Volume 15, Issue 4

We calculate the fractional Laplacian Δα/2 for functions of the form u(x) = (1 − |x|2)+p and v(x) = x d u(x). As an application, we estimate the first eigenvalues of the fractional Laplacian in a ball.

In this paper, a novel numerical analysis is introduced and performed to obtain the numerical solution of the fractional heat- and wave-like equations. A general formulation for the Bernstein fractional derivatives operational matrix is given. In this approach, a truncated Bernstein series together with the Bernstein operational matrix of fractional derivatives are used to reduce the solution of fractional differential problems to the solution of a system of algebraic equations. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.

In this paper we study a fuzzy fractional integral equation. The fractional derivative is considered in the sense of Riemann-Liouville and we establish existence of the solutions of fuzzy fractional integral equations using the Hausdorff measure of noncompactness.

In this paper, we study the existence of mild solutions for a class of semilinear fractional differential equations with nonlocal conditions in Banach spaces. The results are obtained by using convex-power condensing operator and fixed point theory. An example is presented to illustrate the main result.

We obtain a uniqueness result for a differential equation depending on Caputo’s fractional derivative. An example is included to illustrate our result.

We present a definition of the Riemann-Liouville fractional calculus for arbitrary time scales through the use of time scales power functions, unifying a number of theories including continuum, discrete and fractional calculus. Basic properties of the theory are introduced including integrability conditions and index laws. Special emphasis is given to extending Taylor’s theorem to incorporate our theory.

In this paper we investigate Cauchy problem for a class of time-fractional differential equation (0.1)$$\begin{gathered} D_t^\alpha u(t) + c_1 D_t^{\beta _1 } u(t) + \cdots + c_d D_t^{\beta _d } u(t) = Au(t), t > 0, \hfill \\ u^{(j)} (0) = x_j , j = 0, \cdots ,m - 1, \hfill \\ \end{gathered}$$ where A is a closed densely defined linear operator in a Banach space X, α > β 1 > ... > βd > 0, c j are constants and m = ⌈α⌊. A new type of resolvent family corresponding to well-posedness of (0.1) is introduced. We derive the generation theorems, algebraic equations and approximation theorems for such resolvent families. Moreover, we give the exact solution for a kind of generalized fractional telegraph equations. Some examples are given as illustrations.

In this paper, the explicit finite difference method (FDM) is used to study the variable order nonlinear fractional wave equation. The fractional derivative is described in the Riesz sense. Special attention is given to study the stability analysis and the convergence of the proposed method. Numerical test examples are presented to show the efficiency of the proposed numerical scheme.

The purpose of this paper is to obtain solutions for both linear and nonlinear initial value problems (IVPs) for fractional transport equations and fractional diffusion-wave equations using the iterative method.

In this paper, we survey some generalizations of fractional integrals and derivatives and present some of their properties. Using these properties, we show that many integral equations can be solved in a much elegant way. We believe that this will blur the distinction between the integral and differential equations, and provide a systematic approach for the two of these classes.

The article provides an historical survey of the early contributions on the applications of fractional calculus in linear viscoelasticty. The period under examination covers four decades, since 1930’s up to 1970’s, and authors are from both Western and Eastern countries. References to more recent contributions may be found in the bibliography of the author’s book. This paper reproduces, with Publisher’s permission, Section 3.5 of the book: F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press-London and World Scienific-Singapore, 2010.

The generalized functional equations describing a wide class of different self-similar processes are derived. These equations follow from the observation that microscopic function describing an initial self-similar process increases monotonically or even cannot have a certain value. The last case implies the behavior of trigonometric functions cos(zζ n), sin(zζ n) at ζ > 1 and n ≫ 1 that can enter to the microscopic function and when the limits of the initial scaling region are increasing and becoming large. The idea to obtain the desired functional equations is based on the approximate decoupling procedure reducing the increasing microscopic function to the linear combination of the same microscopic functions but having smaller scales. Based on this idea the new solutions for the well-known Weierstrass-Mandelbrot function were obtained. The generalized functional equations derived in this paper will help to increase the limits of applicability in description of a wide class of self-similar processes that exist in nature. The procedure that is presented in this paper allows to understand deeper the relationship between the procedure of the averaging of the smoothed functions on discrete self-similar structures and continuous fractional integrals.