Katugampola fractional integral is a generalization of Riemann–Liouvill...

In the oil industry, many reservoirs produce from partially penetrated wells, either to postpone the arrival of undesirable fluids or to avoid problems during drilling operations. The majority of these reservoirs are heterogeneous and anisotropic, su...

In this paper, the Schrödinger equation involving a fractal time derivative is solved and corresponding eigenvalues and eigenfunctions are given. A partition function for fractal eigenvalues is defined. For generalizing thermodynamics, fractal t...

In this paper, we discuss the existence and uniqueness of solutions for a new class of single and multi-valued boundary value problems involving both Riemann–Liouville and Caputo fractional derivatives, and nonlocal fractional integro-different...

The Sumudu transform of the Dixon elliptic function with non-zero modulus α ≠ 0 for arbitrary powers N is given by the product of quasi C fractions. Next, by assuming the denominators of quasi C fractions as one and applying the Heliermannco...

In this paper, we derive a new Hermite–Hadamard inequality for exponentially convex functions via $\alpha$-fractional integral. We also prove a new integral identity. Using this identity, we establish several Hermite–Hadamard type inequ...

We study a class of conformable time-fractional stochastic equation $T_{\alpha ,t}^{a}u(x,t)=\sigma \left(u(x,t)\right){\dot{W}}_t,x\in \mathbb{R},t\in [a,T],T<\infty ,0<\alpha <1.$ The initial con...

Fractal dimension (FD) is a critical parameter in the characterization of a rock fracture network system. This parameter represents the distribution pattern of fractures in rock media. Moreover, it can be used for the modeling of fracture networks wh...

The Sonine–Letnikov fractional derivative provides the generalized Leibniz rule and, some singular differential equations with integer order can be transformed into the fractional differential equations. The solutions of these equations obtaine...

In this paper, some real world modeling problems: vertical motion of a falling body problem in a resistant medium, and the Malthusian growth equation, are considered by the newly defined Liouville–Caputo fractional conformable derivative and the modi...

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The...